process

EMS Series of Congress ReportsEMS Series of Congress Reports publishes volumes originating from conferences or seminars focusingon any field of pure or applied mathematics. The individual volumes include an introduction into theirsubject and review of the contributions in this context. Articles are required to undergo a refereeing processand are accepted only if they contain a survey or significant results not published elsewhere in theliterature.Previously published:Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowroński (ed.)

K-Theory andNoncommutativeGeometryGuillermo CortiñasJoachim CuntzMax KaroubiRyszard NestCharles A. WeibelEditors

Editors:Guillermo CortiñasDepartamento de MatemáticaFacultad de CienciasExactas y NaturalesUniversidad de Buenos AiresCiudad Universitaria Pab. 11428 Buenos Aires, ArgentinaEmail: gcorti@dm.uba.arJoachim CuntzMathematisches InstitutWestfälische Wilhelms-UniversitätMünsterEinsteinstraße 6248149 Münster, GermanyEmail: cuntz@math.uni-muenster.deMax KaroubiUniversité Paris 72 place Jussieu75251 Paris, FranceEmail: max.karoubi@gmail.comRyszard NestDepartment of MathematicsCopenhagen UniversityUniversitetsparken 52100 Copenhagen, DenmarkEmail: rnest@math.ku.dkCharles A. WeibelDepartment of MathematicsRutgers UniversityHill Center – Busch Campus110 Frelinghuysen RoadPiscataway, NJ 08854-8019, U.S.A.Email: weibel@math.rutgers.edu2000 Mathematics Subject Classification: 19-06, 58-06; 14A22, 14Fxx, 46Lxx, 53D55, 58BxxKey words: Isomorphism conjectures, Waldhausen K-theory, twisted K-theory, equivariant cyclichomology, index theory, Bost–Connes algebra, C*-manifolds, T-duality, deformation of gerbes, torsiontheories, Parshin conjecture, Bloch–Kato conjectureISBN 978-3-03719-060-9The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation,broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind ofuse permission of the copyright owner must be obtained.© 2008 European Mathematical SocietyContact address:European Mathematical Society Publishing HouseSeminar for Applied MathematicsETH-Zentrum FLI C4CH-8092 ZürichSwitzerlandPhone: +41 (0)44 632 34 36Email: info@ems-ph.orgHomepage: www.ems-ph.orgTypeset using the authors’ T E X files: I. Zimmermann, FreiburgPrinted in Germany9 8 7 6 5 4 3 2 1

PrefaceThis volume contains the proceedings ofVASBI, the ICM2006 satellite on K-theory andNoncommutative Geometry which took place in Valladolid, Spain, from August 31 toSeptember 6, 2006. The conference’s Scientific Committee was composed of JoachimCuntz, Max Karoubi, Ryszard Nest and Charles A. Weibel. The conference was madepossible through a grant by Caja Duero; additional funding was provided by the Universityof Valladolid. Funding to cover expenses of US based participants was providedby NSF, through a grant to C. A. Weibel, who was responsible, first of all, for preparinga succesful funding application, and then for managing the funds. The local committee,composed of N. Abad and E. Ellis, were in charge of conference logistics.I am indebted to all these people and institutions for their support.Guillermo Cortiñas, Organizer

ContentsPreface......................................................................vIntroduction.................................................................ixProgram list of speakers and topics...........................................xiiiCategorical aspects of bivariant K-theoryRalf Meyer ..................................................................1Inheritance of isomorphism conjectures under colimitsArthur Bartels, Siegfried Echterhoff, and Wolfgang Lück ........................41Coarse and equivariant co-assembly mapsHeath Emerson and Ralf Meyer ..............................................71On K 1 of a Waldhausen categoryFernando Muro and Andrew Tonks ............................................91Twisted K-theory – old and newMax Karoubi ..............................................................117Equivariant cyclic homology for quantum groupsChristian Voigt ............................................................151A Schwartz type algebra for the tangent groupoidPaulo Carrillo Rouse .......................................................181C -algebras associated with the ax C b-semigroup over NJoachim Cuntz.............................................................201On a class of Hilbert C*-manifoldsWend Werner ..............................................................217Duality for topological abelian group stacks and T -dualityUlrich Bunke, Thomas Schick, Markus Spitzweck, and Andreas Thom ...........227

viiiContentsDeformations of gerbes on smooth manifoldsPaul Bressler, Alexander Gorokhovsky, Ryszard Nest, and Boris Tsygan .........349Torsion classes of finite type and spectraGrigory Garkusha and Mike Prest ...........................................393Parshin’s conjecture revisitedThomas Geisser ...........................................................413Axioms for the norm residue isomorphismCharles Weibel ............................................................427List of contributors ........................................................437List of participants .........................................................439

IntroductionSince its inception 50 years ago, K-theory has been a tool for understanding a widerangingfamily of mathematical structures and their invariants: topological spaces,rings, algebraic varieties and operator algebras are the dominant examples. The invariantsrange from characteristic classes in cohomology, through determinants of matrices,to Chow groups of varieties, as well as to traces and indices of elliptic operators. ThusK-theory is notable for its connections with other branches of mathematics.Noncommutative geometry, on the other hand, develops tools which allow oneto think of noncommutative algebras in the same footing as commutative ones; asalgebras of functions on (noncommutative) spaces. The algebras in question come fromproblems in various areas of mathematics and mathematical physics; typical examplesinclude algebras of pseudodifferential operators, group algebras, and other algebrasarising from quantum field theory.To study noncommutative geometric problems one considers invariants of the relevantnoncommutative algebras. These invariants include algebraic and topologicalK-theory, and also cyclic homology, discovered independently by Alain Connes andBoris Tsygan, which can be regarded both as a noncommutative version of de Rhamcohomology, and as an additive version of K-theory. There are primary and secondaryChern characters which pass from K-theory to cyclic homology. These characters arerelevant both to noncommutative and commutative problems, and have applicationsranging from index theorems to the detection of singularities of commutative algebraicvarieties.The contributions to this volume represent this range of connections betweenK-theory, noncommmutative geometry, and other branches of mathematics.An important connection between K-theory, topology, geometric group theory andnoncommutative geometry is given by the isomorphism conjectures, such as those dueto Baum–Connes, Bost–Connes and Farrell–Jones, which predict certain topologicalformulas for various kinds of K-theory of a crossed product, in both the C -algebraicand the purely algebraic contexts. These problems have received a lot of attention overthe past twenty years. Not surprisingly, then, three of the articles in this volume discussthese problems. The first of these, by R. Meyer, is a survey on bivariant Kasparov theoryand E-theory. Both bivariant K-theories are approached via their universal propertiesand equipped with extra structure such as a tensor product and a triangulated categorystructure. The construction of the Baum–Connes assembly map via localisation ofcategories is reviewed and the relation with the purely topological construction byDavis and Lück is explained. In the second article, A. Bartels, S. Echterhoff andW. Lück investigate when Isomorphism Conjectures, such as the ones due to Baum–Connes, Bost and Farrell–Jones, are stable under colimits of groups over directed sets(with not necessarily injective structure maps). They show in particular that boththe K-theoretic Farrell–Jones Conjecture and the Bost Conjecture with coefficients

xIntroductionhold for those groups for which Higson, Lafforgue and Skandalis have disproved theBaum–Connes Conjecture with coefficients. H. Emerson and R. Meyer study in theircontribution an equivariant co-assembly map that is dual to the usual Baum–Connesassembly map and closely related to coarse geometry, equivariant Kasparov theory,and the existence of dual Dirac morphisms. As applications, they prove the existenceof dual Dirac morphisms for groups with suitable compactifications, that is, satisfyingthe Carlsson–Pedersen condition, and study a K-theoretic counterpart to the properLipschitz cohomology of Connes, Gromov and Moscovici.Many applications of K-theory to geometric topology come from the K-theory ofWaldhausen categories. The article by F. Muro and A. Tonks is about the K-theory of aWaldhausen category. They give a simple representation of all elements in K 1 of sucha category and prove relations between these representatives which hold in K 1 .Another connection between K-theory, topology and analysis comes from twistedK-theory, which has received renewed attention in the last decade, in the light of newdevelopments inspired by Mathematical Physics. The next article is a survey on thissubject, written by M. Karoubi, one of its founders. The author also proves some newresults in the subject: a Thom isomorphism, explicit computations in the equivariantcase and new cohomology operations.As mentioned above, one of the most interesting points of contact between K-theoryand noncommutative geometry comes through cyclic homology. C.Voigt’s contributionis about this theory. He defines equivariant periodic cyclic homology for bornologicalquantum groups. Generalizing corresponding results from the group case, he shows thatthe theory is homotopy invariant, stable, and satisfies excision in both variables. Alongthe way he proves Radford’s formula for the antipode of a bornological quantum group.Moreover he discusses anti-Yetter–Drinfeld modules and establishes an analogue of theTakesaki–Takai duality theorem in the setting of bornological quantum groups.A central theme in noncommutative geometry is index theory. A basic constructionin this area consists of associating a C -algebra to a geometric problem, and thencompute an index using K-theory. P. Carrillo Rouse’s article deals with the problem ofindex theory on singular spaces, and in particular with the construction of algebras andindices between the enveloping C -algebra and the convolution algebra of compactlysupported functions.The article by J. Cuntz is also about C -algebras. It presents a C -algebra whichis naturally associated to the ax C b-semigroup over N. It is simple and purely infiniteand can be obtained from the algebra considered by Bost and Connes by adding oneunitary generator which corresponds to addition. Its stabilization can be described asa crossed product of the algebra of continuous functions, vanishing at infinity, on thespace of finite adeles for Q by the natural action of the ax C b-group over Q.W. Werner applies C -algebraic methods in infinite dimensional differential geometryin his contribution. It deals with a special class of ‘non-compact’ hermitian infinitedimensional symmetric spaces, generically denoted by U . The author calculates theirinvariant connection very explicitly and uses the concept of a Hilbert C*-manifold sothat the Banach manifold in question is of the form Aut U=H, where Aut U is theautomorphism group of the Hilbert C*-manifold. Using results previously obtained

Introductionxiwith D. Blecher, he characterizes causal structure on U that comes from interpretingthe elements of U as bounded Hilbert space operators.The subject of the next article, by U. Bunke, T. Schick, M. Spitzweck and A. Thom,connected with group theory, topology, as well as to mathematical physics and noncommutativegeometry, is Pontrjagin duality. The authors extend Pontrjagin dualityfrom topological abelian groups to certain locally compact group stacks. To this endthey develop a sheaf theory on the big site of topological spaces S in order to prove thatthe sheaves Ext i Sh Ab S .G; T/, i D 1; 2, vanish, where G is the sheaf represented by alocally compact abelian group and T is the circle. As an application of the theory theyinterpret topological T -duality of principal T n -bundles in terms of Pontrjagin dualityof abelian group stacks.An important source of examples and problems in noncommutative geometry comesfrom deformations of commutative structures. In this spirit, the article by P. Bressler,A. Gorokhovsky, R. Nest and B. Tsygan investigates the 2-groupoid of deformationsof a gerbe on a C 1 manifold. They identify the latter with the Deligne 2-groupoid of acorresponding twist of the differential graded Lie algebra of local Hochschild cochainson C 1 functions.The next article, by G. Garkusha and M. Prest, provides a bridge between thenoncommutative and the commutative worlds. A common approach in noncommutativegeometry is to study abelian or triangulated categories and to think of them as thereplacement of an underlying scheme. This idea goes back to work of Grothendieckand Gabriel and continues to be of interest. The approach is justified by the fact thata (commutative noetherian) scheme can be reconstructed from the abelian category ofquasi-coherent sheaves (Gabriel) or from the category of perfect complexes (Balmer). Anatural question to ask, is whether the hypothesis on the scheme to be noetherian is reallynecessary. This is precisely the theme of recent work of Garkusha and Prest. They showhow to deal with affine schemes, without imposing any finiteness assumptions. Morespecifically, the article in this volume discusses for module categories over commutativerings the classification of torsion classes of finite type. For instance, the prime idealspectrum can be reconstructed from this classification.The final two contributions are about K-theory and (commutative) algebraic geometry.T. Geisser’s article is about Parshin’s conjecture. The latter states that K i .X/ Q D 0for i > 0 and X smooth and projective over a finite field F q . The purpose ofthis article is to break up Parshin’s conjecture into several independent statements,in the hope that each of them is easier to attack individually. If CH n .X; i/ is Bloch’shigherLChow group of cycles of relative dimension n, then in view of K i .X/ Q Šn CH n.X; i/ Q , Parshin’s conjecture is equivalent to Conjecture P.n/ for all n, statingthat CH n .X; i/ Q D 0 for i > 0, and all smooth and projective X. Assumingresolution of singularities, the author shows that Conjecture P.n/ is equivalent to theconjunction of three conjectures A.n/, B.n/ and C.n/, and gives several equivalentversions of these conjectures.Finally, C. Weibel’s article gives an axiomatic framework for proving that the normresidue map is an isomorphism (i.e., for settling the motivic Bloch–Kato conjecture).

xiiIntroductionThis framework is a part of the Voevodsky–Rost program.In conclusion, this volume presents a good sample of the wide range of aspects ofcurrent research in K-theory, noncommutative geometry and their applications.Guillermo Cortiñas, Joachim Cuntz, Max Karoubi,Ryszard Nest, and Charles A. Weibel

Program list of speakers and topicsThere were four courses – each consisting of three lectures – nine invited talks, andtwelve short communications.CoursesJoachim Cuntz Cyclic homologyBernhard Keller DG-categoriesBoris Tsygan Characteristic classes in noncommutative geometryCharles Weibel Motivic cohomologyInvited talksPaul Balmer Triangular GeometryStefan Gille A Gersten–Witt complex for Azumaya algebras with involutionVictor Ginzburg Calabi–Yau algebrasAlexander Gorokhovsky Deformation quantization of étale groupoids and gerbesMaxim Kontsevich On the degeneration of the Hodge to de Rham spectralsequenceMarc Levine Algebraic cobordismWolfgang Lück The Farrell–Jones Conjecture for algebraic K-theory holdsfor word-hyperbolic groups and arbitrary coefficientsRalf Meyer Bivariant K-theory: where topology and analysis meetAmnon Neeman The homotopy category of flat modules and Grothendieck’slocal dualityRyszard Nest Applications of triangulated structure of KK-theory to quantumgroupsAndreas Thom Comparison between algebraic and topological K-theoryMariusz Wodzicki MegatraceCommunicationsMarian Anton Homological symbolsGrzegorz Banaszak An imbedding property of K-groupsUlrich Bunke Duality of topological abelian groups stacks and T -dualityLeandro Cagliero The cohomology ring of truncated quiver algebrasPaulo Carrillo Rouse An analytical index for Lie groupoids

xivProgram list of speakers and topicsGuillermo Cortiñas K-theory and homological obstructions to regularityMathias Fuchs Cyclic cohomology associated to higher rank latticesand the idempotents of the group ringImmaculada Gálvez Up-to-homotopy structures on vertex algebrasGrigory Garkusha Triangulated categories of ringsJuan José Guccione Relative cyclic homology of square zero extensionsFernando Muro On the 1-type of Waldhausen K-theoryWend Werner Ternary rings of operators, affine manifolds and causal structure

Categorical aspects of bivariant K-theoryRalf Meyer1 IntroductionNon-commutative topology deals with topological properties of C -algebras. Alreadyin the 1970s, the classification of AF-algebras by K-theoretic data [15] and the workof Brown–Douglas–Fillmore on essentially normal operators [6] showed clearly thattopology provides useful tools to study C -algebras. A breakthrough was Kasparov’sconstruction of a bivariant K-theory for separable C -algebras. Besides its applicationswithin C -algebra theory, it also yields results in classical topology that are hard or evenimpossible to prove without it. A typical example is the Novikov conjecture, whichdeals with the homotopy invariance of certain invariants of smooth manifolds with agiven fundamental group. This conjecture has been verified for many groups usingKasparov theory, starting with [31]. The C -algebraic formulation of the Novikovconjecture is closely related to the Baum–Connes conjecture, which deals with thecomputation of the K-theory K .C red G/ of reduced group C -algebras and has beenone of the centres of attention in non-commutative topology in recent years.The Baum–Connes conjecture in its original formulation [4] only deals with a singleK-theory group; but a better understanding requires a different point of view. Theapproach by Davis and Lück in [14] views it as a natural transformation between twohomology theories for G-CW-complexes. An analogous approach in the C -algebraframework appeared in [38]. These approaches to the Baum–Connes conjecture showthe importance of studying not just single C -algebras, but categories of C -algebrasand their properties. Older ideas like the universal property of Kasparov theory areof the same nature. Studying categories of objects instead of individual objects isbecoming more and more important in algebraic topology and algebraic geometry aswell.Several mathematicians have suggested, therefore, to apply general constructionswith categories (with additional structure) like generators, Witt groups, the centre, andsupport varieties to the C -algebra context. Despite the warning below, this seems apromising project, where little has been done so far. To prepare for this enquiry, wesummarise some of the known properties of categories of C -algebras; we cover tensorproducts, some homotopy theory, universal properties, and triangulated structures.In addition, we examine the Universal Coefficient Theorem and the Baum–Connesassembly map.Despite many formal similarities, the homotopy theory of spaces and non-commutativetopology have a very different focus.On the one hand, most of the complexities of the stable homotopy category of spacesvanish for C -algebras because only very few homology theories for spaces have anon-commutative counterpart: any functor on C -algebras satisfying some reasonable

2 R. Meyerassumptions must be closely related to K-theory. Thus special features of topologicalK-theory become more transparent when we work with C -algebras.On the other hand, analysis may create new difficulties, which appear to be very hardto study topologically. For instance, there exist C -algebras with vanishing K-theorywhich are nevertheless non-trivial in Kasparov theory; this means that the UniversalCoefficient Theorem fails for them. I know no non-trivial topological statement aboutthe subcategory of the Kasparov category consisting of C -algebras with vanishingK-theory; for instance, I know no compact objects.It may be necessary, therefore, to restrict attention to suitable “bootstrap” categoriesin order to exclude pathologies that have nothing to do with classical topology. Moreor less by design, the resulting categories will be localisations of purely topologicalcategories, which we can also construct without mentioning C -algebras. For instance,we know that the Rosenberg–Schochet bootstrap category is equivalent to a full subcategoryof the category of BU-module spectra. But we can hope for more interestingcategories when we work equivariantly with respect to, say, discrete groups.2 Additional structure in C -algebra categoriesWe assume that the reader is familiar with some basic properties of C -algebras, includingthe definition (see for instance [2], [13]). As usual, we allow non-unital C -algebras.We define some categories of C -algebras in §2.1 and consider group C -algebras andcrossed products in §2.2. Then we discuss C -tensor products and mention the notionsof nuclearity and exactness in §2.3. The upshot is that C alg and G-C alg carry twostructures of symmetric monoidal category, which coincide for nuclear C -algebras.We prove in §2.4 that C alg and G-C alg are bicomplete, that is, all diagrams in themhave both a limit and a colimit. We equip morphism spaces between C -algebras with acanonical base point and topology in §2.5; thus the category of C -algebras is enrichedover the category of pointed topological spaces. In §2.6, we define mapping cones andcylinders in categories of C -algebras; these rudimentary tools suffice to carry oversome basic homotopy theory.2.1 Categories of C -algebrasDefinition 1. The category of C -algebras is the category C alg whose objects arethe C -algebras and whose morphisms A ! B are the -homomorphisms A ! B;wedenote this set of morphisms by Hom.A; B/.AC -algebra is called separable if it has a countable dense subset. We often restrictattention to the full subcategory C sep C alg of separable C -algebras.Examples of C -algebras are group C -algebras and C -crossed products. Webriefly recall some relevant properties of these constructions. A more detailed discussioncan be found in many textbooks such as [44].

Categorical aspects of bivariant K-theory 3Definition 2. We write A 22 C to denote that A is an object of the category C. Thenotation f 2 C means that f is a morphism in C; but to avoid confusion we alwaysspecify domain and target and write f 2 C.A; B/ instead of f 2 C.2.2 Group actions, and crossed products. For any locally compact group G, wehave a reduced group C -algebra C red .G/ and a full group C -algebra C .G/. Both aredefined as completions of the group Banach algebra .L 1 .G/; / for suitable C -norms.They are related by a canonical surjective -homomorphism C .G/ ! C red.G/, whichis an isomorphism if and only if G is amenable.The norm on C .G/ is the maximal C -norm, so that any strongly continuousunitary representation of G on a Hilbert space induces a -representation of C .G/.The norm on C red .G/ is defined using the regular representation of G on L2 .G/; hence arepresentation of G only induces a -representation of C red.G/ if it is weakly containedin the regular representation. For reductive Lie groups and reductive p-adic groups,these representations are exactly the tempered representations, which are much easierto classify than all unitary representations.Definition 3. A G-C -algebra isaC -algebra A with a strongly continuous representationof G by C -algebra automorphisms. The category of G-C -algebras is the categoryG-C alg whose objects are the G-C -algebras and whose morphisms A ! Bare the G-equivariant -homomorphisms A ! B; we denote this morphism set byHom G .A; B/.Example 4. If G D Z, then a G-C -algebra is nothing but a pair .A; ˛/ consisting ofaC -algebra A and a -automorphism ˛ W A ! A: let ˛ be the action of the generator1 2 Z.Equipping C -algebras with a trivial action provides a functor W C alg ! G-C alg; A 7! A : (1)Since C has only the identity automorphism, the trivial action is the only way to turn Cinto a G-C -algebra.The full and reduced C -crossed products are versions of the full and reduced groupC -algebras with coefficients in G-C -algebras (see [44]). They define functorsG Ë ;GË r W G-C alg ! C alg; A 7! G Ë A; G Ë r A;such that G Ë C D C .G/ and G Ë r C D C red .G/.Definition 5. A diagram I ! E ! Q in C alg is an extension if it is isomorphic tothe canonical diagram I ! A ! A=I for some ideal I inaC -algebra A; extensionsin G-C alg are defined similarly, using G-invariant ideals in G-C -algebras. We writeI E Q to denote extensions.Although C -algebra extensions have some things in common with extensions of,say, modules, there are significant differences because C alg is not Abelian, not evenadditive.

4 R. MeyerProposition 6. The full crossed product functor G Ë W G-C alg ! C alg is exactin the sense that it maps extensions in G-C alg to extensions in C alg.Proof. This is Lemma 4.10 in [21].Definition 7. A locally compact group G is called exact if the reduced crossed productfunctor G Ë r W G-C alg ! C alg is exact.Although this is not apparent from the above definition, exactness is a geometricproperty of a group: it is equivalent to Yu’s property (A) or to the existence of anamenable action on a compact space [43].Most groups you know are exact. The only source of non-exact groups known at themoment are Gromov’s random groups. Although exactness might remind you of thenotion of flatness in homological algebra, it has a very different flavour. The differenceis that the functor G Ë r always preserves injections and surjections. What may gowrong for non-exact groups is exactness in the middle (compare the discussion beforeProposition 18). Hence we cannot study the lack of exactness by derived functors.Even for non-exact groups, there is a class of extensions for which reduced crossedproducts are always exact:Definition 8. A section for an extensionI i E p Q (2)in G-C alg is a map (of sets) s W Q ! E with p ıs D id Q . We call (2) split if there is asection that is a G-equivariant -homomorphism. We call (2) G-equivariantly cp-splitif there is a G-equivariant, completely positive, contractive, linear section.Sections are also often called lifts, liftings,orsplittings.Proposition 9. Both the reduced and the full crossed product functors map split extensionsin G-C alg again to split extensions in C alg and G-equivariantly cp-splitextensions in G-C alg to cp-split extensions in C alg.Proof. Let K i E p Q be an extension in G-C alg. Proposition 6 shows thatG Ë K G Ë E G Ë Q is again an extension. Since reduced and full crossedproducts are functorial for equivariant completely positive contractions, this extensionis split or cp-split if the original extension is split or equivariantly cp-split, respectively.This yields the assertions for full crossed products.Since a -homomorphism with dense range is automatically surjective, the inducedmap G Ë r p W G Ë r E ! G Ë r Q is surjective. It is evident from the definition ofreduced crossed products that G Ë r i is injective. What is unclear is whether therange of G Ë r i and the kernel of G Ë r p coincide. As for the full crossed product, aG-equivariant completely positive contractive section s W Q ! E induces a completelypositive contractive section G Ë r s for G Ë r p. The linear map' WD id GËr E.G Ë r s/ ı .G Ë r p/W G Ë r E ! G Ë r E

Categorical aspects of bivariant K-theory 5is a retraction from G Ë r E onto the kernel of G Ë r p by construction. Furthermore, itmaps the dense subspace L 1 .G; E/ into L 1 .G; K/. Hence it maps all of G Ë r E intoG Ë r K. This implies G Ë r K D ker.G Ë r p/ as desired.2.3 Tensor products and nuclearity. Most results in this section are proved in detailin [42], [60]. Let A 1 and A 2 be two C -algebras. Their (algebraic) tensor productA 1 ˝ A 2 is still a -algebra. A C -tensor product of A 1 and A 2 is a C -completionof A 1 ˝ A 2 , that is, a C -algebra that contains A 1 ˝ A 2 as a dense -subalgebra. AC -tensor product is determined uniquely by the restriction of its norm to A 1 ˝ A 2 .Anorm on A 1 ˝ A 2 is allowed if it is a C -norm, that is, multiplication and involutionhave norm 1 and kx xkDkxk 2 for all x 2 A 1 ˝ A 2 .There is a maximal C -norm on A 1 ˝ A 2 . The resulting C -tensor product iscalled maximal C -tensor product and denoted A 1 ˝max A 2 . It is characterised by thefollowing universal property:Proposition 10. There is a natural bijection between non-degenerate -homomorphismsA 1 ˝max A 2 ! B.H/ and pairs of commuting non-degenerate -homomorphismsA 1 ! B.H/ and A 2 ! B.H/; here we may replace B.H/ by any multiplier algebraM.D/ of a C -algebra D.A -representation A ! B.H/ is non-degenerate if A H is dense in H; we needthis to get representations of A 1 and A 2 out of a representation of A 1 ˝max A 2 because,for non-unital algebras, A 1 ˝max A 2 need not contain copies of A 1 and A 2 .The maximal tensor product is natural, that is, it defines a bifunctor˝max W C alg C alg ! C alg:If A 1 and A 2 are G-C -algebras, then A 1 ˝max A 2 inherits two group actions of G bynaturality; these are again strongly continuous, so that A 1 ˝max A 2 becomes a G G-C -algebra. Restricting the action to the diagonal in G G, we turn A 1 ˝max A 2 intoa G-C -algebra. Thus we get a bifunctor˝max W G-C alg G-C alg ! G-C alg:The following lemma asserts, roughly speaking, that this tensor product has thesame formal properties as the usual tensor product for vector spaces:Lemma 11. There are canonical isomorphisms.A ˝max B/ ˝max C Š A ˝max .B ˝max C/;A ˝max B Š B ˝max A;C ˝max A Š A Š A ˝max Cfor all objects of G-C alg (and, in particular, of C alg). These define a structure ofsymmetric monoidal category on G-C alg (see [35], [52]).

6 R. MeyerA functor between symmetric monoidal categories is called symmetric monoidal ifit is compatible with the tensor products in a suitable sense [52]. A trivial example isthe functor W C alg ! G-C alg that equips a C -algebra with the trivial G-action.It follows from the universal property that ˝max is compatible with full crossedproducts: if A 22 G-C alg, B 22 C alg, then there is a natural isomorphismG Ë A ˝max .B/ Š .G Ë A/ ˝max B: (3)Like full crossed products, the maximal tensor product may be hard to describe becauseit involves a maximum of all possible C -tensor norms. There is another C -tensor normthat is defined more concretely and that combines well with reduced crossed products.Recall that any C -algebra A can be represented faithfully on a Hilbert space. Thatis, there is an injective -homomorphism A ! B.H/ for some Hilbert space H; hereB.H/ denotes the C -algebra of bounded operators on H. IfA is separable, we can findsuch a representation on the separable Hilbert space H D `2.N/. The tensor product oftwo Hilbert spaces H 1 and H 2 carries a canonical inner product and can be completedto a Hilbert space, which we denote by H 1 x˝ H 2 . If H 1 and H 2 support faithfulrepresentations of C -algebras A 1 and A 2 , then we get an induced -representation ofA 1 ˝ A 2 on H 1 x˝ H 2 .Definition 12. The minimal tensor product A 1 ˝min A 2 is the completion of A 1 ˝ A 2with respect to the operator norm from B.H 1 x˝ H 2 /.It can be check that this is well-defined, that is, the C -norm on A 1 ˝ A 2 does notdepend on the chosen faithful representations of A 1 and A 2 . The same argument alsoyields the naturality of A 1 ˝min A 2 . Hence we get a bifunctor˝min W G-C alg G-C alg ! G-C algIit defines another symmetric monoidal category structure on G-C alg.We may also call A 1 ˝min A 2 the spatial tensor product. It is minimal in the sensethat it is dominated by any C -tensor norm on A 1 ˝A 2 that is compatible with the givennorms on A 1 and A 2 . In particular, we have a canonical surjective -homomorphismA 1 ˝max A 2 ! A 1 ˝min A 2 : (4)Definition 13. AC -algebra A 1 is nuclear if the map in (4) is an isomorphism for allC -algebras A 2 .The name comes from an analogy between nuclear C -algebras and nuclear locallyconvex topological vector spaces (see [20]). But this is merely an analogy: the onlyC -algebras that are nuclear as locally convex topological vector spaces are the finitedimensionalones.Many important C -algebras are nuclear. This includes the following examples:• commutative C -algebras;• matrix algebras and algebras of compact operators on Hilbert spaces;

Categorical aspects of bivariant K-theory 7• group C -algebras of amenable groups (or groupoids);•C -algebras of type I and, in particular, continuous trace C -algebras.If A is nuclear, then there is only one reasonable C -algebra completion of A ˝ B.Therefore, if we can write down any, it must be equal to both A ˝min B and A ˝max B.Example 14. For a compact space X andaC -algebra A, we let C.X; A/ be theC -algebra of all continuous functions X ! A. IfX is a pointed compact space, welet C 0 .X; A/ be the C -algebra of all continuous functions X ! A that vanish at thebase point of X; this contains C.X; A/ as a special case because C.X; A/ Š C 0 .X C ;A/,where X C D X tf?g with base point ?. WehaveC 0 .X; A/ Š C 0 .X/ ˝min A Š C 0 .X/ ˝max A:Example 15. There is a unique C -norm on M n ˝ A D M n .A/ for all n 2 N.For a Hilbert space H, let K.H/ be the C -algebra of compact operators on H.Then K.H/˝A contains copies of M n .A/, n 2 N, for all finite-dimensional subspacesof H. These carry a unique C -norm. The C -norms on these subspaces are compatibleand extend to the unique C -norm on K.H/ ˝ A.The class of nuclear C -algebras is closed under ideals, quotients (by ideals), extensions,inductive limits, and crossed products by actions of amenable locally compactgroups. In particular, this covers crossed products by automorphisms (see Example 4).C -subalgebras of nuclear C -algebras need not be nuclear any more, but they stillenjoy a weaker property called exactness:Definition 16. AC -algebra A is called exact if the functor A ˝min preservesC -algebra extensions.It is known [33], [43] that a discrete group is exact (Definition 7) if and only if itsgroup C -algebra is exact (Definition 16), if and only if the group has an amenableaction on some compact topological space.Example 17. Let G be the non-Abelian free group on 2 generators. Let G act freely andproperly on a tree as usual. Let X be the ends compactification of this tree, equippedwith the induced action of G. This action is known to be amenable, so that G is anexact group. Since the action is amenable, the crossed product algebras G Ë r C.X/ andG ËC.X/ coincide and are nuclear. The embedding C ! C.X/ induces an embeddingC red .G/ ! G Ë C.X/. But G is not amenable. Hence the C -algebra C red.G/ is exactbut not nuclear.As for crossed products, ˝min respects injections and surjections. The issue withexactness in the middle is the following. Elements of A ˝min B are limits of tensorsof the form P niD1 a i ˝ b i with a 1 ;:::;a n 2 A, b 1 ;:::;b n 2 B. If an elementin A ˝min B is annihilated by the map to .A=I / ˝min B, then we can approximateit by such finite sums for which P niD1 .a i mod I/ ˝ b i goes to 0. But this doesnot suffice to find approximations in I ˝ B. Thus the kernel of the projection mapA ˝min B .A=I / ˝min B may be strictly larger than I ˝min B.

8 R. MeyerProposition 18. The functor ˝max is exact in each variable, that is, ˝max D mapsextensions in G-C alg again to extensions for each D 22 G-C alg.Both ˝min and ˝max map split extensions to split extensions and (equivariantly)cp-split extensions again to (equivariantly) cp-split extensions.The proof is similar to the proofs of Propositions 6 and 9.If A or B is nuclear, we simply write A ˝ B for A ˝max B Š A ˝min B.2.4 Limits and colimitsProposition 19. The categories C alg and G-C alg are bicomplete, that is, any(small) diagram in these categories has both a limit and a colimit.Proof. To get general limits and colimits, it suffices to construct equalisers and coequalisersfor pairs of parallel morphisms f 0 ;f 1 W A B, direct products and coproductsA 1 A 2 and A 1 t A 2 for any pair of objects and, more generally, for arbitrary setsof objects.The equaliser and coequaliser of f 0 ;f 1 W A B areker.f 0f 1 / Dfa 2 A j f 0 .a/ D f 1 .a/g Aand the quotient of A 1 by the closed -ideal generated by the range of f 0 f 1 , respectively.Here we use that quotients of C -algebras by closed -ideals are againC -algebras. Notice that ker.f 0 f 1 / is indeed a C -subalgebra of A.The direct product A 1 A 2 is the usual direct product, equipped with the canonicalC -algebra structure. We can generalise the construction of the direct product to infinitedirect products: let Q i2I A i be the set of all norm-bounded sequences .a i / i2I witha i 2 A i for all i 2 I ; this is a C -algebra with respect to the obvious -algebra structureand the norm k.a i /k WD sup i2I ka i k. It has the right universal property because any -homomorphism is norm-contracting. (A similar construction with Banach algebraswould fail at this point.)The coproduct A 1 tA 2 is also called free product and denoted A 1 A 2 ; its constructionis more involved. The free C-algebra generated by A 1 and A 2 carries a canonicalinvolution, so that it makes sense to study C -norms on it. It turns out that there is amaximal such C -norm. The resulting C -completion is the free product C -algebra.In the equivariant case, A 1 t A 2 inherits an action of G, which is strongly continuous.The resulting object of G-C alg has the correct universal property for a coproduct.An inductive system of C -algebras .A i ;˛ji / i2I is called reduced if all the maps˛ji W A i ! A j are injective; then they are automatically isometric embeddings. Wemay as well assume that these maps are identical inclusions of C -subalgebras. Thenwe can form a -algebra S A i , and the given C -norms piece together to a C -normon S A i . The resulting completion is lim .A ! i ;˛ji/. In particular, we can construct aninfinite coproduct as the inductive limit of its finite sub-coproducts. Thus we get infinitecoproducts.

Categorical aspects of bivariant K-theory 9The category of commutative C -algebras is equivalent to the opposite of the categoryof pointed compact spaces by the Gelfand–Naimark Theorem. It is frequentlyconvenient to replace a pointed compact space X with base point ? by the locallycompact space X nf?g. A continuous map X ! Y extends to a pointed continuousmap X C ! Y C if and only if it is proper. But there are more pointed continuousmaps f W X C ! Y C than proper continuous maps X ! Y because points in X maybe mapped to the point at infinity 12Y C . For instance, the zero homomorphismC 0 .Y / ! C 0 .X/ corresponds to the constant map x 7! 1.Example 20. If U X is an open subset of a locally compact space, then C 0 .U / isan ideal in C 0 .X/. No map X ! U corresponds to the embedding C 0 .U / ! C 0 .X/.Example 21. Products of commutative C -algebras are again commutative and correspondby the Gelfand–Naimark Theorem to coproducts in the category of pointedcompact spaces. The coproduct of a set of pointed compact spaces is the Stone–Čechcompactification of their wedge sum. Thus infinite products in C alg and G-C algdo not behave well for the purposes of homotopy theory.The coproduct of two non-zero C -algebras is never commutative and hence has noanalogue for (pointed) compact spaces. The smash product for pointed compact spacescorresponds to the tensor product of C -algebras becauseC 0 .X ^ Y/Š C 0 .X/ ˝min C 0 .Y /:2.5 Enrichment over pointed topological spaces. Let A and B be C -algebras. It iswell-known that a -homomorphism f W A ! B is automatically norm-contracting andinduces an isometric embedding A= ker f ! B with respect to the quotient norm onA= ker f . The reason for this is that the norm for self-adjoint elements in a C -algebraagrees with the spectral radius and hence is determined by the algebraic structure; bythe C -condition kak 2 Dka ak, this extends to all elements of a C -algebra.It follows that Hom.A; B/ is an equicontinuous set of linear maps A ! B. We alwaysequip Hom.A; B/ with the topology of pointwise norm-convergence. Its subbasicopen subsets are of the formff W A ! B jk.ff 0 /.a/k

10 R. Meyerdefines the topology of uniform convergence on compact subsets on Hom.A; B/ becausethe latter is equicontinuous.There is a distinguished element in Hom.A; B/ as well, namely, the zero homomorphismA ! 0 ! B. Thus Hom.A; B/ becomes a pointed topological space.Proposition 23. The above construction provides an enrichment of C alg over thecategory of pointed Hausdorff topological spaces.Proof. It is clear that 0 ı f D 0 and f ı 0 D 0 for all morphisms f . Furthermore, wemust check that composition of morphisms is jointly continuous. This follows fromthe equicontinuity of Hom.A; B/.This enrichment allows us to carry over some important definitions from categoriesof spaces to C alg. For instance, a homotopy between two -homomorphismsf 0 ;f 1 W A ! B is a continuous path between f 0 and f 1 in the topological spaceHom.A; B/. In the following proposition, Map C .X; Y / denotes the space of morphismsin the category of pointed topological spaces, equipped with the compact-opentopology.Proposition 24 (compare Proposition 3.4 in [28]). Let A and B be C -algebras andlet X be a pointed compact space. ThenMap C X; Hom.A; B/ Š Hom A; C 0 .X; B/ as pointed topological spaces.Proof. If we view A and B as pointed topological spaces with the norm topology andbase point 0, then Hom.A; B/ Map C .A; B/ is a topological subspace with the samebase point because Hom.A; B/ also carries the compact-open topology. Since X iscompact, standard point set topology yields homeomorphismsMap C X; Map C .A; B/ Š Map C .X ^ A; B/Š Map C A; Map C .X; B/ D Map C A; C 0 .X; B/ :These restrict to the desired homeomorphism.In particular, a homotopy between two -homomorphisms f 0 ;f 1 W A B is equivalentto a -homomorphism f W A ! C.Œ0; 1; B/ with ev t ı f D f t for t D 0; 1,where ev t denotes the -homomorphismev t W C.Œ0; 1; B/ ! B;f 7! f.t/:We also haveC 0 X; C 0 .Y; A/ Š C 0 .X ^ Y; A/for all pointed compact spaces X, Y and all G-C -algebras A. Thus a homotopybetween two homotopies can be encoded by a -homomorphismA ! C Œ0; 1; C.Œ0; 1; B/ Š C.Œ0; 1 2 ;B/:

Categorical aspects of bivariant K-theory 11These constructions work only for pointed compact spaces. If we enlarge thecategory of C -algebras to a suitable category of projective limits of C -algebras asin [28], then we can define C 0 .X; A/ for any pointed compactly generated space X.But we lose some of the nice analytic properties of C -algebras. Therefore, I prefer tostick to the category of C -algebras itself.2.6 Cylinders, cones, and suspensions. The following definitions go back to [54],where some more results can be found. The description of homotopies above leads usto define the cylinder over a C -algebra A byCyl.A/ WD C.Œ0; 1; A/:This is compatible with the cylinder construction for spaces becauseCyl C 0 .X/ Š C Œ0; 1; C 0 .X/ Š C 0 .Œ0; 1 C ^ X/for any pointed compact space X; if we use locally compact spaces, we get Œ0; 1 Xinstead of Œ0; 1 C ^ X.The universal property of Cyl.A/ is dual to the usual one for spaces because theidentification between pointed compact spaces and commutative C -algebras is contravariant.Similarly, we may define the cone Cone.A/ and the suspension Sus.A/ byCone.A/ WD C 0 Œ0; 1 nf0g; A/;Sus.A/ WD C 0 Œ0; 1 nf0; 1g;A/Š C 0 .S 1 ; A/;where S 1 denotes the pointed 1-sphere, that is, circle. These constructions are compatiblewith the corresponding ones for spaces as well, that is,Cone C 0 .X/ Š C 0 .Œ0; 1 ^ X/;Sus C 0 .X/ Š C 0 .S 1 ^ X/:Here Œ0; 1 has the base point 0.Definition 25. Let f W A ! B be a morphism in C alg or G-C alg. The mappingcylinder Cyl.f / and the mapping cone Cone.f / of f are the limits of the diagramsMore concretely,A f ! B ev 1Cyl.B/; A f ! B ev 1Cone.B/:Cone.f / D ˚.a; b/ 2 A C 0 .0; 1; B j f.a/D b.1/ ;Cyl.f / D ˚.a; b/ 2 A C 0 .Œ0; 1; B j f.a/D b.1/ :If f W X ! Y is a morphism of pointed compact spaces, then the mapping coneand mapping cylinder of the induced -homomorphism C 0 .f /W C 0 .Y / ! C 0 .X/ agreewith C 0 Cyl.f / and C 0 Cone.f / , respectively.The cylinder, cone, and suspension functors are exact for various kinds of extensions:they map extensions, split extensions, and cp-split extensions again to extensions, split

12 R. Meyerextensions, and cp-split extensions, respectively. Similar remarks apply to mappingcylinders and mapping cones: for any morphism of extensionsI E Qwe get extensions˛ I 0 E0 Q 0 ,ˇCyl.˛/ Cyl.ˇ/ Cyl./; Cone.˛/ Cone.ˇ/ Cone./I (6)if the extensions in (5) are split or cp-split, so are the resulting extensions in (6).The familiar maps relating mapping cones and cylinders to cones and suspensionscontinue to exist in our case. For any morphism f W A ! B in G-C alg, we get amorphism of extensionsSus.B/ Cone.f / ACone.B/ Cyl.f / A.The bottom extension splits and the maps A $ Cyl.f / are inverse to each other upto homotopy. By naturality, the composite map Cone.f / ! A ! B factors throughCone.id B / Š Cone.B/ and hence is homotopic to the zero map.3 Universal functors with certain propertiesWhen we study topological invariants for C -algebras, we usually require homotopyinvariance and some exactness and stability conditions. Here we investigate theseconditions and their interplay and describe some universal functors.We discuss homotopy invariant functors on G-C alg and the homotopy categoryHo.G-C alg/ in §3.1. This is parallel to classical topology. We turn to Morita–Rieffel invariance and C -stability in §3.2–3.3. We describe the resulting localisationusing correspondences. By the way, in a C -algebra context, stability usually refers toalgebras of compact operators instead of suspensions. §3.4 deals with various exactnessconditions: split-exactness, half-exactness, and additivity.Whereas each of the above properties in itself seems rather weak, their combinationmay have striking consequences. For instance, a functor that is both C -stable andsplit-exact is automatically homotopy invariant and satisfies Bott periodicity.Throughout this section, we consider functors S ! C where S is a full subcategoryof C alg or G-C alg for some locally compact group G. The target category Cmay be arbitrary in §3.1–§3.3; to discuss exactness properties, we require C to bean exact category or at least additive. Typical choices for S are the categories ofseparable or separable nuclear C -algebras, or the subcategory of all separable nuclearG-C -algebras with an amenable (or a proper) action of G.(5)

Categorical aspects of bivariant K-theory 13Definition 26. Let P be a property for functors defined on S. A universal functorwith P is a functor uW S ! Univ P .S/ such that• xF ı u has P for each functor xF W Univ P .S/ ! C;• any functor F W S ! C with P factors uniquely as F D xF ı u for some functorxF W Univ P .S/ ! C.Of course, a universal functor with P need not exist. If it does, then it restricts toa bijection between objects of S and Univ P .S/. Hence we can completely describeit by the sets of morphisms Univ P .A; B/ from A to B in Univ P .S/ and the mapsS.A; B/ ! Univ P .A; B/ for A; B 22 S; the universal property means that forany functor F W S ! C with P there is a unique functorial way to extend the mapsHom G .A; B/ ! C F .A/; F .B/ to Univ P .A; B/. There is no a priori reason whythe morphism spaces Univ P .A; B/ for A; B 22 S should be independent of S; butthis happens in the cases we consider, under some assumption on S.3.1 Homotopy invariance. The following discussion applies to any full subcategoryS G-C alg that is closed under the cylinder functor.Definition 27. Let f 0 ;f 1 W A B be two parallel morphisms in S. We write f 0 f 1and call f 0 and f 1 homotopic if there is a homotopy between f 0 and f 1 , that is, amorphism f W A ! Cyl.B/ D C.Œ0; 1; B/ with ev t ı f D f t for t D 0; 1.It is easy to check that homotopy is an equivalence relation on Hom G .A; B/. We letŒA; B be the set of equivalence classes. The composition of morphisms in S descendsto mapsŒB; C ŒA; B ! ŒA; C ; Œf ; Œg 7! Œf ı g;that is, f 1 f 2 and g 1 g 2 implies f 1 ı f 2 g 1 ı g 2 . Thus the sets ŒA; B formthe morphism sets of a category, called homotopy category of S and denoted Ho.S/.The identity maps on objects and the canonical maps on morphisms define a canonicalfunctor S ! Ho.S/. A morphism in S is called a homotopy equivalence if it becomesinvertible in Ho.S/.Lemma 28. The following are equivalent for a functor F W S ! C:(a) F.ev 0 / D F.ev 1 / as maps F C.Œ0; 1; A/ ! F .A/ for all A 22 S;(b) F.ev t / induces isomorphisms F C.Œ0; 1; A/ ! F .A/ for all A 22 S, t 2 Œ0; 1;(c) the embedding as constant functions const W A ! C.Œ0; 1; A/ induces an isomorphismF .A/ ! F C.Œ0; 1; A/ for all A 22 S;(d) F maps homotopy equivalences to isomorphisms;(e) if two parallel morphisms f 0 ;f 1 W A B are homotopic, then F.f 0 / D F.f 1 /;(f) F factors through the canonical functor S ! Ho.S/.Furthermore, the factorisation Ho.S/ ! C in (f) is necessarily unique.

14 R. MeyerProof. We only mention two facts that are needed for the proof. First, we haveev t ı const D id A and const ı ev t id C.Œ0;1;A/ for all A 22 S and all t 2 Œ0; 1.Secondly, an isomorphism has a unique left and a unique right inverse, and these areagain isomorphisms.The equivalence of (d) and (f) in Lemma 28 says that Ho.S/ is the localisation of Sat the family of homotopy equivalences.Since C Œ0; 1; C 0 .X/ Š C 0 .Œ0; 1 X/ for any locally compact space X, ournotion of homotopy restricts to the usual one for pointed compact spaces. Hence theopposite of the homotopy category of pointed compact spaces is equivalent to a fullsubcategory of Ho.C alg/.The sets ŒA; B inherit a base point Œ0 and a quotient topology from Hom.A; B/;thus Ho.S/ is enriched over pointed topological spaces as well. This topology onŒA; B is not so useful, however, because it need not be Hausdorff.A similar topology exists on Kasparov groups and can be defined in various ways,which turn out to be equivalent [12].Let F W G-C alg ! H -C alg be a functor with natural isomorphismsF C.Œ0; 1; A/ Š C.Œ0; 1; F .A/ that are compatible with evaluation maps for all A. The universal property implies that Fdescends to a functor Ho.G-C alg/ ! Ho.H -C alg/. In particular, this applies tothe suspension, cone, and cylinder functors and, more generally, to the functors A˝maxand A˝min on G-C alg for any A 22 G-C alg because both tensor product functorsare associative and commutative andC.Œ0; 1; A/ Š C.Œ0; 1/ ˝max A Š C.Œ0; 1/ ˝min A:The same reasoning applies to the reduced and full crossed product functors G-C alg !C alg.We may stabilise the homotopy category with respect to the suspension functor andconsider a suspension-stable homotopy category with morphism spacesfA; Bg WD lim !k!1ŒSus k A; Sus k Bfor all A; B 22 G-C alg. We may also enlarge the set of objects by adding formaldesuspensions and generalising the notion of spectrum. This is less interesting forC -algebras than for spaces because most functors of interest satisfy Bott Periodicity,so that suspension and desuspension become equivalent.3.2 Morita–Rieffel equivalence and stable isomorphism. One of the basic ideas ofnon-commutative geometry is that G Ë C 0 .X/ (or G Ë r C 0 .X/) should be a substitutefor the quotient space GnX, which may have bad singularities. In the special case of afree and proper G-space X, we expect that G Ë C 0 .X/ and C 0 .GnX/ are “equivalent”

Categorical aspects of bivariant K-theory 15in a suitable sense. Already the simplest possible case X D G shows that we cannotexpect an isomorphism here becauseG Ë C 0 .G/ Š G Ë r C 0 .G/ Š K.L 2 G/:The right notion of equivalence is a C -version of Morita equivalence due to Marc A.Rieffel ([46], [47], [48]); therefore, we call it Morita–Rieffel equivalence.The definition of Morita–Rieffel equivalence involves Hilbert modules over C -algebrasand the C -algebras of compact operators on them; these notions are crucial forKasparov theory as well. We refer to [34] for the definition and a discussion of theirbasic properties.Definition 29. Two G-C -algebras A and B are called Morita–Rieffel equivalent ifthere are a full G-equivariant Hilbert B-module E and a G-equivariant -isomorphismK.E/ Š A.It is possible (and desirable) to express this definition more symmetrically: E is anA; B-bimodule with two inner products taking values in A and B, satisfying variousconditions [46]. Morita–Rieffel equivalent G-C -algebras have equivalent categoriesof G-equivariant Hilbert modules via E ˝B . The converse is unclear.Example 30. The following is a more intricate example of a Morita–Rieffel equivalence.Let and P be two subgroups of a locally compact group G. Then acts onG=P by left translation and P acts on nG by right translation. The correspondingorbit space is the double coset space nG=P . Both ËC 0 .G=P / and P ËC 0 .nG/arenon-commutative models for this double coset space. They are indeed Morita–Rieffelequivalent; the bimodule that implements the equivalence is a suitable completion ofC c .G/, the space of continuous functions with compact support on G.These examples suggest that Morita–Rieffel equivalent C -algebras describe thesame non-commutative space. Therefore, we expect that reasonable functors on C algshould not distinguish between Morita–Rieffel equivalent C -algebras. (We willslightly weaken this statement below.)Definition 31. Two G-C -algebras A and B are called stably isomorphic if there is aG-equivariant -isomorphism A ˝ K.H G / Š B ˝ K.H G /, where H G WD L 2 .G N/is the direct sum of countably many copies of the regular representation of G; we let Gact on K.H G / by conjugation, of course.The following technical condition is often needed in connection with Morita–Rieffelequivalence.Definition 32. AC -algebra is called -unital if it has a countable approximate identityor, equivalently, contains a strictly positive element.Example 33. All separable C -algebras and all unital C -algebras are -unital; thealgebra K.H/ is -unital if and only if H is separable.

16 R. MeyerTheorem 34 ([7]). Two -unital G-C -algebras are G-equivariantly Morita–Rieffelequivalent if and only if they are stably isomorphic.In the non-equivariant case, this theorem is due to Brown–Green–Rieffel [7]. Asimpler proof that carries over to the equivariant case appeared in [41].3.3 C -stable functors. The definition of C -stability is more intuitive in the nonequivariantcase:Definition 35. Fix a rank-one projection p 2 K.`2N/. The resulting embeddingA ! A ˝ K.`2N/, a 7! a ˝ p, is called a corner embedding of A.A functor F W C alg ! C is called C -stable if any corner embedding induces anisomorphism F .A/ Š F A ˝ K.`2N/ .The correct equivariant generalisation is the following:Definition 36 ([36]). A functor F W G-C alg ! C is called C -stable if the canonicalembeddings H 1 ! H 1 ˚ H 2 H 2 induce isomorphismsF A ˝ K.H 1 / Š! F A ˝ K.H1 ˚ H 2 / ŠF A ˝ K.H2 / for all non-zero G-Hilbert spaces H 1 and H 2 .Of course, it suffices to require F A ˝ K.H 1 / Š! F A ˝ K.H1 ˚ H 2 / .Itisnot hard to check that Definitions 35 and 36 are equivalent for trivial G.Remark 37. We have argued in §3.2 why C -stability is an essential property for anydecent homology theory for C -algebras. Nevertheless, it is tempting to assume lessbecause C -stability together with split-exactness has very strong implications.One reasonable way to weaken C -stability is to replace K.`2N/ by M n for n 2 Nin Definition 35 (see [57]). If two unital C -algebras are Morita–Rieffel equivalent,then they are also Morita equivalent as rings, that is, the equivalence is implementedby a finitely generated projective module. This implies that a matrix-stable functor isinvariant under Morita–Rieffel equivalence for unital C -algebras.Matrix-stability also makes good sense in G-C alg for a compact group G: simplyrequire H 1 and H 2 in Definition 36 to be finite-dimensional. But we seem to runinto problems for non-compact groups because they may have few finite-dimensionalrepresentations and we lack a finite-dimensional version of the equivariant stabilisationtheorem.Our next goal is to describe the universal C -stable functor. We abbreviate A K WDK.L 2 G/ ˝ A.Definition 38. A correspondence from A to B (or A Ü B) isaG-equivariantHilbert B K -module E together with a G-equivariant essential (or non-degenerate) -homomorphism f W A K ! K.E/.

Categorical aspects of bivariant K-theory 17Given correspondences E from A to B and F from B to C , their composition isthe correspondence from A to C with underlying Hilbert module E x˝BK F and mapA K ! K.E/ ! K.E x˝BK F /, where the last map sends T 7! T ˝ 1; this yieldscompact operators because B K maps to K.F /. See [34] for the definition of the relevantcompleted tensor product of Hilbert modules.Up to isomorphism, the composition of correspondences is associative and the identitymaps A ! A D K.A/ act as unit elements. Hence we get a category Corr G whosemorphisms are the isomorphism classes of correspondences. It may have advantagesto treat Corr G as a 2-category.Any -homomorphism ' W A ! B yields a correspondence f W A ! K.E/ from Ato B, so that we get a canonical functor \W G-C alg ! Corr G . We let E be the rightideal '.A K / B K in B K , viewed as a Hilbert B-module. Then f.a/ b WD '.a/ brestricts to a compact operator f.a/ on E and f W A ! K.E/ is essential. It can bechecked that this construction is functorial.In the following proposition, we require that the category of G-C -algebras S beclosed under Morita–Rieffel equivalence and consist of -unital G-C -algebras. Welet Corr S be the full subcategory of Corr G with object class S.Proposition 39. The functor \W S ! Corr S is the universal C -stable functor on S;that is, it is C -stable, and any other such functor factors uniquely through \.Proof. First we sketch the proof in the non-equivariant case.We verify that \ is C -stable. The Morita–Rieffel equivalence between K.`2N/ ˝A Š K `2.N;A/ and A is implemented by the Hilbert module `2.N;A/, whichyields a correspondence id;`2.N;A/ from K.`2N/ ˝ A to A; this is inverse to thecorrespondence induced by a corner embedding A ! K.`2N/ ˝ A.A Hilbert B-module E with an essential -homomorphism A ! K.E/ is countablygenerated because A is assumed -unital. Kasparov’s Stabilisation Theorem yields anisometric embedding E ! `2.N;B/. Hence we get -homomorphismsA ! K.`2N/˝ B B:This diagram induces a map F .A/ ! F.K.`2N/ ˝ B/ Š F.B/ for any C -stablefunctor F . Now we should check that this well-defines a functor xF W Corr S ! C withxF ı \ D F , and that this yields the only such functor. We omit these computations.The generalisation to the equivariant case uses the crucial property of the left regularrepresentation that L 2 .G/ ˝ H Š L 2 .G N/ for any countably infinite-dimensionalG-Hilbert space H. Since we replace A and B by A K and B K in the definition ofcorrespondence right away, we can use this to repair a possible lack of G-equivariance;similar ideas appear in [36].Example 40. Let u be a G-invariant multiplier of B with u u D 1; such u arealso called isometries. Then b 7! ubu defines a -homomorphism B ! B. Theresulting correspondence B Ü B is isomorphic as a correspondence to the identity

18 R. Meyercorrespondence: the isomorphism is given by left multiplication with u, which definesa G-equivariant unitary operator from B to the closure of uBu B D u B.Hence inner endomorphisms act trivially on C -stable functors. Actually, this isone of the computations that we have omitted in the proof above; the argument can befound in [11].Now we make the definition of a correspondence more concrete if A is unital. Wehave an essential -homomorphism ' W A ! K.E/ for some G-equivariant HilbertB-module E. Since A is unital, this means that K.E/ is unital and ' is a unital -homomorphism. Then E is finitely generated. Thus E D B 1 p for some projectionp 2 M 1 .B/ and ' is a -homomorphism ' W A ! M 1 .B/ with '.1/ D p. Two -homomorphisms ' 1 ;' 2 W A M 1 .B/ yield isomorphic correspondences if andonly if there is a partial isometry v 2 M 1 .B/ with v' 1 .x/v D ' 2 .x/ and v ' 2 .a/v D' 1 .a/ for all a 2 A.Finally, we combine homotopy invariance and C -stability and consider the universalC -stable homotopy-invariant functor. This functor is much easier to characterise:the morphisms in the resulting universal category are simply the homotopy classes ofG-equivariant -homomorphisms K L 2 .G N/ ˝ A ! K L 2 .G N/ ˝ B/ (see[36], Proposition 6.1). Alternatively, we get the same category if we use homotopyclasses of correspondences A Ü B instead.3.4 Exactness properties. Throughout this subsection, we consider functors F W S !C with values in an exact category C. If C is merely additive to begin with, we canequip it with the trivial exact category structure for which all extensions split. We alsosuppose that S is closed under the kinds of C -algebra extensions that we consider;depending on the notion of exactness, this means: direct product extensions, split extensions,cp-split extensions, or all extensions, respectively. Recall that split extensionsin G-C alg are required to split by a G-equivariant -homomorphism.3.4.1 Additive functors. The most trivial split extensions in G-C alg are the productextensions A A B B for two objects A; B. In this case, the coordinateembeddings and projections provide mapsA A B B: (7)Definition 41. We call F additive if it maps product diagrams (7) in S to direct sumdiagrams in C.There is a partially defined addition on -homomorphisms: call two parallel -homomorphisms'; W A B orthogonal if '.a 1 / .a 2 / D 0 for all a 1 ;a 2 2 A. Equivalently,' C W a 7! '.a/ C .a/ is again a -homomorphism.Lemma 42. The functor F is additive if and only if, for all A; B 22 S, the mapsHom.A; B/ ! C F .A/; F .B/ satisfy F.' C / D F.'/ C F. / for all pairs oforthogonal parallel -homomorphisms '; .

Categorical aspects of bivariant K-theory 19Alternatively, we may also require additivity for coproducts (that is, free products).Of course, this only makes sense if S is closed under coproducts in G-C alg. Thecoproduct A t B in G-C alg comes with canonical maps A A t B B as well;the maps A W A ! A t B and B W B ! A t B are the coordinate embeddings, themaps A W A t B ! A and B W A t B ! B restrict to .id A ;0/ and .0; id B / on Aand B, respectively.Definition 43. We call F additive on coproducts if it maps coproduct diagrams A A t B B to direct sum diagrams in C.The coproduct and product are related by a canonical G-equivariant -homomorphism' W A t B A B that is compatible with the maps to and from A and B, that is,' ı A D A , A ı ' D A , and similarly for B. There is no map backwards, but thereis a correspondence W A B Ü A t B, which is induced by the G-equivariant -homomorphism A .a/ 0A B ! M 2 .A t B/; .a;b/ 7!:0 B .b/It is easy to see that the composite correspondence ' ı is equal to the identitycorrespondence on AB. The other composite ı' is not the identity correspondence,but it is homotopic to it (see [9], [10]). This yields:Proposition 44. If F is C -stable and homotopy invariant, then the canonical mapF.'/W F.At B/ ! F.A B/ is invertible. Therefore, additivity and additivity forcoproducts are equivalent for such functors.The correspondence exists because the stabilisation creates enough room to replace A and B by homotopic homomorphisms with orthogonal ranges. We can achievethe same effect by a suspension (shift A and B to the open intervals .0; 1=2/ and .1=2;1/,respectively). Therefore, any homotopy invariant functor satisfies F Sus.A t B/ ŠF Sus.A B/ .3.4.2 Split-exact functorsDefinition 45. We call F split-exact if, for any split extension K i E p Q withsection s W Q ! E, the map F .i/; F .s/ W F.K/˚ F.Q/ ! F.E/is invertible.It is clear that split-exact functors are additive.Split-exactness is useful because of the following construction of Joachim Cuntz [9].Let B G E be a G-invariant ideal and let f C ;f W A E be G-equivariant -homomorphisms with f C .a/ f .a/ 2 B for all a 2 A. Equivalently, f C and fboth lift the same morphism f N W A ! E=B. The data .A; f C ;f ;E;B/ is called aquasi-homomorphism from A to B.Pulling back the extension B E E=B along f N , we get an extension B E 0 A with two sections fC 0 ;f0 W A E 0 . The split-exactness of F shows that

20 R. MeyerF.B/ F.E 0 / F .A/ is a split extension in C. Since both F.f 0 / and F.fC 0 /are sections for it, we get a map F.fC 0 / F.f 0 /W F .A/ ! F.B/. Thus a quasihomomorphisminduces a map F .A/ ! F.B/if F is split-exact. The formal propertiesof this construction are summarised in [11].GivenaC -algebra A, there is a universal quasi-homomorphism out of A. LetQ.A/ WD A t A be the coproduct of two copies of A and let A W Q.A/ ! A be thefolding homomorphism that restricts to id A on both factors. Let q.A/ be its kernel.The two canonical embeddings A ! A t A are sections for the folding homomorphism.Hence we get a quasi-homomorphism A Q.A/ F q.A/. The universalproperty of the free product shows that any quasi-homomorphism yields a G-equivariant -homomorphism q.A/ ! B.Theorem 46. Suppose S is closed under split extensions and tensor products withC.Œ0; 1/ and K.`2N/. IfF W S ! C is C -stable and split-exact, then F is homotopyinvariant.This is a deep result of Nigel Higson [24]; a simple proof can be found in [11].Besides basic properties of quasi-homomorphisms, it uses that inner endomorphismsact identically on C -stable functors (Example 40).Actually, the literature only contains Theorem 46 for functors on C alg. But theproof in [11] works for functors on categories S as above.3.4.3 Exact functorsDefinition 47. We call F exact if F.K/ ! F.E/ ! F.Q/ is exact (at F.E/) forany extension K E Q in S. More generally, given a class E of extensionsin S like, say, the class of equivariantly cp-split extensions, we define exactness forextensions in E.It is easy to see that exact functors are additive.Most functors we are interested in satisfy homotopy invariance and Bott periodicity,and these two properties prevent a non-zero functor from being exact in the strongersense of being left or right exact. This explains why our notion of exactness is muchweaker than usual in homological algebra.It is reasonable to require that a functor be part of a homology theory, that is,a sequence of functors .F n / n2Z together with natural long exact sequences for allextensions [54]. We do not require this additional information because it tends to behard to get a priori but often comes for free a posteriori:Proposition 48. Suppose that F is homotopy invariant and exact (or exact for equivariantlycp-split extensions). Then F has long exact sequences of the form!F Sus.K/ ! F Sus.E/ ! F Sus.Q/ ! F.K/ ! F.E/ ! F.Q/for any (equivariantly cp-split) extension K E Q. In particular, F is splitexact.

Categorical aspects of bivariant K-theory 21See §21.4 in [5] for the proof.There probably exist exact functors that are not split-exact. It is likely that thealgebraic K 1 -functor provides a counterexample: it is exact but not split-exact onthe category of rings [49]; but I do not know a counterexample to its split-exactnessinvolving only C -algebras.Proposition 48 and Bott periodicity yield long exact sequences that are infinite inboth directions. Thus an exact homotopy invariant functor that satisfies Bott periodicityis part of a homology theory in a canonical way.For universal constructions, we should replace a single functor by a homologytheory, that is, a sequence of functors. The universal functors in this context are nonstableversions of E-theory and KK-theory. We refer to [27] for details.A weaker property than exactness is the existence of Puppe exact sequences formapping cones. The Puppe exact sequence is the special case of the long exact sequenceof Proposition 48 for extensions of the form Sus.B/ Cone.f / A for a morphismf W A ! B. In practice, the exactness of a functor is often established by reducing itto the Puppe exact sequence. Let K i E p Q be an extension. The Puppe exactsequence yields the long exact sequence for the extension Cone.p/ Cyl.p/ Q.There is a canonical morphism of extensionsK EQCone.p/ Cyl.p/ Q,where the vertical map E ! Cyl.p/ is a homotopy equivalence. Hence a functor withPuppe exact sequences is exact for K E Q if and only if it maps the verticalmap K ! Cone.p/ to an isomorphism.4 Kasparov theoryWe define KK G as the universal split-exact C -stable functor on G-C sep; since splitexactand C -stable functors are automatically homotopy invariant, KK G is the universalsplit-exact C -stable homotopy functor as well. The universal property of Kasparovtheory due to Cuntz and Higson asserts that this is equivalent to Kasparov’s definition.We examine some basic properties of Kasparov theory and, in particular, show how toget functors between Kasparov categories.We let E G be the universal exact C -stable homotopy functor on G-C sep or,equivalently, the universal exact, split-exact, and C -stable functor.Kasparov’s own definition of his theory is inspired by previous work of Atiyah [1]on K-homology; later, he also interacted with the work of Brown–Douglas–Fillmore[6] on extensions of C -algebras. A construction in abstract homotopy theory providesa homology theory for spaces that is dual to K-theory. Atiyah realized that certainabstract elliptic differential operators provide cycles for this dual theory; but he did

22 R. Meyernot know the equivalence relation to put on these cycles. Brown–Douglas–Fillmorestudied extensions of C 0 .X/ (and more general C -algebras) by the compact operatorsand found that the resulting structure set is naturally isomorphic to a K-homology group.Kasparov unified and vastly generalised these two results, defining a bivariant functorKK .A; B/ that combines K-theory and K-homology and that is closely related tothe classification of extensions B ˝ K E A (see [29], [30]). A deep theorem ofKasparov shows that two reasonable equivalence relations for these cycles coincide; thisclarifies the homotopy invariance of the extension groups of Brown–Douglas–Fillmore.Furthermore, he constructed an equivariant version of his theory in [31] and applied itto prove the Novikov conjecture for discrete subgroups of Lie groups.The most remarkable feature of Kasparov theory is an associative product on KKcalled Kasparov product. This generalises various known product constructions inK-theory and K-homology and allows to view KK as a category.In applications, we usually need some non-obvious KK-element, and we mustcompute certain Kasparov products explicitly. This requires a concrete description ofKasparov cycles and their products. Since both are somewhat technical, we do notdiscuss them here and merely refer to [5] for a detailed treatment and to [56] for a veryuseful survey article. Instead, we use Higson’s characterisation of KK by a universalproperty [23], which is based on ideas of Cuntz ([10], [9]). The extension to theequivariant case is due to Thomsen [58]. A simpler proof of Thomsen’s theorem andvarious related results can be found in [36].We do not discuss KK G for Z=2-graded G-C -algebras here because it does not fit sowell with the universal property approach, which would simply yield KK GZ=2 becauseZ=2-graded G-C -algebras are the same as G Z=2-C -algebras. The relationshipbetween the two theories is explained in [36], following Ulrich Haag [22]. The gradedversion of Kasparov theory is often useful because it allows us to treat even and oddKK-cycles simultaneously.Fix a locally compact group G. The Kasparov groups KK G 0 .A; B/ for A; B 2G-C sep form the morphism sets A ! B of a category, which we denote by KK G ;the composition in KK G is the Kasparov product. The categories G-C sep and KK Ghave the same objects. We have a canonical functorKK G W G-C sep ! KK Gthat acts identically on objects. This functor contains all information about equivariantKasparov theory for G.Definition 49. A G-equivariant -homomorphism f W A ! B is called a KK G -equivalence if KK G .f / is invertible in KK G .Theorem 50. Let S be a full subcategory of G-C sep that is closed under suspensions,G-equivariantly cp-split extensions, and Morita–Rieffel equivalence. Let KK G .S/ bethe full subcategory of KK G with object class S and let KK G S W S ! KKG .S/ be therestriction of KK G .

Categorical aspects of bivariant K-theory 23The functor KK G S W S ! KKG S is the universal split-exact C -stable functor; inparticular, KK G .S/ is an additive category. In addition, it has the following propertiesand is, therefore, universal among functors on S with some of these extra properties:• it is homotopy invariant;• it is exact for G-equivariantly cp-split extensions;• it satisfies Bott periodicity, that is, in KK G there are natural isomorphismsSus 2 .A/ Š A for all A 22 KK G .Corollary 51. Let F W S ! C be split-exact and C -stable. Then F factors uniquelythrough KK G S , is homotopy invariant, and satisfies Bott periodicity. A KKG -equivalenceA ! B in S induces an isomorphism F .A/ ! F.B/.We will view the universal property of Theorem 50 as a definition of KK G and thusof the groups KK G 0 .A; B/. We also letKK G n .A; B/ WD KKG A; Sus n .B/ Isince the Bott periodicity isomorphism identifies KK G 2 Š KKG 0 , this yields a Z=2-gradedtheory.Now we describe KK G 0 .A; B/ more concretely. Recall A K WD A ˝ K.L 2 G/.Proposition 52. Let A and B be two G-C -algebras. There is a natural bijectionbetween the morphism sets KK G 0 .A; B/ in KKG and the set Œq.A K /; B K ˝ K.`2N/ ofhomotopy classes of G-equivariant -homomorphisms from q.A K / to B K ˝ K.`2N/.Proof. The canonical functor G-C sep ! KK G is C -stable and split-exact, andtherefore homotopy invariant by Theorem 46 (this is already asserted in Theorem 50).Proposition 44 yields that it is additive for coproducts. Split-exactness for the splitextension q.A/ Q.A/ A shows that id A 0W Q.A/ ! A restricts to a KK G -equivalence q.A/ A. Similarly, C -stability yields KK G -equivalences A A K andB B K ˝ K.`2N/. Hence homotopy classes of -homomorphisms from q.A K / toB K˝K.`2N/ yield classes in KK G 0 .A; B/. Using the concrete description of Kasparovcycles, which we have not discussed, it is checked in [36] that this map yields a bijectionas asserted.Another equivalent description isKK G 0 .A; B/ Š Œq.A K/ ˝ K.`2N/;q.B K / ˝ K.`2N/Iin this approach, the Kasparov product becomes simply the composition of morphisms.Proposition 52 suggests that q.A K / and B K ˝K.`2N/ may be the cofibrant and fibrantreplacement of A and B in some model category related to KK G . But it is not clearwhether this is the case. The model category structure constructed in [28] is certainlyquite different.

24 R. MeyerBy the universal property, K-theory descends to a functor on KK, that is, we getcanonical mapsKK 0 .A; B/ ! Hom K .A/; K .B/ for all separable C -algebras A; B, where the right hand side denotes grading-preservinggroup homomorphisms. For A D C, this yields a map KK 0 .C;B/ !Hom Z; K 0 .B/ Š K 0 .B/. Using suspensions, we also get a corresponding mapKK 1 .C;B/! K 1 .B/.Theorem 53. The maps KK .C;B/ ! K .B/ constructed above are isomorphismsfor all B 22 C sep.Thus Kasparov theory is a bivariant generalisation of K-theory. Roughly speaking,KK .A; B/ is the place where maps between K-theory groups live. Most constructionsof such maps, say, in index theory can, in fact, be improved to yield elements ofKK .A; B/. One reason why this has to be so is the Universal Coefficient Theorem(UCT), which computes KK .A; B/ from K .A/ and K .B/ for many C -algebrasA; B. If A satisfies the UCT, then any grading preserving group homomorphismK .A/ ! K .B/ lifts to an element of KK 0 .A; B/.4.1 Extending functors and identities to KK G . We can use the universal propertyto extend various functors G-C sep ! H -C sep to functors KK G ! KK H . Weexplain this by an example:Proposition 54. The full and reduced crossed product functorsG Ë r;GË W G-C alg ! C algextend to functors G Ë r ;GË W KK G ! KK called descent functors.Gennadi Kasparov [31] constructs these functors directly using the concrete descriptionof Kasparov cycles. This requires a certain amount of work; in particular, checkingfunctoriality involves knowing how to compute Kasparov products. The constructionvia the universal property is formal:Proof. We only write down the argument for reduced crossed products, the other caseis similar. It is well-known that G Ë r A ˝ K.H/ Š .G Ë r A/ ˝ K.H/ for anyG-Hilbert space H. Therefore, the composite functorG-C sep GË r! C sep KK ! KKis C -stable. Proposition 9 shows that this functor is split-exact as well (regardless ofwhether G is an exact group). Now the universal property provides an extension to afunctor KK G ! KK.Similarly, we get functorsA ˝min ;A˝max W KK G ! KK G

Categorical aspects of bivariant K-theory 25for any G-C -algebra A. Since these extensions are natural, we even get bifunctors˝min ; ˝max W KK G KK G ! KK G :The associativity, commutativity, and unit constraints in G-C alg induce correspondingconstraints in KK G , so that both ˝min and ˝max turn KK G into a symmetricmonoidal category.Another example is the functor W C alg ! G-C alg that equips a C -algebrawith the trivial G-action; it extends to a functor W KK ! KK G .The universal property also allows us to prove identities between functors. Forinstance, we have natural isomorphisms G Ë r ..A/ ˝min B/ D A ˝min .G Ë r B/ forall G-C -algebras B. Naturality means, to begin with, that the diagramG Ë r ..A 1 / ˝min B 1 /ŠA 1 ˝min .G Ë r B 1 /GË r .˛/˝minˇG Ë r ..A 2 / ˝min B 2 /˛˝min .GË rˇ/Š A 2 ˝min .G Ë r B 2 /commutes if ˛ W A 1 ! A 2 and ˇ W B 1 ! B 2 are a -homomorphism and a G-equivariant -homomorphism, respectively. Two applications of the uniqueness part ofthe universal property show that this diagram remains commutative in KK if ˛ 2KK 0 .A 1 ;A 2 / and ˇ 2 KK G 0 .B 1;B 2 /. Similar remarks apply to the natural isomorphismG Ë ..A/ ˝max B/ Š A ˝max .G Ë B/ and hence to the isomorphismsG Ë .A/ Š C .G/ ˝max A and G Ë r .A/ Š C red .G/ ˝min A.Adjointness relations in Kasparov theory are usually proved most easily by constructingthe unit and counit of the adjunction. For instance, if G is a compact groupthen the functor is left adjoint to G Ë D G Ë r , that is, for all A 22 KK andB 22 KK G , we have natural isomorphismsKK G ..A/; B/ Š KK .A; G Ë B/: (8)This is also known as the Green–Julg Theorem. ForA D C, it specialises to a naturalisomorphism K G .B/ Š K .G Ë B/; this was one of the first appearances of noncommutativealgebras in topological K-theory.Proof of (8). We already know that and G Ë are functors between KK and KK G .It remains to construct natural elements˛A 2 KK 0 A; G Ë .A/ ;ˇB 2 KK G 0 ..G Ë B/;B/for all A 22 KK, B 22 KK G that satisfy the conditions for unit and counit of adjunction[35].The main point is that .G Ë B/ is the G-fixed point subalgebra of B K D B ˝K.L 2 G/. The embedding .G Ë B/ ! B K provides a G-equivariant correspondenceˇB from .G Ë B/ to B and thus an element of KK G 0 ..G Ë B/;B/. This

26 R. Meyerconstruction is certainly natural for G-equivariant -homomorphisms and hence forKK G -morphisms by the uniqueness part of the universal property of KK G .Let e W C ! C .G/ be the embedding that corresponds to the trivial representationof G. Recall that G Ë .A/ Š C .G/ ˝ A. Hence the exterior product of the identitymap on A and KK.e / provides ˛A 2 KK 0 A; G Ë .A/ . Again, naturality for -homomorphisms is clear and implies naturality for morphisms in KK.Finally, it remains to check that.A/ .˛A/! G Ë .A/ ˇ.A/! .A/;G Ë B ˛GËB! G Ë .G Ë B/ GˡB! G Ë Bare the identity morphisms in KK G . Then we get the desired adjointness using a generalconstruction from category theory (see [35]). In fact, both composites are equal to theidentity already as correspondences, so that we do not have to know anything aboutKasparov theory except its C -stability to check this.A similar argument yields an adjointness relationKK G 0 A; .B/ Š KK 0 .G Ë A; B/ (9)for a discrete group G. More conceptually, (9) corresponds via Baaj–Skandalis duality[3] to the Green–Julg Theorem for the dual quantum group of G, which is compactbecause G is discrete. But we can also write down unit and counit of adjunction directly.The trivial representation C .G/ ! C yields natural -homomorphismsG Ë .B/ Š C .G/ ˝max B ! Band hence ˇB 2 KK 0 .G Ë .B/;B/. The canonical embedding A ! G Ë A isG-equivariant if we let G act on G Ë A by conjugation; but this action is inner, sothat G Ë A and .G Ë A/ are G-equivariantly Morita–Rieffel equivalent. Thus thecanonical embedding A ! G Ë A yields a correspondence A Ü .G Ë A/ and˛A 2 KK G 0 A; .G Ë A/ . We must check that the compositesG Ë A G˲A! G Ë .G Ë A/ ˇGËA! G Ë A;.B/ ˛.B/! G Ë .B/ .ˇB /! .B/are identity morphisms in KK and KK G , respectively. Once again, this holds alreadyon the level of correspondences.4.2 Triangulated category structure. We turn KK G into a triangulated category byextending standard constructions for topological spaces [38]. Some arrows changedirection because the functor C 0 from spaces to C -algebras is contravariant. We havealready observed that KK G is additive. The suspension is given by † 1 .A/ WD Sus.A/.Since Sus 2 .A/ Š A in KK G by Bott periodicity, we have † D † 1 . Thus we do notneed formal desuspensions as for the stable homotopy category.

Categorical aspects of bivariant K-theory 27Definition 55. A triangle A ! B ! C ! †A in KK G is called exact if it isisomorphic as a triangle to the mapping cone triangleSus.B/ ! Cone.f / ! A f ! Bfor some G-equivariant -homomorphism f .Alternatively, we can use G-equivariantly cp-split extensions in G-C sep. Anysuch extension I E Q determines a class in KK G 1 .Q; I / Š KKG 0 .Sus.Q/; I /,so that we get a triangle Sus.Q/ ! I ! E ! Q in KK G . Such triangles are calledextension triangles. A triangle in KK G is exact if and only if it is isomorphic to theextension triangle of a G-equivariantly cp-split extension [38].Theorem 56. With the suspension automorphism and exact triangles defined above,KK G is a triangulated category. So is KK G .S/ if S G-C sep is closed undersuspensions, G-equivariantly cp-split extensions, and Morita–Rieffel equivalence as inTheorem 50.Proof. That KK G is triangulated is proved in detail in [38]. We do not discuss thetriangulated category axioms here. Most of them amount to properties of mappingcone triangles that can be checked by copying the corresponding arguments for thestable homotopy category (and reverting arrows). These axioms hold for KK G .S/because they hold for KK G . The only axiom that requires more care is the existenceaxiom for exact triangles; it requires any morphism to be part of an exact triangle. Wecan prove this as in [38] using the concrete description of KK G 0 .A; B/ in Proposition 52.For some applications like the generalisation to KK G .S/, it is better to use extensiontriangles instead. Any f 2 KK G 0 .A; B/ Š KKG 1 .Sus.A/; B/ can be represented bya G-equivariantly cp-split extension K.H B / E Sus.A/, where H B is a fullG-equivariant Hilbert B-module, so that K.H B / is G-equivariantly Morita–Rieffelequivalent to B. The extension triangle of this extension contains f and belongs toKK G .S/ by our assumptions on S.Since model category structures related to C -algebras are rather hard to get (compare[28]), triangulated categories seem to provide the most promising formal setupfor extending results from classical spaces to C -algebras. An earlier attempt can befound in [54]. Triangulated categories clarify the basic bookkeeping with long exactsequences. Mayer–Vietoris exact sequences and inductive limits are discussed fromthis point of view in [38]. More importantly, this framework sheds light on more advancedconstructions like the Baum–Connes assembly map. We will briefly discussthis below.4.3 The Universal Coefficient Theorem. There is a very close relationship betweenK-theory and Kasparov theory. We have already seen that K .A/ Š KK .C;A/ isa special case of KK. Furthermore, KK inherits deep properties of K-theory such asBott periodicity. Thus we may hope to express KK .A; B/ using only the K-theory of

28 R. MeyerA and B – at least for many A and B. This is the point of the Universal CoefficientTheorem.The Kasparov product provides a canonical homomorphism of graded groups W KK .A; B/ ! Hom K .A/; K .B/ ;where Hom denotes the Z=2-graded Abelian group of all group homomorphismsK .A/ ! K .B/. There are topological reasons why cannot always be invertible:since Hom is not exact, the bifunctor Hom K .A/; K .B/ would not be exact oncp-split extensions. A construction of Lawrence Brown provides another natural map W ker ! Ext K C1 .A/; K .B/ :The following theorem is due to Jonathan Rosenberg and Claude Schochet [53], [51];see also [5].Theorem 57. The following are equivalent for a separable C -algebra A:(a) KK .A; B/ D 0 for all B 22 KK with K .B/ D 0;(b) the map is surjective and is bijective for all B 22 KK;(c) for all B 22 KK, there is a short exact sequence of Z=2-graded Abelian groupsExt K C1 .A/; K .B/ KK .A; B/ Hom K .A/; K .B/ :(d) A belongs to the smallest class of C -algebras that contains C and is closed underKK-equivalence, suspensions, countable direct sums, and cp-split extensions;(e) A is KK-equivalent to C 0 .X/ for some pointed compact metrisable space X.If these conditions are satisfied, then the extension in (c) is natural and splits, but thesection is not natural.The class of C -algebras with these properties is also called the bootstrap classbecause of description (d). Alternatively, we may say that they satisfy the UniversalCoefficient Theorem because of (c). Since commutative C -algebras are nuclear, (e)implies that the natural map A ˝max B ! A ˝min B is a KK-equivalence if A or Bbelongs to the bootstrap class [55]. This fails for some A, so that the Universal CoefficientTheorem does not hold for all A. Remarkably, this is the only obstruction to theUniversal Coefficient Theorem known at the moment: we know no nuclear C -algebrathat does not satisfy the Universal Coefficient Theorem. As a result, we can expressKK .A; B/ using only K .A/ and K .B/ for many A and B.When we restrict attention to nuclear C -algebras, then the bootstrap class is closedunder various operations like tensor products, arbitrary extensions and inductive limits(without requiring any cp-sections), and under crossed products by torsion-freeamenable groups. Remarkably, there are no general results about crossed products byfinite groups.

Categorical aspects of bivariant K-theory 29The Universal Coefficient Theorem and the universal property of KK imply thatvery few homology theories for (pointed compact metrisable) spaces can extend to thenon-commutative setting. More precisely, if we require the extension to be split-exact,C -stable, and additive for countable direct sums, then only K-theory with coefficientsis possible. Thus we rule out most of the difficult (and interesting) problems in stablehomotopy theory. But if we only want to study K-theory, anyway, then the operatoralgebraic framework usually provides very good analytical tools. This is most valuablefor equivariant generalisations of K-theory.Jonathan Rosenberg and Claude Schochet [50] have also constructed a spectralsequence that, in favourable cases, computes KK G .A; B/ from K G .A/ and KG .B/;they require G to be a compact Lie group with torsion-free fundamental group andA and B to belong to a suitable bootstrap class. This equivariant UCT is clarifiedin [39], [40].4.4 E-theory and asymptotic morphisms. Recall that Kasparov theory is only exactfor (equivariantly) cp-split extensions. E-theory is a similar theory that is exact for allextensions.Definition 58. We let E G W G-C sep ! E G be the universal C -stable, exact homotopyfunctor.Lemma 59. The functor E G is split-exact and factors through KK G W G-C sep !KK G . Hence it satisfies Bott periodicity.Proof. Proposition 48 shows that any exact homotopy functor is split-exact. The remainingassertions now follow from Corollary 51.The functor E (for trivial G) was first defined as above by Nigel Higson [25]. ThenAlain Connes and Nigel Higson [8] found a more concrete description using asymptoticmorphisms. This is what made the theory usable. The equivariant generalisation of thetheory is due to Erik Guentner, Nigel Higson, and Jody Trout [21].We write E G n .A; B/ for the space of morphisms A ! Susn .B/ in E G . Bott periodicityshows that there are only two different groups to consider.Definition 60. The asymptotic algebra ofaC -algebra B is the C -algebraAsymp.B/ WD C b .R C ;B/=C 0 .R C ;B/:An asymptotic morphism A ! B is a -homomorphism f W A ! Asymp.B/.Representing elements of Asymp.B/ by bounded functions Œ0; 1/ ! B, we canrepresent f by a family of maps f t W A ! B such that f t .a/ 2 C b .R C ;B/for each a 2A and the map a 7! f t .a/ satisfies the conditions for a -homomorphism asymptoticallyfor t !1. This provides a concrete description of asymptotic morphisms and explainsthe name.If a locally compact group G acts on B, then Asymp.B/ inherits an action of G bynaturality (which need not be strongly continuous).

30 R. MeyerDefinition 61. Let A and B be two G-C -algebras for a locally compact group G. AG-equivariant asymptotic morphism from A to B is a G-equivariant -homomorphismf W A ! Asymp.B/. We write A; B for the set of homotopy classes of G-equivariantasymptotic morphisms from A to B. Here a homotopy is a G-equivariant -homomorphism A ! Asymp C.Œ0; 1; B/ .The asymptotic algebra fits, by definition, into an extensionC 0 .R C ;B/ C b .R C ;B/ Asymp.B/:Notice that C 0 .R C ;B/ Š Cone.B/ is contractible. If f W A ! Asymp.B/ is aG-equivariant asymptotic morphism, then we can use it to pull back this extensionto an extension Cone.B/ E A in G-C alg; the G-action on E is automaticallystrongly continuous. If F is exact and homotopy invariant, then F Sus n .E/ !F Sus n .A/ is an isomorphism for all n 1 by Proposition 48. The evaluationmap C b .R C ;B/ ! B at some t 2 R C pulls back to a morphism E ! B, andthese morphisms for different t are all homotopic. Hence we get a well-defined mapF Sus n .A/ Š F Sus n .E/ ! F Sus n .B/ for each asymptotic morphism A ! B.This explains how asymptotic morphisms are related to exact homotopy functors. Thisobservation leads to the following theorem:Theorem 62. There are natural bijectionsE0 G .A; B/ Š Sus A K ˝ K.`2N/ ; Sus B K ˝ K.`2N/ for all separable G-C -algebras A; B.An important step in the proof of Theorem 62 is the Connes–Higson construction,which to an extension I E Q in C sep associates an asymptotic morphismSus.Q/ ! I .AG-equivariant generalisation of this construction is discussed in [59].Thus any extension in G-C sep gives rise to an exact triangle Sus.Q/ ! I ! E ! Qin E G .This also leads to the triangulated category structure of E G . As for KK G , we candefine it using mapping cone triangles or extension triangles – both approaches yieldthe same class of exact triangles. The canonical functor KK G ! E G is exact becauseit evidently preserves mapping cone triangles.Now that we have two bivariant homology theories with apparently very similarformal properties, we must ask which one we should use. It may seem that the betterexactness properties of E-theory raise it above KK-theory. But actually, these strongexactness properties have a drawback: for a general group G, the reduced crossedproduct functor need not be exact, so that there is no guarantee that it extends to afunctor E G ! E. Only full crossed products exist for all groups by Proposition 6; theconstruction of G Ë W E G ! E is the same as in KK-theory. Similar problems occurwith ˝min but not with ˝max .Furthermore, since KK G has a weaker universal property, it acts on more functors,so that results about KK G have stronger consequences. A good example of a functor

Categorical aspects of bivariant K-theory 31that is split-exact but probably not exact is local cyclic cohomology (see [37], [45]).Therefore, the best practice seems to prove results in KK G if possible.Many applications can be done with either E or KK, we hardly notice any difference.An explanation for this is the work of Houghton-Larsen and Thomsen ([27], [59]),which describes KK G 0 .A; B/ in the framework of asymptotic morphisms. Recall thatasymptotic morphisms A ! B generate extensions Cone.B/ E A. If thisextension is G-equivariantly cp-split, then the projection map E A isaKK G -equivalence. A G-equivariant completely positive contractive section for the extensionexists if and only if we can represent our asymptotic morphism by a continuous familyof G-equivariant, completely positive contractions f t W A ! B, t 2 Œ0; 1/.Definition 63. Let A; B cp be the set of homotopy classes of asymptotic morphismsfrom A to B that can be lifted to a G-equivariant, completely positive, contractive mapA ! C b .R C ;B/; of course, we only use homotopies with the same kind of lifting.Theorem 64 ([59]). There are natural bijectionsKK G 0 .A; B/ Š Sus A K ˝ K.`2N/ ; Sus B K ˝ K.`2N/ cpfor all separable G-C -algebras A; B; the canonical functor KK G ! E G correspondsto the obvious map that forgets the additional constraints.Corollary 65. If A is a nuclear C -algebra, then KK .A; B/ Š E .A; B/.Proof. The Choi–Effros Lifting Theorem asserts that any extension of A has a completelypositive contractive section.In the equivariant case, the same argument yields KK G .A; B/ Š EG .A; B/ if Ais nuclear and G acts properly on A (see also [55]). It should be possible to weakenproperness to amenability here, but I am not aware of a reference for this.5 The Baum–Connes assembly map for spaces and operatoralgebrasThe Baum–Connes conjecture is a guess for the K-theory K .C redG/ of reduced groupC -algebras [4]. We shall compare the approach of Davis and Lück [14] using homotopytheory for G-spaces and its counterpart in bivariant K-theory formulated in [38]. Toavoid technical difficulties, we assume that the group G is discrete.The first step in the Davis–Lück approach is to embed the groups of interest suchas K .C redG/ in a G-homology theory, that is, a homology theory on the categoryof (spectra of) G-CW-complexes. For the Baum–Connes assembly map we need ahomology theory for G-CW-complexes with F .G=H / Š K .C redH/. This amountsto finding a G-equivariant spectrum with appropriate homotopy groups [14] and is themost difficult part of the construction. Other interesting invariants like the algebraicK- and L-theory of group rings can be treated using other spectra instead.

32 R. MeyerIn the world of C -algebras, we cannot treat algebraic K- and L-theory; but wehave much better tools to study K .C redG/. We do not need a G-homology theory buta homological functor on the triangulated category KK G . More precisely, we need ahomological functor that takes the value K .C red H/on C 0.G=H / for all subgroups H .The functor A 7! K .G Ë r A/ works fine here because G Ë r C 0 .G=H; A/ is Morita–Rieffel equivalent to H Ë r A for any H -C -algebra A. The corresponding assertionfor full crossed products is known as Green’s Imprimitivity Theorem; reduced crossedproducts can be handled similarly. Thus a topological approach to the Baum–Connesconjecture forces us to consider K .G Ë r A/ for all G-C -algebras A, which leads tothe Baum–Connes conjecture with coefficients.5.1 Assembly maps via homotopy theory. Recall that a homology theory on pointedCW-complexes is determined by its value on S 0 . Similarly, a G-homology theory F isdetermined by its values F .G=H / on homogeneous spaces for all subgroups H G.This does not help much because these groups – which are K .C redH/ in the case ofinterest – are very hard to compute.The idea behind assembly maps is to approximate a given homology theory bya simpler one that only depends on F .G=H / for H 2 F for some family of subgroupsF . The Baum–Connes assembly map uses the family of finite subgroups here;other families like virtually cyclic subgroups appear in isomorphism conjectures forother homology theories. We now fix a family of subgroups F , which we assume tobe closed under conjugation and subgroups.A G; F -CW-complex is a G-CW-complex in which the stabilisers of cells belongto F . The universal G; F -CW-complex is a G; F -CW-complex E.G; F / with theproperty that, for any G; F -CW-complex X there is a G-map X ! E.G; F /, whichis unique up to G-homotopy. This universal property determines E.G; F / uniquely upto G-homotopy. It is easy to see that E.G; F / is H -equivariantly contractible for anyH 2 F . Conversely, a G; F -CW-complex with this property is universal.Example 66. Let G D Z and let F be the family consisting only of the trivial subgroup;this agrees with the family of finite subgroups because G is torsion-free. A G; F -CWcomplexis essentially the same as a CW-complex with a free cellular action of Z. Itiseasy to check that R with the action of Z by translation and the usual cell decompositionis a universal G; F -CW-complex.Given any G-CW-complex X, the canonical map E.G; F / X ! X has thefollowing properties:• E.G; F / X is a G; F -CW-complex;•ifY is a G; F -CW-complex, then any G-map Y ! X lifts uniquely up toG-homotopy to a map Y ! E.G; F / X;• for any H 2 F , the map E.G; F / X ! X becomes a homotopy equivalencein the category of H -spaces.

Categorical aspects of bivariant K-theory 33The first two properties make precise in what sense E.G; F / X is the best approximationto X among G; F -CW-complexes.Definition 67. The assembly map with respect to F is the map F E.G; F / ! F .?/induced by the constant map E.G; F / D E.G; F / ? ! ?.More generally, the assembly map with coefficients in a pointed G-CW-complex(or spectrum) X is the map F .E.G; F / C ^ X/ ! F .S 0 ^ X/ D F .X/ induced bythe map E.G; F / C ! ? C D S 0 .In the stable homotopy category of pointed G-CW-complexes (or spectra), we getan exact triangle E.G; F / C ^ X ! X ! N ! S 1 ^ E.G; F / C ^ X, where Nis H -equivariantly contractible for each H 2 F . This means that the domain of theassembly map F .E.G; F / C ^ X/ is the localisation of F at the class of all objectsthat are H -equivariantly contractible for each H 2 F .Thus the assembly map is an isomorphism for all X if and only if F .N / D 0whenever N is H -equivariantly contractible for each H 2 F . Thus an isomorphismconjecture can be interpreted in two equivalent ways. First, it says that we can reconstructthe homology theory from its restriction to G; F -CW-complexes. Secondly, itsays that the homology theory vanishes for spaces that are H -equivariantly contractiblefor H 2 F .5.2 From spaces to operator algebras. We can carry over the construction of assemblymaps above to bivariant Kasparov theory; we continue to assume G discrete tosimplify some statements. From now on, we let F be the family of finite subgroups.This is the family that appears in the Baum–Connes assembly map. Other families ofsubgroups can also be treated, but some proofs have to be modified and are not yetwritten down.First we need an analogue of G; F -CW-complexes. These are constructible out ofsimpler “cells” which we describe first, using the induction functorsInd G H W KKH ! KK Gfor subgroups H G. For a finite group H , IndH G .A/ is the H -fixed point algebra ofC 0 .G; A/, where H acts by h f.g/D ˛h f.gh/ . For infinite H ,wehaveIndH G .A/ Dff 2 C b.G; A/ j ˛hf.gh/D f.g/for all g 2 G, h 2 H ,and gH 7! kf.g/k is in C 0 .G=H /gIthe group G acts by translations on the left.This construction is functorial for equivariant -homomorphisms. Since it commuteswith C -stabilisations and maps split extensions again to split extensions, itdescends to a functor KK H ! KK G by the universal property (compare §4.1).We also have the more trivial restriction functors Res H G W KKG ! KK H for subgroupsH G. The induction and restriction functors are adjoint:KK G .Ind G H A; B/ Š KKH .A; Res H G B/

34 R. Meyerfor all A 22 KK G ; this can be proved like the similar adjointness statements in §4.1,using the embedding A ! Res H G IndG H.A/ as functions supported on H G andthe correspondence IndH G ResH G .A/ Š C 0.G=H; A/ ! K.`2G=H / ˝ A A. It isimportant here that H G is an open subgroup. By the way, if H G is a cocompactsubgroup (which means finite index in the discrete case), then Res H Gis the left-adjointof IndH G instead.Definition 68. We let CIbe the subcategory of all objects of KK G of the form Ind G H .A/for A 22 KK H and H 2 F . Let hCIi be the smallest class in KK G that contains CIand is closed under KK G -equivalence, countable direct sums, suspensions, and exacttriangles.Equivalently, hCIi is the localising subcategory generated by CI. This is oursubstitute for the category of .G; F /-CW-complexes.Definition 69. Let CC be the class of all objects of KK G with Res H G.A/ Š 0 for allH 2 F .Theorem 70. If P 2hCIi, N 2 CC, then KK G .P; N / D 0. Furthermore, for anyA 22 KK G there is an exact triangle P ! A ! N ! †P with P 2hCIi, N 2 CC.Definitions 68–69 and Theorem 70 are taken from [38]. The map F .P / ! F .A/for a functor F W KK G ! C is analogous to the assembly map in Definition 67 anddeserves to be called the Baum–Connes assembly map for F .We can use the tensor product in KK G to simplify the proof of Theorem 70: oncewe have a triangle P C ! C ! N C ! †P C with P C 2hCIi, N C 2 CC, thenA ˝ P C ! A ˝ C ! A ˝ N C ! †A ˝ P Cis an exact triangle with similar properties for A. It makes no difference whether weuse ˝min or ˝max here. The map P C ! C in KK G .P C ; C/ is analogous to the mapE.G; F / ! ?. It is also called a Dirac morphism for G because the K-homologyclasses of Dirac operators on smooth spin manifolds provided the first important examples[31].The two assembly map constructions with spaces and C -algebras are not justanalogous but provide the same Baum–Connes assembly map. To see this, we mustunderstand the passage from the homotopy category of spaces to KK. Usually, wemap spaces to operator algebras using the commutative C -algebra C 0 .X/. But thisconstruction is only functorial for proper continuous maps, and the functoriality iscontravariant. The assembly map for, say, G D Z is related to the non-proper mapp W R ! ?, which does not induce a map C ! C 0 .R/; even if it did, this map wouldstill go in the wrong direction. The wrong-way functoriality in KK provides an elementp Š 2 KK 1 .C 0 .R/; C/ instead, which is the desired Dirac morphism up to a shift in thegrading. This construction only applies to manifolds with a Spin c -structure, but it canbe generalised as follows.On the level of Kasparov theory, we can define another functor from suitable spacesto KK that is a covariant functor for all continuous maps. The definition uses a notion

Categorical aspects of bivariant K-theory 35of duality due to Kasparov [31] that is studied more systematically in [17]. It requiresyet another version RKK G .XI A; B/ of Kasparov theory that is defined for a locallycompact space X and two G-C -algebras A and B. Roughly speaking, the cycles forthis theory are G-equivariant families of cycles for KK .A; B/ parametrised by X. Thegroups RKK G .XI A; B/ are contravariantly functorial and homotopy invariant in X (forG-equivariant continuous maps).We have RKK G .?I A; B/ D KKG .A; B/ and, more generally, RKKG .XI A; B/ ŠKK G A; C.X; B/ if X is compact. The same statement holds for non-compact X,but the algebra C.X; B/ is not a C -algebra any more: it is an inverse system ofC -algebras.Definition 71 ([17]). A G-C -algebra P X is called an abstract dual for X if, forall second countable locally compact G-spaces Y and all separable G-C -algebras Aand B, there are natural isomorphismsRKK G .X Y I A; B/ Š RKK G .Y I P X ˝ A; B/that are compatible with tensor products.Abstract duals exist for many spaces. For trivial reasons, C is an abstract dualfor the one-point space. For a smooth manifold X with an isometric action of G, bothC 0 .T X/and the algebra of C 0 -sections of the Clifford algebra bundle on X are abstractduals for X; ifX has a G-equivariant Spin c -structure – as in the example of Z actingon R – we may also use a suspension of C 0 .X/. For a finite-dimensional simplicialcomplex with a simplicial action of G, an abstract dual is constructed by GennadiKasparov and Georges Skandalis in [32] and in more detail in [17]. It seems likelythat the construction can be carried over to infinite-dimensional simplicial complexesas well, but this has not yet been written down.There are also spaces with no abstract dual. A prominent example is the Cantor set:it has no abstract dual, even for trivial G (see [17]).Let D be the class of all G-spaces that admit a dual. Recall that X 7! RKK G .X Y I A; B/ is a contravariant homotopy functor for continuous G-maps. Passing to corepresentingobjects, we get a covariant homotopy functorD ! KK G ; X 7! P X :This functor is very useful to translate constructions from homotopy theory to bivariantK-theory. An instance of this is the comparison of the Baum–Connes assembly mapsin both setups:Theorem 72. Let F be the family of finite subgroups of a discrete group G, and letE.G; F / be the universal .G; F /-CW-complex. Then E.G; F / has an abstract dual P ,and the map E.G; F / ! ? induces a Dirac morphism in KK G 0 .P; C/.Theorem 72 should hold for all families of subgroups F , but only the above specialcase is treated in [17], [38].

36 R. Meyer5.3 The Dirac-dual-Dirac method and geometry. Let us compare the approachesin §5.1 and §5.2! The bad thing about the C -algebraic approach is that it applies tofewer theories. The good thing about it is that Kasparov theory is so flexible that anycanonical map between K-theory groups has a fair chance to come from a morphismin KK G which we can construct explicitly.For some groups, the Dirac morphism in KK G .P; C/ is a KK-equivalence:Theorem 73 (Higson–Kasparov [26]). Let the group G be amenable or, more generally,a-T-menable. Then the Dirac morphism for G is a KK G -equivalence, so that G satisfiesthe Baum–Connes conjecture with coefficients.The class of groups for which the Dirac morphism has a one-sided inverse is evenlarger. This is the point of the Dirac-dual-Dirac method. The following definitionin [38] is based on a simplification of this method:Definition 74. A dual Dirac morphism for G is an element 2 KK G .C;P/ with ı D D id P .If such a dual Dirac morphism exists, then it provides a section for the assemblymap F .P ˝ A/ ! F .A/ for any functor F W KK G ! C and any A 22 KK G , so thatthe assembly map is a split monomorphism. Currently, we know no group without adual Dirac morphism. It is shown in [16], [18], [19] that the existence of a dual Diracmorphism is a geometric property of G because it is related to the invertibility of anotherassembly map that only depends on the coarse geometry of G (in the torsion-free case).Instead of going into this construction, we briefly indicate another point of view thatalso shows that the existence of a dual Dirac morphism is a geometric issue. Let P bean abstract dual for some space X (like E.G; F /). The duality isomorphisms in Definition71 are determined by two pieces of data: a Dirac morphism D 2 KK G .P; C/ anda local dual Dirac morphism ‚ 2 RKK G .XI C;P/. The notation is motivated by thespecial case of a Spin c -manifold X with P D C 0 .X/, where D is the K-homology classdefined by the Dirac operator and is defined by a local construction involving pointwiseClifford multiplications. If X D E.G; F /, then it turns out that 2 KK G .C;P/is adual Dirac morphism if and only if the canonical map KK G .C;P/! RKK G .XI C;P/maps 7! ‚. Thus the issue is to globalise the local construction of ‚. This is possibleif we know, say, that X has non-positive curvature. This is essentially how Kasparovproves the Novikov conjecture for fundamental groups of non-positively curved smoothmanifolds in [31].References[1] M. F. Atiyah, Global theory of elliptic operators, in Proc. Internat. Conf. on FunctionalAnalysis and Related Topics (Tokyo, 1969), University of Tokyo Press, Tokyo 1970, 21–30.[2] William Arveson, An invitation to C -algebras, Grad. Texts in Math. 39, Springer-VerlagNew York 1976.[3] Saad Baaj, Georges Skandalis, C -algèbres de Hopf et théorie de Kasparov équivariante,K-Theory 2 (1989), 683–721.

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38 R. Meyer[25] —–, Categories of fractions and excision in KK-theory, J. Pure Appl. Algebra 65 (1990),119–138.[26] Nigel Higson, Gennadi Kasparov, E-theory and KK-theory for groups which act properlyand isometrically on Hilbert space, Invent. Math. 144 (2001), 23–74.[27] T. G. Houghton-Larsen, Klaus Thomsen, Universal (co)homology theories, K-Theory 16(1999), 1–27.[28] Michael Joachim, Mark W. Johnson, Realizing Kasparov’s KK-theory groups as the homotopyclasses of maps of a Quillen model category, in An alpine anthology of homotopytheory, Contemp. Math. 399, Amer. Math. Soc. Providence, RI, 2006, 163–197.[29] G. G. Kasparov, Topological invariants of elliptic operators. I. K-homology, Izv. Akad. NaukSSSR Ser. Mat. 39 (1975), 796–838; English transl. Math. USSR-Izv. 9 (1975), 751–792.[30] —–, The operator K-functor and extensions of C -algebras, Izv. Akad. Nauk SSSR Ser. Mat.44 (1980), 571–636 (in Russian).[31] —–, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147–201.[32] G. G. Kasparov, Georges Skandalis, Groups acting on buildings, operator K-theory, andNovikov’s conjecture, K-Theory 4 (1991), 303–337.[33] Eberhard Kirchberg, Simon Wassermann, Permanence properties of C -exact groups, Doc.Math. 4 (1999), 513–558.[34] E. C. Lance, Hilbert C -modules, A toolkit for operator algebraists, London Math. Soc.Lecture Note Ser. 210, Cambridge University Press, Cambridge 1995.[35] Saunders Mac Lane, Categories for the working mathematician, Grad. Texts in Math. 5,2nd ed., Springer-Verlag, New York 1998.[36] Ralf Meyer, Equivariant Kasparov theory and generalized homomorphisms, K-Theory 21(2000), 201–228.[37] —–, Local and Analytic Cyclic Homology, EMS Tracts Math. 3, Europ. Math. Soc. Publ.House, Zürich 2007.[38] Ralf Meyer, Ryszard Nest, The Baum–Connes conjecture via localisation of categories,Topology 45 (2006), 209–259.[39] —–, Homological algebra in bivariant K-theory and other triangulated categories. I, preprint2007 http://www.arxiv.org/math.KT/0702146.[40] —–, An analogue of the Baum–Connes conjecture for coactions of compact groups, Math.Scand. 100 (2007), 301–316.[41] J. A. Mingo, W. J. Phillips, Equivariant triviality theorems for Hilbert C -modules, Proc.Amer. Math. Soc. 91 (1984), 225–230.[42] Gerard J. Murphy, C -algebras and operator theory, Academic Press Inc. Boston, MA,1990.[43] Narutaka Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci.Paris Sér. I Math. 330 (2000), 691–695.[44] Gert K. Pedersen, C -algebras and their automorphism groups, London Math. Soc.Monogr. 14, Academic Press Inc., London 1979.[45] Michael Puschnigg, Diffeotopy functors of ind-algebras and local cyclic cohomology, Doc.Math. 8 (2003), 143–245.

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Inheritance of isomorphism conjectures under colimitsArthur Bartels, Siegfried Echterhoff, and Wolfgang Lück1 Introduction1.1 Assembly maps. We want to study the following assembly maps:asmb G n W H G n .E VCyc.G/I K R / ! H G n .fgI K R/ D K n .R Ì G/I (1.1)asmb G n W H G n .E F in.G/I KH R / ! H G n .fgI KH R/ D KH n .R Ì G/I (1.2)asmb G n W H n G .E VCyc.G/I L h 1iR/ ! Hn G.fgI Lh1iR/ D L h 1in .R Ì G/I (1.3)asmb G n W H n G .E F in.G/I K top / ! H G A;l 1 n .fgI Ktop / D KA;l 1 n .A Ì l 1 G/I (1.4)asmb G n W H n G .E F in.G/I K topA;r / ! H n G .fgI KtopA;r / D K n.A Ì r G/I (1.5)asmb G n W H n G .E F in.G/I K topA;m / ! H n G .fgI KtopA;m / D K n.A Ì m G/: (1.6)Some explanations are in order. A family of subgroups of G is a collection ofsubgroups of G which is closed under conjugation and taking subgroups. Examplesare the family F in of finite subgroups and the family VCyc of virtually cyclic subgroups.Let E F .G/ be the classifying space associated to F . It is uniquely characterizedup to G-homotopy by the properties that it is a G-CW-complex and that E F .G/ H iscontractible if H 2 F and is empty if H … F . For more information about thesespaces E F .G/ we refer for instance to the survey article [29].Given a group G acting on a ring (with involution) by structure preserving maps, letRÌG be the twisted group ring (with involution) and denote by K n .RÌG/, KH n .RÌG/and L hn 1i .RÌG/its algebraic K-theory in the non-connective sense (see Gersten [17]or Pedersen–Weibel [34]), its homotopy K-theory in the sense of Weibel [41], and itsL-theory with decoration 1 in the sense of Ranicki [37, Chapter 17]. Given a groupG acting on a C -algebra A by automorphisms of C -algebras, let A Ì l 1 G be theBanach algebra obtained from A Ì G by completion with respect to the l 1 -norm, letA Ì r G be the reduced crossed product C -algebra, and let A Ì m G be the maximalcrossed product C -algebra and denote by K n .AÌ l 1 G/, K n .AÌ r G/and K n .AÌ m G/their topological K-theory.The source and target of the assembly maps are given by G-homology theories(see Definition 2.1 and Theorem 6.1) with the property that for every subgroup H G

42 A. Bartels, S. Echterhoff, and W. Lückand n 2 ZH G n .G=H I K R/ Š K n .R Ì H/IH G n .G=H I KH R/ Š KH n .R Ì H/IH G nH G nH G nH G n.G=H I Lh1iR/ Š L h n1i .R Ì H/I.G=H I KtopA;l 1 / Š K n .A Ì l 1 H/I.G=H I KtopA;r / Š K n.A Ì r H/I.G=H I KtopA;r / Š K n.A Ì m H/:All the assembly maps are induced by the projection from E F in .G/ or E VCyc .G/respectively to the one-point-space fg.Remark 1.7. It might be surprising to the reader that we restrict to C*-algebra coefficientsA in the assembly map (1.4). Indeed, our main results rely heavily on thevalidity of the conjecture for hyperbolic groups, which, so far, is only known for C*-algebra coefficients. Moreover we also want to study the passage from the l 1 -setting tothe C -setting. Hence we decided to restrict ourselves to the case of C -coefficientsthroughout. We mention that on the other hand the assembly map (1.4) can also bedefined for Banach algebra coefficients [33].1.2 Conventions. Before we go on, let us fix some conventions. A group G is alwaysdiscrete. Hyperbolic group is to be understood in the sense of Gromov (see forinstance [11], [12], [18], [19]). Ring means associative ring with unit and ring homomorphismspreserve units. Homomorphisms of Banach algebras are assumed to benorm decreasing.1.3 Isomorphism conjectures. The Farrell–Jones conjecture for algebraic K-theoryfor a group G and a ring R with G-action by ring automorphisms says that the assemblymap (1.1) is bijective for all n 2 Z. Its version for homotopy K-theory says that theassembly map (1.2) is bijective for all n 2 Z. If R is a ring with involution and Gacts on R by automorphism of rings with involutions, the L-theoretic version of theFarrell–Jones conjecture predicts that the assembly map (1.3) is bijective for all n 2 Z.The Farrell–Jones conjecture for algebraic K- and L-theory was originally formulatedin the paper by Farrell–Jones [15, 1.6 on page 257] for the trivial G-action on R. Itshomotopy K-theoretic version can be found in [4, Conjecture 7.3], again for trivialG-action on R.Let G be a group acting on the C -algebra A by automorphisms of C -algebras.The Bost conjecture with coefficients and the Baum–Connes conjecture with coefficientsrespectively predict that the assembly map (1.4) and (1.5) respectively are bijectivefor all n 2 Z. The original statement of the Baum–Connes conjecture with trivialcoefficients can be found in [9, Conjecture 3.15 on page 254].

Inheritance of isomorphism conjectures under colimits 43Our formulation of these conjectures follows the homotopy theoretic approach in[13]. The original assembly maps are defined differently. We do not give the proofthat our maps agree with the original ones but at least refer to [13, page 239], wherethe Farrell–Jones conjecture is treated and to Hambleton–Pedersen [21], where suchidentification is given for the Baum–Connes conjecture with trivial coefficients.1.4 Inheritance under colimits. The main purpose of this paper is to prove thatthese conjectures are inherited under colimits over directed systems of groups (with notnecessarily injective structure maps). We want to show:Theorem 1.8 (Inheritance under colimits). Let fG i j i 2 I g be a directed systemof groups with (not necessarily injective) structure maps i;j W G i ! G j . Let G Dcolim i2I G i be its colimit with structure maps i W G i ! G. Let R be a ring (withinvolution) and let A be a C -algebra with structure preserving G-action. Given i 2 Iand a subgroup H G i , we let H act on R and A by restriction with the grouphomomorphism . i /j H W H ! G. Fixn 2 Z. Then:(i) If the assembly mapasmb H n W H H n .E VCyc.H /I K R / ! H H n .fgI K R/ D K n .R Ì H/of (1.1) is bijective for all n 2 Z, all i 2 I and all subgroups H G i , then forevery subgroup K G of G the assembly mapasmb K n W H K n .E VCyc.K/I K R / ! H K n .fgI K R/ D K n .R Ì K/of (1.1) is bijective for all n 2 Z.The corresponding version is true for the assembly maps given in (1.2), (1.3),(1.4), and (1.6).(ii) Suppose that all structure maps i;j are injective and that the assembly mapasmb G inW H G in .E VCyc.G i /I K R / ! H G in .fgI K R/ D K n .R Ì G i /of (1.1) is bijective for all n 2 Z and i 2 I . Then the assembly mapasmb G n W H G n .E VCyc.G/I K R / ! H G n .fgI K R/ D K n .R Ì G/of (1.1) is bijective for all n 2 Z.The corresponding statement is true for the assembly maps given in (1.2), (1.3),(1.4), (1.5), and (1.6).Theorem 1.8 will follow from Theorem 4.5 and Lemma 6.2 as soon as we haveproved Theorem 6.1. Notice that the version (1.5) does not appear in assertion (i). Acounterexample will be discussed below. The (fibered) version of Theorem 1.8 (i) inthe case of algebraic K-theory and L-theory with coefficients in Z with trivial G-actionhas been proved by Farrell–Linnell [16, Theorem 7.1].

44 A. Bartels, S. Echterhoff, and W. Lück1.5 Colimits of hyperbolic groups. In [23, Section 7] Higson, Lafforgue and Skandalisconstruct counterexamples to the Baum–Connes conjecture with coefficients, actuallywith a commutative C -algebra as coefficients. They formulate precise propertiesfor a group G which imply that it does not satisfy the Baum–Connes conjecture withcoefficients. Gromov [20] describes the construction of such a group G as a colimitover a directed system of groups fG i j i 2 I g, where each G i is hyperbolic.This construction did raise the hope that these groups G may also be counterexamplesto the Baum–Connes conjecture with trivial coefficients. But – to the authors’knowledge – this has not been proved and no counterexample to the Baum–Connesconjecture with trivial coefficients is known.Of course one may wonder whether such counterexamples to the Baum–Connesconjecture with coefficients or with trivial coefficients respectively may also be counterexamplesto the Farrell–Jones conjecture or the Bost conjecture with coefficients orwith trivial coefficients respectively. However, this can be excluded by the followingresult.Theorem 1.9. Let G be the colimit of the directed system fG i j i 2 I g of groups (withnot necessarily injective structure maps). Suppose that each G i is hyperbolic. LetK G be a subgroup. Then:(i) The group K satisfies for every ring R on which K acts by ring automorphismsthe Farrell–Jones conjecture for algebraic K-theory with coefficients in R, i.e.,the assembly map (1.1) is bijective for all n 2 Z.(ii) The group K satisfies for every ring R on which K acts by ring automorphismsthe Farrell–Jones conjecture for homotopy K-theory with coefficients in R, i.e.,the assembly map (1.2) is bijective for all n 2 Z.(iii) The group K satisfies for every C -algebra A on which K acts by C -automorphismsthe Bost conjecture with coefficients in A, i.e., the assembly map (1.4)is bijective for all n 2 Z.Proof. If G is the colimit of the directed system fG i j i 2 I g, then the subgroup K Gis the colimit of the directed system fi 1 .K/ j i 2 I g, where i W G i ! G is thestructure map. Hence it suffices to prove Theorem 1.9 in the case G D K. This casefollows from Theorem 1.8 (i) as soon as one can show that the Farrell–Jones conjecturefor algebraic K-theory, the Farrell–Jones conjecture for homotopy K-theory, or theBost conjecture respectively holds for every subgroup H of a hyperbolic group G witharbitrary coefficients R and A respectively.Firstly we prove this for the Bost conjecture. Mineyev and Yu [30, Theorem 17]show that every hyperbolic group G admits a G-invariant metric d O which is weaklygeodesic and strongly bolic. Since every subgroup H of G clearly acts properly on Gwith respect to any discrete metric, it follows that H belongs to the class C 0 as describedby Lafforgue in [27, page 5] (see also the remarks at the top of page 6 in [27]). Nowthe claim is a direct consequence of [27, Theorem 0.0.2].

Inheritance of isomorphism conjectures under colimits 45The claim for the Farrell–Jones conjecture is proved for algebraic K-theory andhomotopy K-theory in Bartels–Lück–Reich [6] which is based on the results of [5].There are further groups with unusual properties that can be obtained as colimits ofhyperbolic groups. This class contains for instance a torsion-free non-cyclic group allwhose proper subgroups are cyclic constructed by Ol’shanskii [32]. Further examplesare mentioned in [31, p.5] and [38, Section 4].We mention that if one can prove the L-theoretic version of the Farrell–Jones conjecturefor subgroups of hyperbolic groups with arbitrary coefficients, then it is alsotrue for subgroups of colimits of hyperbolic groups by the argument above.1.6 Discussion of (potential) counterexamples. If G is an infinite group which satisfiesKazhdan’s property (T), then the assembly map (1.6) for the maximal group C -algebra fails to be an isomorphism if the assembly map (1.5) for the reduced groupC -algebra is injective (which is true for a very large class of groups and in particularfor all hyperbolic groups by [25]). The reason is that a group has property (T) ifand only if the trivial representation 1 G is isolated in the dual yG of G. This impliesthat Cm .G/ has a splitting C ˚ ker.1 G/, where we regard 1 G as a representationof Cm .G/.If G is infinite, then the first summand is in the kernel of the regularrepresentation W Cm .G/ ! C r .G/ (see for instance [14]), hence the K-theory mapW K 0 .Cm .G// ! K 0.Cr .G// is not injective. For a finite group H we have AÌ r H DA Ì m H and hence we can apply [13, Lemma 4.6] to identify the domains of (1.5)and (1.6). Under this identification the composition of the assembly map (1.6) with is the assembly map (1.5) and the claim follows.Hence the Baum–Connes conjecture for the maximal group C -algebras is not truein general since the Baum–Connes conjecture for the reduced group C -algebras hasbeen proved for some groups with property (T) by Lafforgue [26] (see also [39]). So inthe sequel our discussion refers always to the Baum–Connes conjecture for the reducedgroup C -algebra.One may speculate that the Baum–Connes conjecture with trivial coefficients isless likely to be true for a given group G than the Farrell–Jones conjecture or theBost conjecture. Some evidence for this speculation comes from lack of functorialityof the reduced group C -algebra. A group homomorphism ˛ W H ! G induces ingeneral not a C -homomorphism Cr .H / ! C r .G/, one has to require that its kernelis amenable. Here is a counterexample, namely, if F is a non-abelian free group, thenCr .F / is a simple algebra [35] and hence there is no unital algebra homomorphismCr .F / ! C r .f1g/ D C. On the other hand, any group homomorphism ˛ W H ! Ginduces a homomorphismH H n .E F in.H /I K topC;r / ind˛! H G n .˛E F in .H /I K topC;r / H G n .f / ! H G n .E F in.G/I K topC;r /where G acts trivially on C and f W ˛E F in .H / ! E F in .G/ is the up to G-homotopyunique G-map. Notice that the induction map ind˛ exists since the isotropy groupsof E F in .H / are finite. Moreover, this map is compatible under the assembly maps

46 A. Bartels, S. Echterhoff, and W. Lückfor H and G with the map K n .Cr .˛//W K n.Cr .H // ! K n.Cr .G// provided that˛ has amenable kernel and hence Cr .˛/ is defined. So the Baum–Connes conjectureimplies that every group homomorphism ˛ W H ! G induces a group homomorphism˛ W K n .Cr .H // ! K n.Cr .G//, although there may be no C -homomorphismCr .H / ! C r .G/ induced by ˛. No such direct construction of ˛ is known in general.Here is another failure of the reduced group C -algebra. Let G be the colimit of thedirected system fG i j i 2 I g of groups (with not necessarily injective structure maps).Suppose that for every i 2 I and preimage H of a finite group under the canonical mapi W G i ! G the Baum–Connes conjecture for the maximal group C -algebra holds(This is for instance true by [22] if ker. i / has the Haagerup property). Thencolim i2I H G in .E F in.G i /I K topC;m / Š ! colim i2I H G inŠ! HGn .E F in .G/I K topC;m /.E i F in.G i /I K topC;m /is a composition of two isomorphisms. The first map is bijective by the TransitivityPrinciple 4.3, the second by Lemma 3.4 and Lemma 6.2. This implies that the followingcomposition is an isomorphismcolim i2I H G in .E F in.G i /I K topC;r / ! colim i2I H G in! H G n .E F in.G/I K topC;r /.E i F in.G i /I K topC;r /Namely, these two compositions are compatible with the passage from the maximal tothe reduced setting. This passage induces on the source and on the target isomorphismssince E F in .G i / and E F in .G/ have finite isotropy groups, for a finite group H we haveCr .H / D C m .H / and hence we can apply [13, Lemma 4.6]. Now assume furthermorethat the Baum–Connes conjecture for the reduced group C -algebra holds for G i foreach i 2 I and for G. Then we obtain an isomorphismcolim i2I K n .C r .G i// Š ! K n .C r .G//:Again it is in general not at all clear whether there exists such a map in the case, wherethe structure maps i W G i ! G do not have amenable kernels and hence do not inducemaps Cr .G i/ ! Cr .G/.These arguments do not apply to the Farrell–Jones conjecture or the Bost conjecture.Namely any group homomorphism ˛ W H ! G induces maps R Ì H ! R Ì G,A Ì l 1 H ! A Ì l 1 G, and A Ì m H ! A Ì m G for a ring R or a C -algebra A withstructure preserving G-action, where we equip R and A with the H -action comingfrom ˛. Moreover we will show for a directed system fG i j i 2 I g of groups (withnot necessarily injective structure maps) and G D colim i2I G i that there are canonical

Inheritance of isomorphism conjectures under colimits 47isomorphisms (see Lemma 6.2)colim i2I K n .R Ì G i / Š ! K n .R Ì G/Icolim i2I KH n .R Ì G i / Š ! KH n .R Ì G/Icolim i2I L hn 1i .R Ì G i / Š ! L h n1i .R Ì G/Icolim i2I K n .A Ì l 1 G i / Š ! K n .A Ì l 1 G/Icolim i2I K n .A Ì m G i / Š ! K n .A Ì m G/:Let A be a C -algebra with G-action by C -automorphisms. We can consider Aas a ring only. Notice that we get a commutative diagramH G n .E VCyc.G/I K A / KH n .A Ì G/Hn G.E VCyc.G/I KH A / KH n .A Ì G/ŠidH G n .E F in.G/I KH A / KH n .A Ì G/Hn G.E F in.G/I K top /A;l 1ŠHn G.E F in.G/I K topA;m /K n .A Ì l 1 G/ K n .A Ì m G/ŠH G n .E F in.G/I K topA;r / K n .A Ì r G/,where the horizontal maps are assembly maps and the vertical maps are change oftheory and rings maps or induced by the up to G-homotopy unique G-map E F in .G/ !E VCyc .G/. The second left vertical map, which is marked with Š, is bijective. Thisis shown in [4, Remark 7.4] in the case, where G acts trivially on R, the proof carriesdirectly over to the general case. The fourth and fifth vertical left arrow, which aremarked with Š, are bijective, since for a finite group H we have A Ì H D A Ì l 1H D A Ì r H D A Ì m H and hence we can apply [13, Lemma 4.6]. In particularthe Bost conjecture and the Baum–Connes conjecture together imply that the mapK n .A Ì l 1 G/ ! K n .A Ì r G/ is bijective, the map K n .A Ì l 1 G/ ! K n .A Ì m G/ issplit injective and the map K n .A Ì m G/ ! K n .A Ì r G/ is split surjective.The upshot of this discussions is:

48 A. Bartels, S. Echterhoff, and W. Lück• The counterexamples of Higson, Lafforgue and Skandalis [23, Section 7] tothe Baum–Connes conjecture with coefficients are not counterexamples to theFarrell–Jones conjecture or the Bost conjecture.• The counterexamples of Higson, Lafforgue and Skandalis [23, Section 7] showthat the map K n .A Ì l 1 G/ ! K n .A Ì r G/ is in general not bijective.• The passage from the topological K-theory of the Banach algebra l 1 .G/ to thereduced group C -algebra is problematic and may cause failures of the Baum–Connes conjecture.• The Bost conjecture and the Farrell–Jones conjecture are more likely to be truethan the Baum–Connes conjecture.• There is – to the authors’ knowledge – no promising candidate of a group G forwhich a strategy is in sight to show that the Farrell–Jones conjecture or the Bostconjecture are false. (Whether it is reasonable to believe that these conjecturesare true for all groups is a different question.)1.7 Homology theories and spectra. The general strategy of this paper is to presentmost of the arguments in terms of equivariant homology theories. Many of the argumentsfor the Farrell–Jones conjecture, the Bost conjecture or the Baum–Connesconjecture become the same, the only difference lies in the homology theory we applythem to. This is convenient for a reader who is not so familiar with spectra and prefersto think of K-groups and not of K-spectra.The construction of these equivariant homology theories is a second step and donein terms of spectra. Spectra cannot be avoided in algebraic K-theory by definitionand since we want to compare also algebraic and topological K-theory, we need spectradescriptions here as well. Another nice feature of the approach to equivarianttopological K-theory via spectra is that it yields a theory which can be applied toall G-CW-complexes. This will allow us to consider in the case G D colim i2I G i theequivariant K-homology of the G i -CW-complexi E F in.G/ D E F in.G i / althoughi E F in.G/ has infinite isotropy groups if the structure map i W G i ! G has infinitekernel.Details of the constructions of the relevant spectra, namely, the proof of Theorem 8.1,will be deferred to [2]. We will use the existence of these spectra as a black box. Theseconstructions require some work and technical skills, but their details are not at allrelevant for the results and ideas of this paper and their existence is not at all surprising.1.8 Twisting by cocycles. In the L-theory case one encounters also non-orientablemanifolds. In this case twisting with the first Stiefel–Whitney class is required. Ina more general setup one is given a group G, a ring R with involution and a grouphomomorphism w W G ! cent.R / to the center of the multiplicative group of unitsin R. So far we have used the standard involution on the group ring RG, which isgiven by r g D r g 1 . One may also consider the w-twisted involution given by

Inheritance of isomorphism conjectures under colimits 49r g D rw.g/ g 1 . All the results in this paper generalize directly to this casesince one can construct a modified L-theory spectrum functor (over G) using the w-twisted involution and then the homology arguments are just applied to the equivarianthomology theory associated to this w-twisted L-theory spectrum.Acknowledgements. The work was financially supported by the Sonderforschungsbereich478 – Geometrische Strukturen in der Mathematik – and the Max-Planck-Forschungspreis of the third author.2 Equivariant homology theoriesIn this section we briefly explain basic axioms, notions and facts about equivarianthomology theories as needed for the purposes of this article. The main examples whichwill play a role in connection with the Bost, the Baum–Connes and the Farrell–Jonesconjecture will be presented later in Theorem 6.1.Fix a group G and a ring ƒ. In most cases ƒ will be Z. The following definitionis taken from [28, Section 1].Definition 2.1 (G-homology theory). A G-homology theory H G with values in ƒ-modules is a collection of covariant functors HnG from the category of G-CW-pairs tothe category of ƒ-modules indexed by n 2 Z together with natural transformations@ G n .X; A/W H n G.X; A/ ! H n G 1 .A/ WD H n G 1.A; ;/ for n 2 Z such that the followingaxioms are satisfied:• G-homotopy invariance. Iff 0 and f 1 are G-homotopic maps .X; A/ ! .Y; B/of G-CW-pairs, then H G n .f 0/ D H G n .f 1/ for n 2 Z.• Long exact sequence of a pair. Given a pair .X; A/ of G-CW-complexes, thereis a long exact sequenceH G nC1 .j / ! H G nC1 .X; A/ @G nC1! H G n .A/H G n .i/ ! H G n .X/ H G n .j / ! H G n .X; A/ @G n! ;where i W A ! X and j W X ! .X; A/ are the inclusions.• Excision. Let .X; A/ be a G-CW-pair and let f W A ! B be a cellular G-mapof G-CW-complexes. Equip .X [ f B;B/ with the induced structure of a G-CW-pair. Then the canonical map .F; f /W .X; A/ ! .X [ f B;B/ induces anisomorphismH G n .F; f /W H G n .X; A/ Š ! H G n .X [ f B;B/:

50 A. Bartels, S. Echterhoff, and W. Lück• Disjoint union axiom. Let fX i j i 2 I g be a family of G-CW-complexes. Denoteby j i W X i ! `i2I X i the canonical inclusion. Then the mapMHn G .j i/W M Hn G .X i/ Š a ! HnG X ii2I i2Ii2Iis bijective.Let H G and KG be G-homology theories. A natural transformation T W H G !K G of G-homology theories is a sequence of natural transformations T n W H G n ! KG nof functors from the category of G-CW-pairs to the category of ƒ-modules which arecompatible with the boundary homomorphisms.Lemma 2.2. Let T W HG ! KG be a natural transformation of G-homology theories.Suppose that T n .G=H / is bijective for every homogeneous space G=H and n 2 Z.Then T n .X; A/ is bijective for every G-CW-pair .X; A/ and n 2 Z.Proof. The disjoint union axiom implies that both G-homology theories are compatiblewith colimits over directed systems indexed by the natural numbers (such as the systemgiven by the skeletal filtration X 0 X 1 X 2 [ n0 X n D X). The argumentfor this claim is analogous to the one in [40, 7.53]. Hence it suffices to prove thebijectivity for finite-dimensional pairs. Using the axioms of a G-homology theory, thefive lemma and induction over the dimension one reduces the proof to the special case.X; A/ D .G=H; ;/.Next we present a slight variation of the notion of an equivariant homology theoryintroduced in [28, Section 1]. We have to treat this variation since we later want to studycoefficients over a fixed group which we will then pullback via group homomorphismswith as target. Namely, fix a group . A group .G; / over is a group G togetherwith a group homomorphism W G ! . A map ˛ W .G 1 ; 1 / ! .G 2 ; 2 / of groupsover is a group homomorphisms ˛ W G 1 ! G 2 satisfying 2 ı ˛ D 1 .Let ˛ W H ! G be a group homomorphism. Given an H -space X, define theinduction of X with ˛ to be the G-space denoted by ˛X which is the quotient of G Xby the right H -action .g; x/ h WD .g˛.h/; h 1 x/ for h 2 H and .g; x/ 2 G X.If ˛ W H ! G is an inclusion, we also write ind G H instead of ˛. If .X; A/ is anH -CW-pair, then ˛.X; A/ is a G-CW-pair.Definition 2.3 (Equivariant homology theory over a group ). An equivariant homologytheory H ‹ with values in ƒ-modules over a group assigns to every group .G; /over a G-homology theory HG with values in ƒ-modules and comes with the followingso called induction structure: given a homomorphism ˛ W .H; / ! .G; / ofgroups over and an H -CW-pair .X; A/, there are for each n 2 Z natural homomorphismssatisfyingind˛ W H H n .X; A/ ! H G n .˛.X; A// (2.4)

Inheritance of isomorphism conjectures under colimits 51• Compatibility with the boundary homomorphisms. @ G n ı ind˛ D ind˛ ı@ H n .• Functoriality. Let ˇ W .G; / ! .K; / be another morphism of groups over .Then we have for n 2 Zindˇı˛ D H K n .f 1/ ı indˇ ı ind˛ W H H n .X; A/ ! H K n ..ˇ ı ˛/ .X; A//;where f 1 W ˇ˛.X; A/ Š ! .ˇ ı ˛/ .X; A/, .k;g;x/7! .kˇ.g/; x/ is the naturalK-homeomorphism.• Compatibility with conjugation. Let .G; / be a group over and let g 2 G be anelement with .g/ D 1. Then the conjugation homomorphisms c.g/W G ! Gdefines a morphism c.g/W .G; / ! .G; / of groups over . Let f 2 W .X; A/ !c.g/ .X; A/ be the G-homeomorphism which sends x to .1; g 1 x/ in G c.g/.X; A/.Then for every n 2 Z and every G-CW-pair .X; A/ the homomorphismagrees with H G n .f 2/.ind c.g/ W H G n .X; A/ ! H G n .c.g/ .X; A//• Bijectivity. If ˛ W .H; / ! .G; / is a morphism of groups over such thatthe underlying group homomorphism ˛ W H ! G is an inclusion of groups, thenind˛ W Hn H .fg/ ! H n G.˛fg/ D Hn G .G=H / is bijective for all n 2 Z.Definition 2.3 reduces to the one of an equivariant homology in [28, Section 1] ifone puts Df1g.Lemma 2.5. Let ˛ W .H; / ! .G; / be a morphism of groups over . Let .X; A/ bean H -CW-pair such that ker.˛/ acts freely on X A. Thenis bijective for all n 2 Z.ind˛ W H H n .X; A/ ! H G n .˛.X; A//Proof. Let F be the set of all subgroups of H whose intersection with ker.˛/ is trivial.Obviously, this is a family, i.e., closed under conjugation and taking subgroups. A H -CW-pair .X; A/ is called a F -H -CW-pair if the isotropy group of any point in X Abelongs to F .AH -CW-pair .X; A/ is a F -H -CW-pair if and only if ker.˛/ acts freelyon X A.The n-skeleton of ˛.X; A/ is ˛ applied to the n-skeleton of .X; A/. Let .X; A/ bean H -CW-pair and let f W A ! B be a cellular H -map of H -CW-complexes. Equip.X [ f B;B/ with the induced structure of a H -CW-pair. Then there is an obviousnatural isomorphism of G-CW-pairs˛.X [ f B;B/ Š ! .˛X [˛f ˛B;˛B/:

52 A. Bartels, S. Echterhoff, and W. LückNow we proceed as in the proof of Lemma 2.2 but now considering the transformationsind˛ W H H n .X; A/ ! H G n .˛.X; A//only for F -H -CW-pairs .X; A/. Thus we can reduce the claim to the special case.X; A/ D H=L for some subgroup L H with L \ ker.˛/ Df1g. This specialcase follows from the following commutative diagram whose vertical arrows are bijectiveby the axioms and whose upper horizontal arrow is bijective since ˛ induces anisomorphism ˛j L W L ! ˛.L/:H L n .fg/ind˛jL W L!˛.L/Hn˛.L/.fg/H H nind H Lind˛.H=L/ ind G˛.L/HGn .˛H=L/ D Hn G.G=˛.L//.3 Equivariant homology theories and colimitsFix a group and an equivariant homology theory H ‹ with values in ƒ-modules over .Let X be a G-CW-complex. Let ˛ W H ! G be a group homomorphism. Denoteby ˛X the H -CW-complex obtained from X by restriction with ˛. We have alreadyintroduced the induction ˛Y of an H -CW-complex Y . The functors ˛ and ˛ areadjoint to one another. In particular the adjoint of the identity on ˛X is a naturalG-mapf.X;˛/W ˛˛X ! X: (3.1)It sends an element in G ˛ ˛X given by .g; x/ to gx.Consider a map ˛ W .H; / ! .G; / of groups over . Define the ƒ-mapa n D a n .X; ˛/W Hn H ind˛.˛X/ ! Hn G.˛˛X/ H n G.f .X;˛// ! Hn G.X/:If ˇ W .G; / ! .K; / is another morphism of groups over , then by the axioms ofan induction structure the composite Hn H .˛ˇX/ a n.ˇ X;˛/! Hn G.ˇX/ a n.X;ˇ/!Hn K.X/ agrees with a n.X; ˇ ı ˛/W Hn H .˛ˇX/ D Hn H ..ˇ ı ˛/ X/ ! Hn G .X/ fora K-CW-complex X.Consider a directed system of groups fG i j i 2 I g with G D colim i2I G i andstructure maps i W G i ! G for i 2 I and i;j W G i ! G j for i;j 2 I;i j .Weobtain for every G-CW-complex X a system of ƒ-modules fH G i .i X/ j i 2 I gwith structure maps a n .j X; i;j/W H G i .i X/ ! H G j .j X/. We get a map ofƒ-modulestn G .X; A/ WD colim i2I a n .X; i /W colim i2I H G in . i .X; A// ! H n G .X; (3.2) A/:The next definition is an extension of [4, Definition 3.1].

Inheritance of isomorphism conjectures under colimits 53Definition 3.3 ((Strongly) continuous equivariant homology theory). An equivarianthomology theory H ‹ over is called continuous if for every group .G; / over andevery directed system of subgroups fG i j i 2 I g of G with G D S i2I G i the ƒ-map(see (3.2))tn G .fg/W colim i2I H G in .fg/ ! H n G .fg/is an isomorphism for every n 2 Z.An equivariant homology theory H ‹ over is called strongly continuous if forevery group .G; / over and every directed system of groups fG i j i 2 I g withG D colim i2I G i and structure maps i W G i ! G for i 2 I the ƒ-mapis an isomorphism for every n 2 Z.tn G .fg/W colim i2I H G in .fg/ ! H n G .fg/Here and in the sequel we view G i as a group over by ı i W G ii W G i ! G as a morphism of groups over .! andLemma 3.4. Let .G; / be a group over . Consider a directed system of groupsfG i j i 2 I g with G D colim i2I G i and structure maps i W G i ! G for i 2 I . Let.X; A/ be a G-CW-pair. Suppose that H ‹ is strongly continuous.Then the ƒ-homomorphism (see (3.2))is bijective for every n 2 Z.t G n .X; A/W colim i2I H G in . i .X; A// Š! HGn .X; A/Proof. The functor sending a directed systems of ƒ-modules to its colimit is an exactfunctor and compatible with direct sums over arbitrary index maps. If .X; A/ is a pairof G-CW-complexes, then .i X; i A/ is a pair of G i-CW-complexes. Hence thecollection of maps ftn G .X; A/ j n 2 Zg is a natural transformation of G-homologytheories of pairs of G-CW-complexes which satisfy the disjoint union axiom. Hence inorder to show that tn G .X; A/ is bijective for all n 2 Z and all pairs of G-CW-complexes.X; A/, it suffices by Lemma 2.2 to prove this in the special case .X; A/ D .G=H; ;/.For i 2 I let k i W G i =i 1 .H / !i .G=H / be the G i-map sending g 1 i i.H /to i .g i /H . Consider a directed system of ƒ-modules fH G in .G i =i1 .H // j i 2 I gwhose structure maps for i;j 2 I;i j are given by the compositeH G in .G i= i1 .H // ind i;j! H G jn .G j i;j G i = i 1 .H //H G jn .f i;j /! H G jn .G j = 1j .H //for the G j -map f i;j W G j i;j G i = 1i.H / ! G j = 1j.H / sending .g j ;g i 1i.H //

54 A. Bartels, S. Echterhoff, and W. Lückto g j i;j .g i /j1 .H /. Then the following diagram commutes:colim i2I H 1i .H /n.fg/colim i2I ind G i1.H /iŠcolim i2I H G in .G i =i 1 .H //t H n .fg/Šcolim i2I H G in .i colim i2I H G in .k i /.G=H //Hn H .fg/ind G HŠt G n .G=H / HGn .G=H /,where the horizontal maps are the isomorphism given by induction. For the directedsystem fi 1 .H / j i 2 I g with structure maps i;j j 1i .H / W i 1 .H / !j 1 .H /,the group homomorphism colim i2I i j 1i .H / W colim i2I i1 .H / ! H is an isomorphism.This follows by inspecting the standard model for the colimit over a directedsystem of groups. Hence the left vertical arrow is bijective since H ‹ is strongly continuousby assumption. Therefore it remains to show that the mapcolim i2I H G in .k i/W colim i2I H G in .G i= 1 .H // !colim i2I H G iin . iis surjective.Notice that the map given by the direct sum of the structure mapsMH G in . i G=H / ! colim i2I H G in . i G=H /i2IG=H / (3.5)is surjective. Hence it remains to show for a fixed i 2 I that the image of the structuremapH G in . i G=H / ! colim i2I H G in . i G=H /is contained in the image of the map (3.5).We have the decomposition of the G i -seti G=H into its G i-orbitsaG i = i 1 .gHg 1 / Š ! i G=H; g i 1 .gHg 1 / 7! i .g i /gH:G i .gH /2G i n. i G=H /It induces an identification of ƒ-modulesMH G in G i = 1 .gHg 1 / D H G iG i .gH /2G i n. i G=H /iin . iG=H /:Hence it remains to show for fixed elements i 2 I and G i .gH / 2 G i n.i G=H / thatthe obvious compositionH G in G i = 1 .gHg 1 / H G in . i G=H / ! colim i2I H G in . i G=H /i

Inheritance of isomorphism conjectures under colimits 55is contained in the image of the map (3.5).Choose an index j with j i and g 2 im. j /. Then the structure map for i j isa map H G in .i G=H / ! H G jn .j G=H / which sends the summand corresponding toG i .gH / 2 G i n.i G=H / to the summand corresponding to G j .1H / 2 G j n.j G=H /which is by definition the image ofH G jn .k j /W H G jn .G j = j1 .H // ! H G jn . j G=H /:Obviously the image of composite of the last map with the structure mapH G jn . j G=H / ! colim i2I H G in . i G=H /is contained in the image of the map (3.5). Hence the map (3.5) is surjective. Thisfinishes the proof of Lemma 3.4.4 Isomorphism conjectures and colimitsA family F of subgroups of G is a collection of subgroups of G which is closed underconjugation and taking subgroups. Let E F .G/ be the classifying space associated toF . It is uniquely characterized up to G-homotopy by the properties that it is a G-CWcomplexand that E F .G/ H is contractible if H 2 F and is empty if H … F . For moreinformation about these spaces E F .G/ we refer for instance to the survey article [29].Given a group homomorphism W K ! G and a family F of subgroups of G, definethe family F of subgroups of K by F D fH K j .H/ 2 F g: (4.1)If is an inclusion of subgroups, we also write F j K instead of F .Definition 4.2 (Isomorphism conjecture for H ‹ ). Fix a group and an equivarianthomology theory H ‹ with values in ƒ-modules over .A group .G; / over together with a family of subgroups F of G satisfies theisomorphism conjecture (for H ‹) if the projection pr W E F .G/ !fg to the one-pointspacefg induces an isomorphismH G n .pr/W H G n .E F .G// Š ! H G n .fg/for all n 2 Z.From now on fix a group and an equivariant homology theory H ‹ over .Theorem 4.3 (Transitivity principle). Let .G; / be a group over . Let F G befamilies of subgroups of G. Assume that for every element H 2 G the group .H; j H /over satisfies the isomorphism conjecture for F j H .Then the up to G-homotopy unique map E F .G/ ! E G .G/ induces an isomorphismHn G.E F .G// ! Hn G.E G .G// for all n 2 Z. In particular, .G; / satisfies theisomorphism conjecture for G if and only if .G; / satisfies the isomorphism conjecturefor F .

56 A. Bartels, S. Echterhoff, and W. LückProof. The proof is completely analogous to the one in [4, Theorem 2.4, Lemma 2.2],where only the case Df1g is treated.Theorem 4.4. Let .G; / be a group over . Let F be a family of subgroups of G.(i) Let G be the directed union of subgroups fG i j i 2 I g. Suppose that H ‹ iscontinuous and for every i 2 I the isomorphism conjecture holds for .G i ;j Gi /and F j Gi .Then the isomorphism conjecture holds for .G; / and F .(ii) Let fG i j i 2 I g be a directed system of groups with G D colim i2I G i andstructure maps i W G i ! G. Suppose that H ‹ is strongly continuous and forevery i 2 I the isomorphism conjecture holds for .G i ; ı i / andi F .Then the isomorphism conjecture holds for .G; / and F .Proof. (i) The proof is analogous to the one in [4, Proposition 3.4].(ii) This follows from the following commutative square whose horizontal arrowsare bijective because of Lemma 3.4 and the identification i E F .G/ D E i F .G i /:colim i2I H G in.E i F .G i // t G n .E F .G//ŠHn G.E F .G//colim i2I H G in .fg/tn G.fg/Š HGn .fg/.Fix a class of groups C closed under isomorphisms, taking subgroups and takingquotients, e.g., the class of finite groups or the class of virtually cyclic groups. For agroup G let C.G/ be the family of subgroups of G which belong to C.Theorem 4.5. Let .G; / be a group over .(i) Let G be the directed union G D S i2I G i of subgroups G i Suppose that H ‹ iscontinuous and that the isomorphism conjecture is true for .G i ;j Gi / and C.G i /for all i 2 I .Then the isomorphism conjecture is true for .G; / and C.G/.(ii) Let fG i j i 2 I g be a directed system of groups with G D colim i2I G i andstructure maps i W G i ! G. Suppose that H ‹ is strongly continuous and thatthe isomorphism conjecture is true for .H; C.H // for every i 2 I and everysubgroup H G i .Then for every subgroup K G the isomorphism conjecture is true for .K; j K /and C.K/.

Inheritance of isomorphism conjectures under colimits 57Proof. (i) This follows from Theorem 4.4 (i) since C.G i / D C.G/j Gi holds for i 2 I .(ii) If G is the colimit of the directed system fG i j i 2 I g, then the subgroup K Gis the colimit of the directed system fi1 .K/ j i 2 I g. Hence we can assume G D Kwithout loss of generality.Since C is closed under quotients by assumption, we have C.G i / i C.G/ forevery i 2 I . Hence we can consider for any i 2 I the compositionH G in.E C.G i /.G i // ! H G in.E i C.G/.G i // ! H G in .fg/:Because of Theorem 4.4 (ii) it suffices to show that the second map is bijective. Byassumption the composition of the two maps is bijective. Hence it remains to show thatthe first map is bijective. By Theorem 4.3 this follows from the assumption that theisomorphism conjecture holds for every subgroup H G i and in particular for anyH 2 i C.G/ for C.G i/j H D C.H /.5 Fibered isomorphism conjectures and colimitsIn this section we also deal with the fibered version of the isomorphism conjectures.(This is not directly needed for the purpose of this paper and the reader may skip thissection.) This is a stronger version of the Farrell–Jones conjecture. The Fibered Farrell–Jones conjecture does imply the Farrell–Jones conjecture and has better inheritanceproperties than the Farrell–Jones conjecture.We generalize (and shorten the proof of) the result of Farrell–Linnell [16, Theorem7.1] to a more general setting about equivariant homology theories as developedin Bartels–Lück [3].Definition 5.1 (Fibered isomorphism conjecture for H ‹ ). Fix a group and an equivarianthomology theory H ‹ with values in ƒ-modules over . A group .G; / over together with a family of subgroups F of G satisfies the fibered isomorphism conjecture(for H ‹ ) if for each group homomorphism W K ! G the group .K; ı / over satisfies the isomorphism conjecture with respect to the family F .Theorem 5.2. Let .G; / be a group over . Let F be a family of subgroups of G. LetfG i j i 2 I g be a directed system of groups with G D colim i2I G i and structure mapsi W G i ! G. Suppose that H ‹ is strongly continuous and for every i 2 I the fiberedisomorphism conjecture holds for .G i ; ı i / andi F .Then the fibered isomorphism conjecture holds for .G; / and F .Proof. Let W K ! G be a group homomorphism. Consider the pullback of groupsK i i G iS iiK G.

58 A. Bartels, S. Echterhoff, and W. LückExplicitly K i Df.k; g i / 2 K G i j .k/ D i .g i /g. Let x i;j W K i ! K j be themap induced by i;j W G i ! G j ,id K and id G and the pullback property. One easilychecks by inspecting the standard model for the colimit over a directed set that weobtain a directed system x i;j W K i ! K j of groups indexed by the directed set I andŠthe system of maps x i W K i ! K yields an isomorphism colim i2I K i ! K. Thefollowing diagram commutes:colim i2I H K inS i E F .G/ t K n . E F .G//ŠHn K. E F .G//colim i2I H K in .fg/tn K.fg/Š HKn .fg/,where the vertical arrows are induced by the obvious projections onto fg and thehorizontal maps are the isomorphisms from Lemma 3.4. Notice that S i E F .G/ is amodel for E Si F .K i/ D E ii F .K i /. Hence each map H K inS i E F .G/ !H K in .fg/ is bijective since .G i ;ı i / satisfies the fibered isomorphism conjecture fori F and hence .K i; ı i ı i / satisfies the isomorphism conjecture for i i F .This implies that the left vertical arrow is bijective. Hence the right vertical arrow isan isomorphism. Since E F .G/ is a model for E F .K/, this means that .K; ı/ satisfies the isomorphism conjecture for F . Since W K ! G is any grouphomomorphism, .G; / satisfies the fibered isomorphism conjecture for F .The proof of the following results are analogous to the one in [3, Lemma 1.6] and [4,Lemma 1.2], where only the case Df1g is treated.Lemma 5.3. Let .G; / be a group over and let F G be families of subgroupsof G. Suppose that .G; / satisfies the fibered isomorphism conjecture for the family F .Then .G; / satisfies the fibered isomorphism conjecture for the family G .Lemma 5.4. Let .G; / be a group over . Let W K ! G be a group homomorphismand let F be a family of subgroups of G. If .G; / satisfies the fibered isomorphismconjecture for the family F , then .K; ı/satisfies the fibered isomorphism conjecturefor the family F .For the remainder of this section fix a class of groups C closed under isomorphisms,taking subgroups and taking quotients, e.g., the families F in or VCyc.Lemma 5.5. Let .G; / be a group over . Suppose that the fibered isomorphismconjecture holds for .G; / and C.G/. Let H G be a subgroup.Then the fibered isomorphism conjecture holds for .H; j H / and C.H /.Proof. This follows from Lemma 5.4 applied to the inclusion H ! G since C.H / DC.G/j H .Theorem 5.6. Let .G; / be a group over .

Inheritance of isomorphism conjectures under colimits 59(i) Let G be the directed union G D S i2I G i of subgroups G i . Suppose that H ‹ is continuous and that the fibered isomorphism conjecture is true for .G i ;j Gi /and C.G i / for all i 2 I .Then the fibered isomorphism conjecture is true for .G; / and C.G/.(ii) Let fG i j i 2 I g be a directed system of groups with G D colim i2I G i andstructure maps i W G i ! G. Suppose that H ‹ is strongly continuous and thatthe fibered isomorphism conjecture is true for .G i ; ı i / and C.G i // for alli 2 I .Then the fibered isomorphism conjecture is true for .G; / and C.G/.Proof. (i) The proof is analogous to the one in [4, Proposition 3.4], where the case Df1g is considered.(ii) Because C is closed under taking quotients we conclude C.G i / i C.G/.Now the claim follows from Theorem 5.2 and Lemma 5.3.Corollary 5.7. (i) Suppose that H ‹ is continuous. Then the ( fibered) isomorphismconjecture for .G; / and C.G/ is true for all groups .G; / over if and only if it istrue for all such groups where G is a finitely generated group.(ii) Suppose that H ‹ is strongly continuous. Then the fibered isomorphism conjecturefor .G; / and C.G/ is true for all groups .G; / over if and only if it is true forall such groups where G is finitely presented.Proof. Let .G; / be a group over where G is finitely generated. Choose a finitelygenerated free group F together with an epimorphism W F ! G. Let K be the kernelof . Consider the directed system of finitely generated subgroups fK i j i 2 I g of K.Let SK i be the smallest normal subgroup of K containing K i . Explicitly SK i is givenby elements which can be written as finite products of elements of the shape fk i f 1for f 2 F and k 2 K i . We obtain a directed system of groups fF=SK i j i 2 I g,where for i j the structure map i;j W F=SK i ! F= SK j is the canonical projection.If i W F=SK i ! F=K D G is the canonical projection, then the collection of mapsŠf i j i 2 I g induces an isomorphism colim i2I F=SK i ! G. By construction for eachi 2 I the group F=SK i is finitely presented and the fibered isomorphism conjectureholds for .F= SK i ; ı i / and C.F= SK i / by assumption. Theorem 5.6 (ii) implies thatthe fibered Farrell–Jones conjecture for .G; / and C.G/ is true.6 Some equivariant homology theoriesIn this section we will describe the relevant homology theories over a group andshow that they are (strongly) continuous. (We have defined the notion of an equivarianthomology theory over a group in Definition 2.3.)

60 A. Bartels, S. Echterhoff, and W. Lück6.1 Desired equivariant homology theories. We will need the followingTheorem 6.1 (Construction of equivariant homology theories). Suppose that we aregiven a group and a ring R (with involution) or a C -algebra A respectively onwhich acts by structure preserving automorphisms. Then:(i) Associated to these data there are equivariant homology theories with values inZ-modules over the group H ‹ . I K R/H ‹ . I KH R/H ‹ . I Lh 1iR/;H ‹ . I Ktop /;A;l 1H ‹ . I Ktop A;r /;H ‹ . I Ktop A;m /;where in the case H ‹. I Ktop A;r/ we will have to impose the restriction to theinduction structure that a homomorphisms ˛ W .H; / ! .G; / over inducesa transformation ind˛ W Hn H .X; X 0/ ! Hn G.˛.X; X 0 // only if the kernel of theunderlying group homomorphism ˛ W H ! G acts with amenable isotropy onX n X 0 .(ii) If .G; / is a group over and H G is a subgroup, then there are for everyn 2 Z identificationsH H n .fgI K R/ Š H G n .G=H I K R/ Š K n .R Ì H/IH H n .fgI KH R/ Š H G n .G=H I KH R/ Š KH n .R Ì H/IH H nH H nH H nH H n.fgI Lh1iR/ Š Hn G 1i.G=H I LhR/ Š L h n1i .R Ì H/I.fgI Ktop / Š H G A;l 1 n .G=H I Ktop / Š KA;l 1 n .A Ì l 1 H/I.fgI KtopA;r / Š H n G .G=H I KtopA;r / Š K n.A Ì r H/I.fgI KtopA;m / Š H n G .G=H I KtopA;r / Š K n.A Ì m H/:Here H and G act on R and A respectively via the given -action, W G ! and the inclusion H G, K n .R Ì H/ is the algebraic K-theory of the twistedgroup ring R Ì H ,KH n .R Ì H/is the homotopy K-theory of the twisted groupring R Ì H , L hn1i .R Ì H/is the algebraic L-theory with decoration h 1i ofthe twisted group ring with involution R Ì H , K n .A Ì l 1 H/ is the topologicalK-theory of the crossed product Banach algebra A Ì l 1 H , K n .A Ì r H/ isthe topological K-theory of the reduced crossed product C -algebra A Ì r H ,and K n .A Ì m H/ is the topological K-theory of the maximal crossed productC -algebra A Ì m H ;

Inheritance of isomorphism conjectures under colimits 61(iii) Let W 0 ! 1 be a group homomorphism. Let R be a ring (with involution)and A be a C -algebra on which 1 acts by structure preserving automorphisms.Let .G; / be a group over 0 . Then in all cases the evaluation at .G; / of theequivariant homology theory over 0 associated to R or A respectivelyagrees with the evaluation at .G; ı / of the equivariant homology theory over 1 associated to R or A respectively.(iv) Suppose the group acts on the rings (with involution) R and S or on theC -algebras A and B respectively by structure preserving automorphisms. Let W R ! S or W A ! B be a -equivariant homomorphism of rings (withinvolution) or C -algebras respectively. Then induces natural transformationsof homology theories over ‹ W H ‹ . I K R/ ! H ‹ . I K S/I ‹ W H ‹ . I KH R/ ! H ‹ . I KH S/I ‹ W H ‹ . I Lh 1iR/ ! H ‹ 1i. I LhS/I ‹ W H ‹ . I Ktop / ! H ‹ A;l 1 . I Ktop /IB;l 1 ‹ W H ‹ . I Ktop A;r / ! H ‹ . I Ktop B;r /I ‹ W H ‹ . I Ktop A;m / ! H ‹ . I Ktop B;m /:They are compatible with the identifications appearing in assertion (ii).(v) Let act on the C -algebra A by structure preserving automorphisms. Wecan consider A also as a ring with structure preserving G-action. Then thereare natural transformations of equivariant homology theories with values in Z-modules over H ‹ . I K A/ ! H ‹ . I KH A/ ! H ‹ . I Ktop /A;l 1! H ‹ . I Ktop A;m / ! H ‹ . I Ktop A;r /:They are compatible with the identifications appearing in assertion (ii).6.2 (Strong) continuity. Next we want to showLemma 6.2. Suppose that we are given a group and a ring R (with involution) or aC -algebra A respectively on which G acts by structure preserving automorphisms.Then the homology theories with values in Z-modules over H ‹ . I K R/; H ‹ . I KH R/; H ‹ 1i. I LhR/; H ‹ . I Ktop /; and H ‹ A;l 1 . I Ktop A;m /(see Theorem 6.1) are strongly continuous in the sense of Definition 3.3, whereasH ‹ . I Ktop A;r /is only continuous.

62 A. Bartels, S. Echterhoff, and W. LückProof. We begin with H ‹. I K R/ and H ‹. I KH R/. We have to show for everydirected systems of groups fG i j i 2 I g with G D colim i2I G i together with a mapW G ! that the canonical mapscolim i2I K n .R Ì G i / ! K n .R Ì G/Icolim i2I KH n .R Ì G i / ! KH n .R Ì G/;are bijective for all n 2 Z. Obviously R Ì G is the colimit of rings colim i2I R Ì G i .Now the claim follows for K n .R Ì G/ for n 0 from [36, (12) on page 20].Using the Bass–Heller–Swan decomposition one gets the results for K n .R Ì G/ forall n 2 Z and that the mapcolim i2I N p K n .R Ì G i / ! N p K n .R Ì G/is bijective for all n 2 Z and all p 2 Z;p 1 for the Nil-groups N p K n .RG/ definedby Bass [8, XII]. Now the claim for homotopy K-theory follows from the spectralsequence due to Weibel [41, Theorem 1.3].Next we treat H ‹. I Lh 1iR/. We have to show for every directed systems of groupsfG i j i 2 I g with G D colim i2I G i together with a map W G ! that the canonicalmapcolim i2I L hn 1i .R Ì G i / ! L h n 1i .R G/is bijective for all n 2 Z. Recall from [37, Definition 17.1 and Definition 17.7] thatL hn 1i .R Ì G/ D colim m!1 Lnh miL h mi.R Ì G/In .R Ì G/ D coker L h mC1inC1.R Ì G/ ! L h mC1inC1.R Ì GŒZ/ for m 0.Since L h1in .R Ì G/ is L h n .R Ì G/, it suffices to show that! n W colim i2I L h n .R Ì G i/ ! L h n .R Ì G/is bijective for all n 2 Z. We give the proof of surjectivity for n D 0 only, the proofsof injectivity for n D 0 and of bijectivity for the other values of n are similar.The ring R Ì G is the colimit of rings colim i2I R Ì G i . Let i W R Ì G i ! R Ì Gand i;j W R Ì G i ! R Ì G j for i;j 2 I;i j be the structure maps. One candefine R Ì G as the quotient of `i2I R Ì G i= , where x 2 R Ì G i and y 2 R Ì G jsatisfy x y if and only if i;k .x/ D j;k .y/ holds for some k 2 I with i;j k.The addition and multiplication is given by adding and multiplying representativesbelonging to the same R Ì G i . Let M.m;nI R Ì G/ be the set of .m; n/-matrices withentries in R Ì G. GivenA i 2 M.m;nI R Ì G i /, define i;j .A i / 2 M.m;nI R Ì G j /and i .A i / 2 M.m;nI R Ì G/ by applying i;j and i to each entry of the matrix A i .We need the following key properties which follow directly from inspecting the modelfor the colimit above:(i) Given A 2 M.m;nI R Ì G/, there exists i 2 I and A i 2 M.m;nI R Ì G i / withi.A i / D A.

Inheritance of isomorphism conjectures under colimits 63(ii) Given A i 2 M.m;nI RÌG i / and A j 2 M.m;nI RÌG j / with i .A i / D j .A j /,there exists k 2 I with i;j k and i;k .A i / D j;k .A j /.An element ŒA in L h 0.R Ì G/ is represented by a quadratic form on a finitelygenerated free R Ì G-module, i.e., a matrix A 2 GL n .R Ì G/ for which there existsa matrix B 2 M.n;nI R Ì G/ with A D B C B , where B is given by transposingthe matrix B and applying the involution of R elementwise. Fix such a choice ofa matrix B. Choose i 2 I and B i 2 M.n;nI R Ì G i / with i .B i / D B. Theni.B i C Bi / D A is invertible. Hence we can find j 2 I with i j such thatA j WD i;j .B i C Bj / is invertible. Put B j D i;j .B i /. Then A j D B j C Bj andj .A j / D A. Hence A j defines an element ŒA j 2 L h n .R Ì G j / which is mapped toŒA under the homomorphism L h n .R Ì G j / ! L n .R Ì G/ induced by j . Hence themap ! 0 of (6.2) is surjective.Next we deal with H ‹ . I Ktop A;l 1 /. We have to show for every directed systems ofgroups fG i j i 2 I g with G D colim i2I G i together with a map W G ! that thecanonical mapcolim i2I K n .A Ì l 1 G i / ! K n .A Ì l 1 G/is bijective for all n 2 Z. Since topological K-theory is a continuous functor, it sufficesto show that the colimit (or sometimes also called inductive limit) of the system ofBanach algebras fA Ì l 1 G i j i 2 I g in the category of Banach algebras with normdecreasing homomorphisms is A Ì l 1 G. So we have to show that for any Banachalgebra B and any system of (norm deceasing) homomorphisms of Banach algebras˛i W AÌ l 1 G i ! B compatible with the structure maps AÌ l 1 i;j W AÌ l 1 G i ! AÌ l 1 G jthere exists precisely one homomorphism of Banach algebras ˛ W A Ì l 1 G ! B withthe property that its composition with the structure map AÌ l 1 i W AÌ l 1 G i ! AÌ l 1 Gis ˛i for i 2 I .It is easy to see that in the category of C-algebras the colimit of the system fAÌG i ji 2 I g is A Ì G with structure maps A Ì i W A Ì G i ! A Ì G. Hence the restrictionsof the homomorphisms ˛i to the subalgebras A Ì G i yields a homomorphism of centralC-algebras ˛0 W A Ì G ! B uniquely determined by the property that the compositionof ˛0 with the structure map A Ì i W A Ì G i ! A Ì G is ˛ij AÌGi for i 2 I .If˛ exists,its restriction to the dense subalgebra A Ì G has to be ˛0. Hence ˛ is unique if it exists.Of course we want to define ˛ to be the extension of ˛0 to the completion A Ì l 1 G ofA Ì G with respect to the l 1 -norm. So it remains to show that ˛0 W A Ì G ! B is normdecreasing. Consider an element u 2 A Ì G which is given by a finite formal sumu D P g2F a g g, where F G is some finite subset of G and a g 2 A for g 2 F .Wecan choose an index j 2 I and a finite set F 0 G j such that j j F 0 W F 0 ! F is oneto-one.For g 2 F let g 0 2 F 0 denote the inverse image of g under this map. Considerthe element v D P g 0 2F 0 a g g 0 in A Ì G j . By construction we have A Ì j .v/ D uand kvk Dkuk D P niD1 ka ik. We concludek˛0.u/kDk˛0 ı .A Ì j /.v/k Dk˛j .v/k kvk Dkuk:The proof for H ‹. I Ktop A;m/ follows similarly, using the fact that by definition of thenorm on AÌ m G every -homomorphism of AÌG into a C -algebra B extends uniquely

64 A. Bartels, S. Echterhoff, and W. Lückto A Ì m G. The proof for the continuity of H ‹. I Ktop A;r/ follows from Theorem 4.1in [10].Notice that we have proved all promised results of the introduction as soon as wehave completed the proof of Theorem 6.1 which we have used as a black box so far.7 From spectra over groupoids to equivariant homology theoriesIn this section we explain how one can construct equivariant homology theories fromspectra over groupoids.A spectrum E Df.E.n/; .n// j n 2 Zg is a sequence of pointed spaces fE.n/ jn 2 Zg together with pointed maps called structure maps .n/W E.n/^S 1 ! E.nC1/.A(strong) map of spectra (sometimes also called function in the literature) f W E ! E 0is a sequence of maps f .n/W E.n/ ! E 0 .n/ which are compatible with the structuremaps .n/, i.e., we have f.nC 1/ ı .n/ D 0 .n/ ı .f .n/ ^ id S 1/ for all n 2 Z. Thisshould not be confused with the notion of a map of spectra in the stable category (see[1, III.2.]). Recall that the homotopy groups of a spectrum are defined by i .E/ WD colim k!1 iCk .E.k//;where the system iCk .E.k// is given by the composition iCk .E.k// S ! iCkC1 .E.k/ ^ S 1 / .k/ ! iCkC1 .E.k C 1//of the suspension homomorphism and the homomorphism induced by the structuremap. We denote by Spectra the category of spectra.A weak equivalence of spectra is a map f W E ! F of spectra inducing an isomorphismon all homotopy groups.Given a small groupoid G , denote by Groupoids # G the category of small groupoidsover G , i.e., an object is a functor F 0 W G 0 ! G with a small groupoid as sourceand a morphism from F 0 W G 0 ! G to F 1 W G 1 ! G is a functor F W G 0 ! G 1 satisfyingF 1 ı F D F 0 . We will consider a group as a groupoid with one object and as set ofmorphisms. An equivalence F W G 0 ! G 1 of groupoids is a functor of groupoids F forwhich there exists a functor of groupoids F 0 W G 1 ! G 0 such that F 0 ı F and F ı F 0 arenaturally equivalent to the identity functor. A functor F W G 0 ! G 1 of small groupoids isan equivalence of groupoids if and only if it induces a bijection between the isomorphismclasses of objects and for any object x 2 G 0 the map aut G0 .x/ ! aut G1 .F .x// inducedby F is an isomorphism of groups.Lemma 7.1. Let be a group. Consider a covariant functorEW Groupoids # ! Spectrawhich sends equivalences of groupoids to weak equivalences of spectra.

Inheritance of isomorphism conjectures under colimits 65Then we can associate to it an equivariant homology theory H ‹.;I E/ (with valuesin Z-modules) over such that for every group .G; / over and subgroup H Gwe have a natural identificationH H n .fgI E/ D H G n .G=H; E/ D n.E.H //:If TW E ! F is a natural transformation of such functors Groupoids # !Spectra, then it induces a transformation of equivariant homology theories over H ‹ . I T/W H ‹ .; I E/ ! H ‹ .; I F/such that for every group .G; / over and subgroup H G the homomorphismHn H .fgI T/W H n H .fgI E/ ! H n H .fgI F/ agrees under the identification above with n .T.H //W n .E.H // ! n .F.H //.Proof. We begin with explaining how we can associate to a group .G; / over aG-homology theory H G . I E/ with the property that for every subgroup H G wehave an identificationHn G .G=H; E/ D n.E.H //:We just follow the construction in [13, Section 4]. Let Or.G/ be the orbit category of G,i.e., objects are homogenous spaces G=H and morphisms are G-maps. Given a G-set S,the associated transport groupoid t G .S/ has S as set of objects and the set of morphismsfrom s 0 2 S to s 1 2 S consists of the subset fg 2 G j gs 1 D s 2 g of G. Compositionis given by the group multiplication. A G-map of sets induces a functor betweenthe associated transport groupoids in the obvious way. In particular the projectionG=H ! G=G induces a functor of groupoids pr S W t G .S/ ! t G .G=G/ D G. Thust G .S/ becomes an object in Groupoids # by the composite ı pr S . We obtaina covariant functor t G W Or.G/ ! Groupoids # . Its composition with the givenfunctor E yields a covariant functorNow defineE G WD E ı t G W Or.G/ ! Spectra:H G .X; AI E/ WD H G . I EG /;where H G . I EG / is the G-homology theory which is associated to E G W Or.G/ !Spectra and defined in [13, Section 4 and 7]. Namely, if X is a G-CW-complex, we canassign to it a contravariant functor map G .G=‹; X/W Or.G/ ! Spaces sending G=Hto map G .G=H; X/ D X H and put H G n .XI EG / WD n .map G .G=‹; X/ C ^Or.G/ E G /for the spectrum map G .G=‹; X/ C ^Or.G/ E G (which is denoted in [13] bymap G .G=‹; X/ C ˝Or.G/ E G ).Next we have to explain the induction structure. Consider a group homomorphism˛ W .H; / ! .G; / of groups over and an H -CW-complex X. We have to constructa homomorphismH H n .XI E/ ! H G n .˛XI E/:

66 A. Bartels, S. Echterhoff, and W. LückThis will be done by constructing a map of spectramap H .H=‹; X/ C ^Or.H / E H ! map G .G=‹; ˛X/ C ^Or.G/ E G :We follow the constructions in [13, Section 1]. The homomorphism ˛ induces a covariantfunctor Or.˛/W Or.H / ! Or.G/ by sending H=L to ˛.H=L/ D G=˛.L/.Given a contravariant functor Y W Or.H / ! Spaces, we can assign to it its inductionwith Or.˛/ which is a contravariant functor ˛Y W Or.G/ ! Spaces. Given a contravariantfunctor Z W Or.G/ ! Spaces, we can assign to it its restriction which isthe contravariant functor ˛Z WD Z ı Or.˛/W Or.G/ ! Spaces. Induction ˛ and˛ form an adjoint pair. Given an H -CW-complex X, there is a natural identification˛.map H .H=‹; X// D map G .G=‹; ˛X/. Using [13, Lemma 1.9] we get for anH -CW-complex X a natural map of spectramap H .H=‹; X/ C ^Or.H / ˛E G ! map G .G=‹; ˛X/ C ^Or.G/ E G :Given an H -set S, we obtain a functor of groupoids t H .S/ ! t G .˛S/sending s 2 Sto .1; s/ 2 G ˛ S and a morphism in t H .S/ given by a group element h to the onein t G .˛S/ given by ˛.h/. This yields a natural transformation of covariant functorsOr.H / ! Groupoids # from t H ! t G ı Or.˛/. Composing with the functor Egives a natural transformation of covariant functors Or.H / ! Spectra from E H to˛E G . It induces a map of spectramap H .H=‹; X/ C ^Or.H / E H ! map H .H=‹; X/ C ^Or.H / ˛E G :Its composition with the maps of spectra constructed beforehand yields the desired mapof spectra map H .H=‹; X/ C ˝Or.H / E H ! map G .G=‹; ˛X/ C ˝Or.G/ E G .We omit the straightforward proof that the axioms of an induction structure aresatisfied. This finishes the proof of Theorem 6.1.The statement about the natural transformation TW E ! F is obvious.8 Some K-theory spectra associated to groupoidsThe last step in completing the proof of Theorem 6.1 is to prove the following Theorem8.1 because then we can apply it in combination with Lemma 7.1. (Actuallywe only need the version of Theorem 8.1, where G is given by a group .) LetGroupoids finker # G be the subcategory of Groupoids # G which has the same objectsand for which a morphism from F 0 W G 0 ! G to F 1 W G 1 ! G given by a functorF W G 0 ! G 1 satisfying F 1 ı F D F 0 has the property that for every object x 2 G 0the group homomorphism aut G0 .x/ ! aut G1 .F .x// induced by F has a finite kernel.Denote by Rings, -Rings, and C -C Algebras the categories of rings, rings withinvolution and C -algebras.Theorem 8.1. Let G be a fixed groupoid. Let R W G ! Rings, R W G !-Rings, orAW G ! C Algebras respectively be a covariant functor. Then there exist covariant

Inheritance of isomorphism conjectures under colimits 67functorsK R W Groupoids # G ! SpectraIKH R W Groupoids # G ! SpectraIL h 1iRtogether with natural transformationsW Groupoids # G ! SpectraIK top W Groupoids # G ! SpectraIA;l 1K topA;r W Groupoidsfinker # G ! SpectraIK topA;mW Groupoids # G ! Spectra;I 1 W K ! KHII 2 W KH ! K top IA;l 1I 3 W K top ! K topA;l 1 A;m II 4 W K topA;m ! Ktop A;r ;of functors from Groupoids # G or Groupoids finker # G respectively to Spectra suchthat the following holds:(i) Let F i W G i ! G be objects for i D 0; 1 and F W F 0 ! F 1 be a morphismbetween them in Groupoids # G or Groupoids finker # G respectively such thatthe underlying functor of groupoids F W G 0 ! G 1 is an equivalence of groupoids.Then the functors send F to a weak equivalences of spectra.(ii) Let F 0 W G 0 ! G be an object in Groupoids # G or Groupoids finker # G respectivelysuch that the underlying groupoid G 0 has only one object x. LetG D mor G0 .x; x/ be its automorphisms group. We obtain a ring R.y/, a ringR.y/ with involution, or a C -algebra B.y/ with G-operation by structure preservingmaps from the evaluation of the functor R or A respectively at y D F.x/.Then: n .K R .F // D K n .R.y/ Ì G/I n .KH R .F // D KH n .R.y/ Ì G/I n .L h 1iR.F // D L h n1i .R.y/ Ì G/I n .K top .F // D KA.y/;l 1 n .A.y/ Ì l 1 G/I n .K topA.y/;r .F // D K n.A.y/ Ì r G/I n .K topA.y/;m .F // D K n.A.y/ Ì m G/;where K n .R.y/ÌG/is the algebraic K-theory of the twisted group ring R.y/ÌG,KH n .R.y/ Ì G/ is the homotopy K-theory of the twisted group ring R.y/ Ì G,

68 A. Bartels, S. Echterhoff, and W. LückL hn1i.R.y/ÌG/ is the algebraic L-theory with decoration h 1i of the twistedgroup ring with involution R.y/ Ì G, K n .A.y/ Ì l 1 G/ is the topological K-theory of the crossed product Banach algebra A.y/ Ì l 1 G, K n .A.y/ Ì r G/ isthe topological K-theory of the reduced crossed product C -algebra A.y/ Ì r G,and K n .A.y/ Ì m G/ is the topological K-theory of the maximal crossed productC -algebra A.y/ Ì m G.The natural transformations I 1 , I 2 , I 3 and I 4 become under this identificationsthe obvious change of rings and theory homomorphisms.(iii) These constructions are in the obvious sense natural in R and A respectively andin G .We defer the details of the proof of Theorem 8.1 in [2]. Its proof requires some workbut there are many special cases which have already been taken care of. If we wouldnot insist on groupoids but only on groups as input, these are the standard algebraic K-and L-theory spectra or topological K-theory spectra associated to group rings, groupBanach algebras and group C -algebras. The construction for the algebraic K- andL-theory and the topological K-theory in the case, where G acts trivially on a ring Ror a C -algebra are already carried out or can easily be derived from [4], [13], and [24]except for the case of a Banach algebra. The case of the K-theory spectrum associatedto an additive category with G-action has already been carried out in [7]. The mainwork which remains to do is to treat the Banach case and to construct the relevantnatural transformation from KH to K topA;l 1References[1] J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Math. Universityof Chicago Press, Chicago, Ill., 1974.[2] A. Bartels, M. Joachim, and W. Lück, From algebraic to topological K-theory spectra, inpreparation, 2007.[3] A. Bartels and W. Lück, Induction theorems and isomorphism conjectures for K- and L-theory, Forum Math. 19 (2007), 379–406.[4] A. Bartels and W. Lück, Isomorphism conjecture for homotopy K-theory and groups actingon trees, J. Pure Appl. Algebra 205 (2006),660–696.[5] A. Bartels, W. Lück, and H. Reich, The K-theoretic Farrell-Jones Conjecture for hyperbolicgroups, Invent. Math. 172 (2008), 29–70.[6] A. Bartels, W. Lück, and H. Reich, On the Farrell-Jones Conjecture and its applications,J. Topol. 1 (2008), 57–86.[7] A. Bartels and H. Reich, Coefficients for the Farrell-Jones conjecture, Adv. Math.,209(1):337–362, 2007.[8] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.[9] P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory ofgroup C -algebras, in C -algebras: 1943–1993 (San Antonio, TX, 1993), Amer. Math.Soc., Providence, RI, 1994, 240–291.

Inheritance of isomorphism conjectures under colimits 69[10] P. Baum, S. Millington, and R. Plymen, Local-global principle for the Baum-Connes conjecturewith coefficients, K-Theory 28 (2003), 1–18.[11] B. H. Bowditch, Notes on Gromov’s hyperbolicity criterion for path-metric spaces, inGroup theory from a geometrical viewpoint (Trieste, 1990), World Scientific Publishing,River Edge, NJ, 1991, 64–167.[12] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, GrundlehrenMath. Wiss. 319, Springer-Verlag, Berlin 1999.[13] J. F. Davis and W. Lück, Spaces over a category and assembly maps in isomorphismconjectures in K- and L-theory, K-Theory 15 (1998), 201–252.[14] P. de la Harpe and A. Valette, La propriété .T / de Kazhdan pour les groupes localementcompacts (avec un appendice de Marc Burger), Astérisque 175 (1989).[15] F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math.Soc. 6 (1993), 249–297.[16] F. T. Farrell and P. A. Linnell, K-theory of solvable groups, Proc. London Math. Soc. (3)87 (2003), 309–336.[17] S. M. Gersten, On the spectrum of algebraic K-theory, Bull. Amer. Math. Soc. 78 (1972),216–219.[18] É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov,Progr. Math. 83, Birkhäuser, Boston, MA, 1990.[19] M. Gromov, Hyperbolic groups, in Essays in group theory, Math. Sci. Res. Inst. Publ. 8,Springer-Verlag, New York 1987, 75–263.[20] M. Gromov, Spaces and questions, Geom. Funct. Anal., special volume, Part I (2000)118–161.[21] I. Hambleton and E. K. Pedersen, Identifying assembly maps in K- and L-theory, Math.Ann. 328 (2004), 27–57.[22] N. Higson and G. Kasparov, E-theory and KK-theory for groups which act properly andisometrically on Hilbert space, Invent. Math. 144 (2001), 23–74.[23] N. Higson, V. Lafforgue, and G. Skandalis, Counterexamples to the Baum-Connes conjecture,Geom. Funct. Anal. 12 (2002), 330–354.[24] M. Joachim, K-homology of C -categories and symmetric spectra representing K-homology,Math. Ann. 327 (2003), 641–670.[25] G. Kasparov and G. Skandalis, Groups acting properly on “bolic” spaces and the Novikovconjecture, Ann. of Math. (2) 158(2003), 165–206.[26] V. Lafforgue, Une démonstration de la conjecture de Baum-Connes pour les groupes réductifssur un corps p-adique et pour certains groupes discrets possédant la propriété (T),C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 439–444.[27] V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes, Invent. Math. 149 (2002),1–95.[28] W. Lück, Chern characters for proper equivariant homology theories and applications toK- and L-theory, J. Reine Angew. Math. 543 (2002), 193–234.[29] W. Lück, Survey on classifying spaces for families of subgroups, in Infinite groups: geometric,combinatorial and dynamical aspects, Progr. Math. 248, Birkhäuser, Basel 2005,269–322 .

70 A. Bartels, S. Echterhoff, and W. Lück[30] I. Mineyev and G. Yu, The Baum-Connes conjecture for hyperbolic groups, Invent. Math.149 (2002), 97–122.[31] A. Y. Ol’shankskii, D. Osin, and M. Sapir, Lacunary hyperbolic groups, preprint 2007,arXiv:math.GR/0701365v1.[32] A. Y. Ol’shanskii, An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSRSer. Mat. 43 (1979), 1328–1393.[33] W. Paravicini, KK-theory for banach algebras and proper groupoids, Ph.D. thesis, 2006.[34] E. K. Pedersen and C. A. Weibel, A non-connective delooping of algebraic K-theory, inAlgebraic and Geometric Topology (New Brunswick 1983), Lecture Notes in Math. 1126,Springer-Verlag, Berlin 1985, 166–181.[35] R. T. Powers, Simplicity of the C -algebra associated with the free group on two generators,Duke Math. J. 42 (1975), 151–156.[36] D. Quillen, Higher algebraic K-theory. I, In Algebraic K-theory, I: Higher K-theories(Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math. 341,Springer-Verlag, Berlin 1973, 85–147.[37] A. A. Ranicki, Lower K- and L-theory, London Math. Soc. Lecture Note Ser. 178, CambridgeUniversity Press, Cambridge 1992.[38] M. Sapir, Some group theory problems, Internat. J. Algebra Comput. 17 (2007), 1189–1214.[39] G. Skandalis, Progrès récents sur la conjecture de Baum-Connes. Contribution de VincentLafforgue, Astérisque 276 (2002), 105–135.[40] R. M. Switzer, Algebraic topology—homotopy and homology, Grundlehren Math. Wiss.212. Springer-Verlag, New York 1975.[41] C. A. Weibel, Homotopy algebraic K-theory, In Algebraic K-theory and algebraic numbertheory (Honolulu, HI, 1987), Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989,461–488.

Coarse and equivariant co-assembly mapsHeath Emerson and Ralf Meyer 1 IntroductionThis is a sequel to the articles [5], [7], which deal with a coarse co-assembly map thatis dual to the usual coarse assembly map. Here we study an equivariant co-assemblymap that is dual to the Baum–Connes assembly map for a group G.A rather obvious choice for such a dual map is the mapp EG W KKG .C; C/ ! RKKG .EGI C; C/ (1)induced by the projection p EG W EG ! point. This map and its application to theNovikov conjecture go back to Kasparov ([10]). Nevertheless, (1) is not quite themap that we consider here. Our map is closely related to the coarse co-assembly mapof [5]. It is an isomorphism if and only if the Dirac-dual-Dirac method applies to G.Hence there are many cases – groups with ¤ 1 – where our co-assembly map is anisomorphism and (1) is not.Most of our results only work if the group G is (almost) totally disconnected and hasa G-compact universal proper G-space EG. We impose this assumption throughoutthe introduction.First we briefly recall some of the main ideas of [5], [7]. The new ingredient inthe coarse co-assembly map is the reduced stable Higson corona c red .X/ of a coarsespace X. Its definition resembles that of the usual Higson corona, but its K-theorybehaves much better. The coarse co-assembly map is a mapW K C1 c red .X/ ! KX .X/; (2)where KX .X/ is a coarse invariant of X that agrees with K .X/ if X is uniformlycontractible.If jGj is the coarse space underlying a group G, then there is a commuting diagramKK G C;C 0.G/ p EGRKK G ŠK C1 c red .jGj/ EGI C;C 0.G/ ŠKX .jGj/.(3) This research was partially carried out at the Westfälische Wilhelms-Universität Münster and supportedby the EU-Network Quantum Spaces and Noncommutative Geometry (Contract HPRN-CT-2002-00280) andthe Deutsche Forschungsgemeinschaft (SFB 478).

72 H. Emerson and R. MeyerIn this situation, KX .jGj/ Š K .EG/because jGj is coarsely equivalent to EG, whichis uniformly contractible. The commuting diagram (3), coupled with the reformulationof the Baum–Connes assembly map in [11], is the source of the relationship betweenthe coarse co-assembly map and the Dirac-dual-Dirac method mentioned above.If G is a torsion-free discrete group with finite classifying space BG, the coarseco-assembly map is an isomorphism if and only if the Dirac-dual-Dirac method appliesto G. A similar result for groups with torsion is available, but this requires workingequivariantly with respect to compact subgroups of G.In this article, we work equivariantly with respect to the whole group G. The actionof G on its underlying coarse space jGj by isometries induces an action on c red .G/.We consider a G-equivariant analogueW K topC1 G; cred .jGj/ ! K .C 0 .EG/ Ì G/ (4)of the coarse co-assembly map (2); here Ktop .G; A/ denotes the domain of the Baum–Connes assembly map for G with coefficients A. We avoid K .c red .X/ Ì G/ andK .c red .X/ Ì r G/ because we can say nothing about these two groups. In contrast, thegroup Ktop G; c red .jGj is much more manageable. The only analytical difficulties inthis group come from coarse geometry.There is a commuting diagram similar to (3) that relates (4) to equivariant Kasparovtheory. To formulate this, we need some results of [11]. There is a certain G-C -algebraP and a class D 2 KK G .P; C/ called Dirac morphism such that the Baum–Connesassembly map for G is equivalent to the mapK .A ˝ P/ Ì r G/ ! K .A Ì r G/induced by Kasparov product with D. The Baum–Connes conjecture holds for G withcoefficients in P ˝ A for any A. The Dirac morphism is a weak equivalence, that is, itsimage in KK H .P; C/ is invertible for each compact subgroup H of G.The existence of the Dirac morphism allows us to localise the (triangulated) categoryKK G at the multiplicative system of weak equivalences. The functor from KK G to itslocalisation turns out to be equivalent to the mapp EG W KKG .A; B/ ! RKK G .EGI A; B/:One of the main results of this paper is a commuting diagramK topC1 G; cred .jGj/ ŠK .EG/ŠKK G .C; P/ pEG RKKG .EGI C; P/.(5)In other words, the equivariant coarse co-assembly (4) is equivalent to the mapp EG W KKG .C; P/ ! RKKG .EGI C; P/:

Coarse and equivariant co-assembly maps 73This map is our proposal for a dual to the Baum–Connes assembly map.We should justify why we prefer the map (4) over (1). Both maps have isomorphictargets:RKK G .EGI C; P/ Š RKKG .EGI C; C/ Š K .C 0 .EG/ Ì G/:Even in the usual Baum–Connes assembly map, the analytical side involves a choicebetween full and reduced group C -algebras and crossed products. Even though thefull group C -algebra has better functoriality properties and is sometimes preferredbecause it gives potentially finer invariants, the reduced one is used because its K-theoryis closer to K top .G/. In formulating a dual version of the assembly map, weare faced with a similar situation. Namely, the topological object that is dual toK top .G/ is RKK G .EGI C; C/. For the analytical side, we have some choices; weprefer KK G .C; P/ over KKG .C; C/ because the resulting co-assembly map is an isomorphismin more cases.Of course, we must check that this choice is analytical enough to be useful forapplications. The most important of these is the Novikov conjecture. Elements ofRKK G .EGI C; C/ yield maps Ktop .G/ ! Z, which are analogous to higher signatures.In particular, (4) gives rise to such objects. The maps Ktop .G/ ! Z that come from aclass in the range of (1) are known to yield homotopy invariants for manifolds becausethere is a pairing between KK G .C; C/ and K .Cmax G/ (see [8]). But since (4) factorsthrough (1), the former also produces homotopy invariants.In particular, surjectivity of (4) implies the Novikov conjecture for G. More istrue: since KK G .C; P/ is the home of a dual-Dirac morphism, (5) yields that G has adual-Dirac morphism and hence a -element if and only if (4) is an isomorphism. Thisobservation can be used to give an alternative proof of the main result of [7].We call elements in the range of (4) boundary classes. These automatically form agraded ideal in the Z=2-graded unital ring RKK G .EGI C; C/. In contrast, the rangeof the unital ring homomorphism (1) need not be an ideal because it always containsthe unit element of RKK G .EGI C; C/. We describe two important constructions ofboundary classes, which are related to compactifications and to the proper Lipschitzcohomology of G studied in [3], [4].Let EG Z be a G-equivariant compactification of EG that is compatible withthe coarse structure in a suitable sense. Since there is a mapK topG; C.Z n EG/ ! K top G; c red .EG/ ;we get boundary classes from the boundary Z n EG. This construction also showsthat G has a dual-Dirac morphism if it satisfies the Carlsson–Pedersen condition. Thisimproves upon a result of Nigel Higson ([9]), which shows split injectivity of theBaum–Connes assembly map with coefficients under the same assumptions.Although we have discussed only KK G .C; P/ so far, our main technical result ismore general and can also be used to construct elements in Kasparov groups of theform KK G C;C 0.X/ for suitable G-spaces X. IfX is a proper G-space, then we canuse such classes to construct boundary classes in RKK G .EGI C; C/. This provides a

74 H. Emerson and R. MeyerK-theoretic counterpart of the proper Lipschitz cohomology of G defined by Connes,Gromov, and Moscovici in [3]. Our approach clarifies the geometric parts of severalconstructions in [3]; thus we substantially simplify the proof of the homotopy invarianceof Gelfand–Fuchs cohomology classes in [3].2 Preliminaries2.1 Dirac-dual-Dirac method and Baum–Connes conjecture. First, we recall theDirac-dual-Dirac method of Kasparov and its reformulation in [11]. This is a techniquefor proving injectivity of the Baum–Connes assembly mapW K top .G; B/ ! K C r .G; B/ ; (6)where G is a locally compact group and B isaC -algebra with a strongly continuousaction of G or, briefly, a G-C -algebra.This method requires a proper G-C -algebra A and classesd 2 KK G .A; C/; 2 KK G .C; A/; WD ˝A d 2 KK G .C; C/;such that p EG ./ D 1 C in RKK G .EGI C; C/. If these data exist, then the Baum–Connes assembly map (6) is injective for all B. If, in addition, D 1 C in KK G .C; C/,then the Baum–Connes assembly map is invertible for all B, so that G satisfies theBaum–Connes conjecture with arbitrary coefficients.Let A and B be G-C -algebras. An element f 2 KK G .A; B/ is called a weak equivalencein [11] if its image in KK H .A; B/ is invertible for each compact subgroup Hof G.The following theorem contains some of the main results of [11].Theorem 1. Let G be a locally compact group. Then there is a G-C -algebra P anda class D 2 KK G .P; C/ called Dirac morphism such that(a) D is a weak equivalence;(b) the Baum–Connes conjecture holds with coefficients in A ˝ P for any A;(c) the assembly map (6) is equivalent to the mapD W K .A ˝ P Ì r G/ ! K .A Ì r G/I(d) the Dirac-dual-Dirac method applies to G if and only if there is a class 2KK G .C; P/ with ˝C D D 1 A , if and only if the mapis an isomorphism.D W KK G .C; P/ ! KKG .P; P/; x 7! D ˝ x; (7)

Coarse and equivariant co-assembly maps 75Whereas [7] studies the invertibility of (7) by relating it to (2), here we are going tostudy the map (7) itself.It is shown in [11] that the localisation of the category KK G at the weak equivalencesis isomorphic to the category RKK G .EG/ whose morphism spaces are the groupsRKK G .EGI A; B/ as defined by Kasparov in [10]. This statement is equivalent to theexistence of a Poincaré duality isomorphismKK G .A ˝ P;B/Š RKKG .EGI A; B/ (8)for all G-C -algebras A and B (this notion of duality is analysed in [6]). The canonicalfunctor from KK G to the localisation becomes the obvious functorp EG W KKG .A; B/ ! RKK G .EGI A; B/:Since D is a weak equivalence, p EG.D/ is invertible. Hence the maps in the followingcommuting square are isomorphisms for all G-C -algebras A and B:RKK G .EGI A; B ˝ P/ŠD RKK G .EGI A; B/ŠD RKK G .EGI A ˝ P;B ˝ P/ŠTogether with (8) this impliesD ŠD RKKG .EGI A ˝ P;B/.KK G .A ˝ P;B/Š KKG .A ˝ P;B ˝ P/:In the following, it will be useful to turn the isomorphismKtop .G; A/ Š K .A ˝ P/ Ì r G in Theorem 1.(c) into a definition.2.2 Group actions on coarse spaces. Let G be a locally compact group and let X bea right G-space and a coarse space. We always assume that G acts continuously andcoarsely on X, that is, the set f.xg; yg/ j g 2 K; .x; y/ 2 Eg is an entourage for anycompact subset K of G and any entourage E of X.Definition 2. We say that G acts by translations on X if f.x; gx/ j x 2 X; g 2 Kgis an entourage for all compact subsets K G. We say that G acts by isometries ifevery entourage of X is contained in a G-invariant entourage.Example 3. Let G be a locally compact group. Then G has a unique coarse structurefor which the right translation action is isometric; the corresponding coarse space isdenoted jGj. The generating entourages are of the form[Kg Kg Df.xg; yg/ j g 2 G; x; y 2 Kgg2Gfor compact subsets K of G. The left translation action is an action by translations forthis coarse structure.

76 H. Emerson and R. MeyerExample 4. More generally, any proper, G-compact G-space X carries a unique coarsestructure for which G acts isometrically; its entourages are defined as in Example 3.With this coarse structure, the orbit map G ! X, g 7! g x, is a coarse equivalencefor any choice of x 2 X. If the G-compactness assumption is omitted, the result is a-coarse space. We always equip a proper G-space with this additional structure.2.3 The stable Higson corona. We next recall the definition of the stable Higsoncorona of a coarse space X from [5], [7]. Let D beaC -algebra.Let M.D ˝ K/ be the multiplier algebra of D ˝ K, and let xB red .X; D/ be theC -algebra of norm-continuous, bounded functions f W X ! M.D ˝ K/ for whichf.x/ f.y/2 D ˝ K for all x;y 2 X. We also letB red .X; D/ WD xB red .X; D/=C 0 .X; D ˝ K/:Definition 5. A function f 2 xB red .X; D/ has vanishing variation if the functionE 3 .x; y/ 7! kf.x/ f.y/k vanishes at 1 for any closed entourage E X X.The reduced stable Higson compactification of X with coefficients D is the subalgebraNc red .X; D/ xB red .X; D/ of vanishing variation functions. The quotientc red .X; D/ WD Nc red .X; D/=C 0 .X; D ˝ K/ B red .X; D/is called reduced stable Higson corona of X. This defines a functor on the coarsecategory of coarse spaces: a coarse map f W X ! X 0 induces a map c red .X 0 ;D/ !c red .X; D/, and two maps X ! X 0 induce the same map c red .X 0 ;D/! c red .X; D/if they are close. Hence a coarse equivalence X ! X 0 induces an isomorphismc red .X 0 ;D/Š c red .X; D/.For some technical purposes, we must allow unions X D S X n of coarse spacessuch that the embeddings X n ! X nC1 are coarse equivalences; such spaces are called-coarse spaces. The main example is the Rips complex P .X/ of a coarse space X,which is used to define its coarse K-theory. More generally, if X is a proper but notG-compact G-space, then X may be endowed with the structure of a -coarse space.For coarse spaces of the form jGj for a locally compact group G with a G-compactuniversal proper G-space EG, we may use EG instead of P .X/ because EG is coarselyequivalent to G and uniformly contractible. Therefore, we do not need -coarse spacesmuch; they only occur in Lemma 7.It is straightforward to extend the definitions of Nc red .X; D/ and c red .X; D/ to-coarse spaces (see [5], [7]). Since we do not use this generalisation much, we omitdetails on this.Let H be a locally compact group that acts coarsely and properly on X. It is crucialfor us to allow non-compact groups here, whereas [7] mainly needs equivariance forcompact groups. Let D be an H -C -algebra, and let K H WD K.`2N ˝L 2 H/. Then Hacts on xB red .X; D ˝ K H / by.h f /.x/ WD h f.xh/ ;

Coarse and equivariant co-assembly maps 77where we use the obvious action of H on D ˝ K H and its multiplier algebra. Theaction of H on xB red .X; D ˝ K H / need not be continuous; we let xB redH.X; D/ be thesubalgebra of H -continuous elements in xB red .X; D ˝ K H /. We let Nc redH.X; D/ bethe subalgebra of vanishing variation functions in xB redH.X; D/. Both algebras containC 0 .X; D ˝ K H / as an ideal. The corresponding quotients are denoted by B redH.X; D/and c redH .X; D/. By construction, we have a natural morphism of extensions of H -C -algebrasC 0 .X; D ˝ K H / NcredH .X; D/ c redH.X; D/C 0 .X; D ˝ K H / x B redH .X; D/ Bred(9)H .X; D/.Concerning the extension of this construction to -coarse spaces, we only mentionone technical subtlety. We must extend the functor Ktop .H; / from C -algebras to-H -C -algebras. Here we use the definitionK top .H; A/ Š K .A ˝ P/ Ì r H ; (10)where D 2 KK H .P; C/ is a Dirac morphism for H . The more traditional definitionas a colimit of KK G .C 0.X/; A/, where X EG is G-compact, yields a wrong resultif A is a -H -C -algebra because colimits and limits do not commute.Let H be a locally compact group, let X be a coarse space with an isometric,continuous, proper action of H , and let D be an H -C -algebra. The H -equivariantcoarse K-theory KXH .X; D/ of X with coefficients in D is defined in [7] byKXH .X; D/ WD Ktop H; C 0 .P .X/; D/ : (11)As observed in [7], we have Ktop H; C 0 .P .X/; D/ Š K .C 0 .P .X/; D/ Ì H/ becauseH acts properly on P .X/.For most of our applications, X will be equivariantly uniformly contractible forall compact subgroups K H , that is, the natural embedding X ! P .X/ is aK-equivariant coarse homotopy equivalence. In such cases, we simply haveKXH .X; D/ Š Ktop H; C 0 .X; D/ : (12)In particular, this applies if X is an H -compact universal proper H -space (again, recallthat the coarse structure is determined by requiring H to act isometrically).The H -equivariant coarse co-assembly map for X with coefficients in D is a certainmap W K topC1H; credH .X; D/ ! KXH .X; D/defined in [7]. In the special case where we have (12), this is simply the boundarymap for the extension C 0 .X; D ˝ K H / Nc redH.X; D/ credH.X; D/. We areimplicitly using the fact that the functor Ktop .H; / has long exact sequences for arbitraryextensions of H -C -algebras, which is proved in [7] using the isomorphismKtop .H; B/ WD K .B ˝ P/ Ì r H Š K .B ˝max P/ Ì H

78 H. Emerson and R. Meyerand exactness properties of maximal C -tensor products and full crossed products.There is also an alternative picture of the co-assembly map as a forget-control map,provided X is uniformly contractible (see [7, §2.8]). We have the following equivariantversion of this result:Proposition 6. Let G be a totally disconnected group with a G-compact universalproper G-space EG. Then the G-equivariant coarse co-assembly map for G is equivalentto the mapj W K topC1G; credG.EG;D/ ! K topC1induced by the inclusion j W c red .EG;D/ ! Bred .EG;D/.GGG; BredG.EG;D/The equivalence of the two maps means that there is a natural commuting diagramK topC1K topC1G; credG .jGj;D/ KX G .jGj;D/ ŠG; credG .EG;D/ j KtopC1ŠG; BredG .EG;D/ .Recall that j is induced by the inclusion Nc redG.EG;D/ ! xB redG.EG;D/, which exactlyforgets the vanishing variation condition. Hence j is a forget-control map.Proof. We may replace jGj by EG because EG is coarsely equivalent to jGj. The coarseK-theory of EG agrees with the usual K-theory of EG (see [7]). A slight elaborationof the proof of [7, Lemma 15] shows thatK H xB redG .EG;D/ Š KK H C; xB redG.EG;D/vanishes for all compact subgroups H of G. This yields K topG; xB redG .EG;D/ D 0by a result of [2]. Now the assertion follows from the Five Lemma and the naturalityof the K-theory long exact sequence for (9) as in [7].3 Classes in Kasparov theory from the stable Higson coronaIn this section, we show how to construct classes in equivariant KK-theory from theK-theory of the stable Higson corona. The following lemma is our main technicaldevice:Lemma 7. Let G and H be locally compact groups and let X be a coarse spaceequipped with commuting actions of G and H . Suppose that G acts by translationsand that H acts properly and by isometries. Let A and D be H -C -algebras, equippedwith the trivial G-action. We abbreviateB X WD C 0 .X; D ˝ K H ˝max A/ Ì H;E X WD .Nc redH.X; D/ ˝max A/ Ì H

Coarse and equivariant co-assembly maps 79and similarly for P .X/ instead of X. There are extensions B X E X E X =B Xand B P .X/ E P .X/ E P .X/ =B P .X/ withE P .X/ =B P .X/ Š E X =B X Š .c redH .X; D/ ˝max A/ Ì H;and a natural commuting diagramK C1 .E X =B X /KK G .C;B X/@K .B P .X/ /p EG RKKG .EGI C;B X /.(13)Proof. The quotients E X =B X and E P .X/ =B P .X/ are as asserted and agree becauseX ! P .X/ is a coarse equivalence and because maximal tensor products and fullcrossed products are exact functors in complete generality, unlike spatial tensor productsand reduced crossed products. We let @ be the K-theory boundary map for the extensionB P .X/ E P .X/ E X =B X .Since we have a natural map c redH.X; D/ ˝max A ! c redH.X; D ˝max A/, we mayreplace the pair .D; A/ by .D ˝max A; C/ and omit A if convenient. Stabilising Dby K H , we can further eliminate the stabilisations.First we lift the K-theory boundary map for the extension B X E X E X =B Xto a map W K C1 .E X =B X / ! KK G .C;B X/. The G-equivariance of the resultingKasparov cycles follows from the assumption that G acts on X by translations.We have to distinguish between the cases D0 and D1. We only write downthe construction for D0. Since the algebra E X =B X is matrix-stable, K 1 .E X =B X /is the homotopy group of unitaries in E X =B X without further stabilisation. A cyclefor KK G 0 .C;B X/ is given by two G-equivariant Hilbert modules E˙ over B X and aG-continuous adjointable operator F W E C ! E for which 1 FF , 1 F F andgF F for g 2 G are compact; we take E˙ D B X and let F 2 E X M.B X / bea lifting for a unitary u 2 E X =B X . Since G acts on X by translations, the inducedaction on Nc red .X; D/ and hence on E X =B X is trivial. Hence u is a G-invariant unitaryin E X =B X . For the lifting F , this means that1 FF ; 1 F F; gF F 2 B X :Hence F defines a cycle for KK G 0 .C;B X/. We get a well-defined map Œu 7! ŒF fromK 1 .E X =B X / to KK G 0 .C;B X/ because homotopic unitaries yield operator homotopicKasparov cycles.Next we have to factor the map p EG ı in (13) through K 0.B P .X/ /. The mainingredient is a certain continuous map Nc W EG X ! P .X/. We use the same descriptionof P .X/ as in [7] as the space of positive measures on X with 1=2 < .X/ 1;this is a -coarse space in a natural way, we write it as P .X/ D S P d .X/.There is a function c W EG ! R C for which R EGc.g/dg D 1 for all 2 EGand supp c \ Y is compact for G-compact Y EG. If 2 EG, x 2 X, then the

80 H. Emerson and R. MeyerconditionZhNc.;x/;˛i WDGc.g/˛.g 1 x/dgfor ˛ 2 C 0 .X/ defines a probability measure on X. Since such measures are containedin P .X/, Nc defines a map Nc W EG X ! P .X/. This map is continuous and satisfiesNc.g;g 1 xh/ DNc.;x/h for all g 2 G, 2 EG, x 2 X, h 2 H .ForaC -algebra Z, let C.EG;Z/ be the -C -algebra of all continuous functionsf W EG ! Z without any growth restriction. Thus C.EG;Z/ D lim C.K;Z/,where K runs through the directed set of compact subsets of EG.We claim that . Nc f /./.x/ WD f Nc.;x/ for f 2 C 0 .P .X/; D/ defines a continuous-homomorphismNc W C 0 .P .X/; D/ ! C EG;C 0 .X; D/ :If K EG is compact, then there is a compact subset L G such that c. g/ D 0for 2 K and g … L. Hence Nc.;x/ is supported in L 1 x for 2 K. Since G actson X by translations, such measures are contained in a filtration level P d .X/. HenceNc .f / restricts to a C 0 -function K X ! D for all f 2 C 0 .P .X/; D/. This provesthe claim. Since Nc is H -equivariant and G-invariant, we get an induced mapB P .X/ D C 0 .P .X/; D/ Ì H ! .C EG;C 0 .X; D/ Ì H/ G D C.EG;B X / G ;where Z G Z denotes the subalgebra of G-invariant elements. We obtain an induced-homomorphism between the stable multiplier algebras as well.An element of K 0 B P .X/ / is represented by a self-adjoint bounded multiplier F 2M.B P .X/ ˝ K/ such that 1 FF and 1 F F belong to B P .X/ ˝ K. NowFQWDNc .F / is a G-invariant bounded multiplier of C.EG;B X ˝ K/ and hence a G-invariantmultiplier of C 0 .EG;B X ˝ K/, such that ˛ .1 FQFQ / and ˛ .1 FQ F/ Q belongto C 0 .EG;B X ˝ K/ for all ˛ 2 C 0 .EG/. This says exactly that FQis a cycle forRKK G 0 .EGI C;B X/. This construction provides the natural map W K 0 .B P .X/ / ! RKK G 0 .EGI C;B X/:Finally, a routine computation, which we omit, shows that the two images of a unitaryu 2 E X =B X differ by a compact perturbation. Hence the diagram (13) commutes.We are mainly interested in the case where A is the source P of a Dirac morphismfor H . Then K C1 .E X =B X / D Ktop H; c redH.X; D/ , and the top row in (13) is theH -equivariant coarse co-assembly map for X with coefficients D. Since we assume Hto act properly on X,wehaveaKK G -equivalence B X C 0 .X; D/ Ì H , and similarlyfor P .X/. Hence we now get a commuting squareK topC1H; credH .X; D/ @KK G .C;C 0.X; D/ Ì H/p EGKX H.X; D/ RKKG .EGI C;C 0 .X; D/ Ì H/.(14)

Coarse and equivariant co-assembly maps 81If, in addition, D D C and the action of H on X is free, then we can further simplifythis toK topC1H; credH .X/ @KXH .X/KK G C;C 0.X=H / p EG RKK G EGI C;C 0.X=H / .We may also specialise the space X to jGj, with G acting by multiplication on theleft, and with H G a compact subgroup acting on jGj by right multiplication. Thisis the special case of (13) that is used in [7]. The following applications will requireother choices of X.3.1 Applications to Lipschitz classes. Now we use Lemma 7 to construct interestingelements in KK G C;C 0.X/ for a G-space X. This is related to the method of Lipschitzmaps developed by Connes, Gromov and Moscovici in [3].3.1.1 Pulled-back coarse structures. Let X be a G-space, let Y be a coarse spaceand let ˛ W X ! Y be a proper continuous map. We pull back the coarse structureon Y to a coarse structure on X, letting E X X be an entourage if and only if˛.E/ Y Y is one. Since ˛ is proper and continuous, this coarse structure iscompatible with the topology on X. For this coarse structure, G acts by translationsif and only if ˛ satisfies the following displacement condition used in [3]: for anycompact subset K G, the set˚˛.gx/; ˛.x/2 Y Y j x 2 X; g 2 K(15)is an entourage of Y . The map ˛ becomes a coarse map. Hence we obtain a commutingdiagramK C1 c red .Y / ˛ K C1 c red .X/ C;C 0.X/ KK G @ YKX .Y /˛@ X KX .X/p EG RKK G EGI C;C 0.X/ :with and as in Lemma 7.The constructions of [3, §I.10] only use Y D R N with the Euclidean coarse structure.The coarse co-assembly map is an isomorphism for R N because R N is scalable.Moreover, R N is uniformly contractible and has bounded geometry. Hence we obtaincanonical isomorphismsK C1 c red .R N / Š KX .R N / Š K .R N /:In particular, K C1 c red .R N / Š Z with generator Œ@R N in K 1 N c red .R N / . Thisclass is nothing but the usual dual-Dirac morphism for the locally compact group R N .

82 H. Emerson and R. MeyerAs a result, any map ˛ W X ! R N that satisfies the displacement condition aboveinducesŒ˛ WD ˛Œ@R N 2 KK G N C;C 0.X/ :The commutative diagram (13) computes p EG Œ˛ 2 RKKG NEGI C;C 0.X/ in purelytopological terms.3.1.2 Principal bundles over coarse spaces. As in [3], we may replace a fixed mapX ! R N by a section of a vector bundle over X. But we need this bundle tohave a G-equivariant spin structure. To encode this, we consider a G-equivariantSpin.N /-principal bundle W E ! B together with actions of G on E and B suchthat is G-equivariant and the action on E commutes with the action of H WD Spin.N /.Let T WD E Spin.N / R N be the associated vector bundle over B. It carries a G-invariantEuclidean metric and spin structure. As is well-known, sections ˛ W B ! Tcorrespond bijectively to Spin.N /-equivariant maps ˛0 W E ! R N ; here a section ˛corresponds to the map ˛0 W E ! R N that sends y 2 E to the coordinates of ˛.y/ inthe orthogonal frame described by y. Since the group Spin.N / is compact, the map ˛0is proper if and only if b 7! k˛.b/k is a proper function on B.As in §3.1.1, a Spin.N /-equivariant proper continuous map ˛0 W E ! Y for a coarsespace Y allows us to pull back the coarse structure of Y to E; then Spin.N / acts byisometries. The group G acts by translations if and only if ˛0 satisfies the displacementcondition from §3.1.1. If Y D R N , we can rewrite this in terms of ˛ W B ! T :weneedsupfkg˛.g 1 b/ ˛.b/kjb 2 B; g 2 Kgto be bounded for all compact subsets K G.If the displacement condition holds, then we are in the situation of Lemma 7 withH D Spin.N / and X D E. Since H acts freely on E, C 0 .E/ Ì H is G-equivariantlyMorita–Rieffel equivalent to C 0 .B/. We obtain canonical mapsK Spin.N /C1c redSpin.N / .RN / .˛0/ ! K Spin.N /C1c redSpin.N / .E/! KK G C;C 0.E/ Ì Spin.N / Š KK G C;C 0.B/ :The Spin.N /-equivariant coarse co-assembly map for R N is an isomorphism by [7]because the group R N Ì Spin.N / has a dual-Dirac morphism. Using also the uniformcontractibility of R N and Spin.N /-equivariant Bott periodicity, we getK Spin.N /C1c redSpin.N / .RN / Š KX Spin.N / .RN / Š K Spin.N / .RN / Š K CNSpin.N / .point/:The class of the trivial representation in Rep.SpinN/ Š K Spin.N /0.C/ is mapped to theusual dual-Dirac morphism Œ@R N 2 K Spin.N /1 Nc redSpin.N / .RN / for R N . As a result, anyproper section ˛ W B ! T satisfying the displacement condition inducesŒ˛ WD ˛Œ@R N 2 KK G N C;C 0.B/ :

Coarse and equivariant co-assembly maps 83Again, the commutative diagram (13) computes p EG Œ˛ 2 RKKG Nin purely topological terms.EGI C;C 0.X/ 3.1.3 Coarse structures on jet bundles. Let M be an oriented compact manifold andlet Diff C .M / be the infinite-dimensional Lie group of orientation-preserving diffeomorphismsof M . Let G be a locally compact group that acts on M by a continuousgroup homomorphism G ! Diff C .M /. The Gelfand–Fuchs cohomology of M ispart of the group cohomology of Diff C .M / and by functoriality maps to the groupcohomology of G. It is shown in [3] that the range of Gelfand–Fuchs cohomology inthe cohomology of G yields homotopy-invariant higher signatures.This argument has two parts; one is geometric and concerns the construction of aclass in KK G C;C 0.X/ for a suitable space X; the other uses cyclic homology toconstruct linear functionals on K .C 0 .X/ Ì r G/ associated to Gelfand–Fuchs cohomologyclasses. We can simplify the first step; the second has nothing to do with coarsegeometry.Let k W JC k .M / ! M be the oriented k-jet bundle over M . That is, a point inJC k .M / is the kth order Taylor series at 0 of an orientation-preserving diffeomorphismfrom a neighbourhood of 0 2 R n into M . This is a principal H -bundle over M ,where H is a connected Lie group whose Lie algebra h is the space of polynomial mapsp W R n ! R n of order k with p.0/ D 0, with an appropriate Lie algebra structure.The maximal compact subgroup K H is isomorphic to SO.n/, acting by isometrieson R n . It acts on h by conjugation.Since our construction is natural, the action of G on M lifts to an action on JC k .M /that commutes with the H -action. We let H act on the right and G on the left. DefineX k WD JC k .M /=K. This is the bundle space of a fibration over M with fibres H=K.Gelfand–Fuchs cohomology can be computed using a chain complex of Diff C .M /-invariant differential forms on X k for k ! 1. Using this description, Connes,Gromov, and Moscovici associate to a Gelfand–Fuchs cohomology class a functionalK .C 0 .X k / Ì r G/ ! C for sufficiently high k in [3].Since JC k .M /=H Š M is compact, there is a unique coarse structure on Jk C.M / forwhich H acts isometrically (see §2.2). With this coarse structure, JC k .M / is coarselyequivalent to H . The compactness of JC k .M /=H Š M also implies easily that Gacts by translations. We have a Morita–Rieffel equivalence C 0 .X k / C 0 .JC k M/Ì Kbecause K acts freely on JC k .M /. We want to study the mapK C1 .c redK .Jk C M/Ì K/ ! KKG .C;C 0.JC k M/Ì K/ Š KKG C;C 0.X k / produced by Lemma 7.Since H is almost connected, it has a dual-Dirac morphism by [10]; hence theK-equivariant coarse co-assembly map for H is an isomorphism by the main resultof [7]. Moreover, H=K is a model for EG by [1]. We getK C1 .c redK.Jk C M/Ì K/ Š K C1.c red .jH j/ Ì K/ Š KX K .jH j/ Š K K .H=K/:K

84 H. Emerson and R. MeyerLet h and k be the Lie algebras of H and K. There is a K-equivariant homeomorphismh=k Š H=K, where K acts on h=k by conjugation. Now we need to know whetherthere is a K-equivariant spin structure on h=k. One can check that this is the case ifk 0; 1 mod 4. Since we can choose k as large as we like, we can always assumethat this is the case. The spin structure allows us to use Bott periodicity to identifyKK .H=K/ Š KH N.C/, which is the representation ring of K in degree N , whereN D dim h=k. Using our construction, the trivial representation of K yields a canonicalelement in KK G N C;C 0.X k / .This construction is much shorter than the corresponding one in [3] because we useKasparov’s result about dual-Dirac morphisms for almost connected groups. Much ofthe corresponding argument in [3] is concerned with proving a variation on this resultof Kasparov.4 Computation of KK G .C; P/So far we have merely used the diagram (13) to construct certain elements in KK G .C;B/.Now we show that this construction yields an isomorphic description of KK G .C; P/.This assertion requires G to be a totally disconnected group with a G-compact universalproper G-space. We assume this throughout this section.Lemma 8. In the situation of Lemma 7, suppose that X DjGj with G acting by lefttranslations and that H G is a compact subgroup acting on X by right translations;here jGj carries the coarse structure of Example 3. Then the maps and areisomorphisms.Proof. We reduce this assertion to results of [7]. The C -algebras c redH.jGj;D/Ì Hand c redH.jGj;D/H are strongly Morita equivalent, whence have isomorphic K-theory.It is shown in [7] thatK C1 c red H .jGj;D/H Š KK G .C; IndG H D/: (16)Finally, IndH G .D/ D C 0.G; D/ HC 0 .G; D/ Ì H . Hence we getis G-equivariantly Morita–Rieffel equivalent toK C1 .c redH.jGj;D/Ì H/ Š K C1 c red H .jGj;D/H Š KK G C; IndG H .D/Š KK G C;C 0.G; D/ Ì H :(17)It is a routine exercise to verify that this composition agrees with the map in (13).Similar considerations apply to the map .We now set X DjGj, and let G D H . The actions of G on jGj on the left and rightare by translations and isometries, respectively. Lemma 7 yields a map‰ W K C1 .c red .jGj;D/˝max A/ Ì G ! KK G C;C 0.jGj;D˝max A/ Ì G : (18)

Coarse and equivariant co-assembly maps 85for all A; D, where we use the G-equivariant Morita-Rieffel equivalence betweenC 0 .jGj;D/Ì G and D. It fits into a commuting diagramK C1 .c redG .jGj;D/˝max A/ Ì G @K .C 0 .EG;D ˝max A/ Ì G/‰ D;AKK G .C;D˝max A/p EGŠ RKKG .EGI C;D˝max A/.Lemma 9. The class of G-C -algebras A for which ‰ D;A is an isomorphism for all Dis triangulated and thick and contains all G-C -algebras of the form C 0 .G=H / forcompact open subgroups H of G.Proof. The fact that this category of algebras is triangulated and thick means that it isclosed under suspensions, extensions, and direct summands. These formal propertiesare easy to check.Since H G is open, there is no difference between H -continuity and G-continuity.Hencec redG.jGj;D/˝max C 0 .G=H / Ì G Š c red .jGj;D/˝max C 0 .G=H / Ì GH c redH.jGj;D/Ì H;where means Morita-Rieffel equivalence. Similar simplifications can be made inother corners of the square. Hence the diagram for A D C 0 .G=H / and G acting on theright is equivalent to a corresponding diagram for trivial A and H acting on the right.The latter case is contained in Lemma 8.Theorem 10. Let G be an almost totally disconnected group with G-compact EG.Then for every B 2 KK G , the mapis an isomorphism and the diagram‰ W K topC1 G; cred .jGj;B/ ! KK G .C;B ˝ P/C1 G; cred .jGj;B/ KX G .jGj;B/K topŠ‰ KK G .C;B ˝ P/ pEG RKKG .EGI C;B ˝ P/commutes. In particular, K topC1 G; cred .jGj/ is naturally isomorphic to KK G .C; P/.Proof. It is shown in [7] that for such groups G, the algebra P belongs to the thick triangulatedsubcategory of KK G that is generated by C 0 .G=H / for compact subgroups Hof G. Hence the assertion follows from Lemma 9 and our definition of K top .Š

86 H. Emerson and R. MeyerCorollary 11. Let D 2 KK G .P; C/ be a Dirac morphism for G. Then the followingdiagram commutes:K topC1G; credŠ G .jGj/ @ D K .C 0 .EG;P/ Ì G/ K .C 0 .EG/ Ì G/‰ ŠKK G .C; P/ pEG RKK G D .EGI C; P/ Š RKKG .EGI C; C/D pEG.C; C/,KK G where ‰ is as in Theorem 10 and is the G-equivariant coarse co-assembly mapfor jGj, and the indicated maps are isomorphisms.Proof. This follows from Theorem 10 and the general properties of the Dirac morphismdiscussed in §2.1.Definition 12. We call a 2 RKK G .EGI C; C/(a) a boundary class if it lies in the range of W K topC1 G; cred .jGj/ ! RKK G .EGI C; C/I(b) properly factorisable if a D p EG .b ˝A c/ for some proper G-C -algebra A andsome b 2 KK G .C;A/, c 2 KKG .A; C/;(c) proper Lipschitz if a D p EG .b ˝C 0 .X/ c/, where b 2 KK G C;C 0.X// is constructedas in §3.1.1 and §3.1.2 and c 2 KK G .C 0.X/; C/ is arbitrary.Proposition 13. Let G be a totally disconnected group with G-compact EG.(a) A class a 2 RKK G .EGI C; C/ is properly factorisable if and only if it is a boundaryclass.(b) Proper Lipschitz classes are boundary classes.(c) The boundary classes form an ideal in the ring RKK G .EGI C; C/.(d) The class 1 EG is a boundary class if and only if G has a dual-Dirac morphism; inthis case, the G-equivariant coarse co-assembly map is an isomorphism.(e) Any boundary class lies in the range ofp EG W KKG .C; C/ ! RKKG .EGI C; C/and hence yields homotopy invariants for manifolds.ŠŠ

Coarse and equivariant co-assembly maps 87Proof. By Corollary 11, the equivariant coarse co-assembly map W K topC1 G; cred .jGj/ ! K C 0 .EG Ì G/ Š RKK G .EGI C; C/is equivalent to the mapKK G .C; P/ p EG! RKK G .EGI C; P/ Š RKKG .EGI C; C/:If we combine this with the isomorphism RKK G .EGI C; C/ Š KKG .P; P/, the resultingmapKK G .C; P/ ! KKG .P; P/is simply the product (on the left) with D 2 KK G .P; C/; this map is known to be anisomorphism if and only if it is surjective, if and only if 1 P is in its range, if and onlyif the H -equivariant coarse co-assembly mapK .c redH.jGj/ Ì H/ ! KX H .jGj/is an isomorphism for all compact subgroups H of G by [7].For any G-C -algebra B, the Z=2-graded group Ktop .G; B/ Š K .B ˝ P/ Ì r G is a graded module over the Z=2-graded ring KK G .P; P/ Š RKKG .EGI C; C/ ina canonical way; the isomorphism between these two groups is a ring isomorphismbecause it is the composite of the two ring isomorphismsKK G .P; P/ p EG!Š RKKG .EGI P; P/ ˝PŠRKKG .EGI C; C/:Hence we get module structures on K topG; c red .jGj/ andK topG; C 0 .EG;K/ Š K .C 0 .EG/ Ì G/ Š RKK G .EGI C; C/:The latter isomorphism is a module isomorphism; thus Ktop G; C 0 .EG;K/ is a freemodule of rank 1. The equivariant co-assembly map is natural in the formal sense, so thatit is an RKK G .EGI C; C/-module homomorphism. Hence its range is a submodule,that is, an ideal in RKK G .EGI C; C/ (since this ring is graded commutative, there isno difference between one- and two-sided graded ideals). This also yields that issurjective if and only if it is bijective, if and only if the unit class 1 EG belongs to itsrange; we already know this from [7].If A is a proper G-C -algebra, then id A ˝ D 2 KK G .A ˝ P;A/is invertible ([11]).If b 2 KK G .C;A/ and c 2 KKG .A; C/, then we can write the Kasparov productb ˝A c asC b ! A id A˝DŠA ˝ P c˝id P! P ! D C;where the arrows are morphisms in the category KK G . Therefore, b ˝A c factorsthrough D and hence is a boundary class by Theorem 10.

88 H. Emerson and R. Meyer5 Dual-Dirac morphisms and the Carlsson–Pedersen conditionNow we construct boundary classes in RKK G .EGI C; C/ from more classical boundaries.We suppose again that EG is G-compact, so that EG is a coarse space.A metrisable compactification of EG is a metrisable compact space Z with a homeomorphismbetween EG and a dense open subset of Z. It is called coarse if allscalar-valued functions on Z have vanishing variation; this implies the correspondingassertion for operator-valued functions because C.Z;D/ Š C.Z/˝ D. Equivalently,the embedding EG ! Z factors through the Higson compactification of EG. A compactificationis called G-equivariant if Z is a G-space and the embedding EG ! Zis G-equivariant. An equivariant compactification is called strongly contractible if itis H -equivariantly contractible for all compact subgroups H of G. The Carlsson–Pedersencondition requires that there should be a G-compact model for EG that has acoarse, strongly contractible, and G-equivariant compactification.Typical examples of such compactifications are the Gromov boundary for a hyperbolicgroup (viewed as a compactification of the Rips complex), or the visibilityboundary of a CAT(0) space on which G acts properly, isometrically, and cocompactly.Theorem 14. Let G be a locally compact group with a G-compact model for EGand let EG Z be a coarse, strongly contractible, G-equivariant compactification.Then G has a dual-Dirac morphism.Proof. We use the C -algebra xB redG.Z/ as defined in §2.3. Since Z is coarse, thereis an embedding xB redG.Z/ NcredG.EG/. Let @Z WD Z n EG be the boundary of thecompactification. Identifyingwe obtain a morphism of extensionsxB redG.@Z/ Š xB redG .Z/=C 0.EG;K G /;0 C 0 .EG;K G / NcredG .EG/ credG .EG/ 00 C 0 .EG;K G / xB redG.Z/ xB redG.@Z/ 0.Let H be a compact subgroup. Since Z is compact, we have xB G .Z/ D C.Z;K/.Since Z is H -equivariantly contractible by hypothesis, xB G .Z/ is H -equivariantlyhomotopy equivalent to C. Hence xB redG.Z/ has vanishing H -equivariant K-theory.This implies Ktop G; xB redG.Z/ D 0 by [2], so that the connecting mapK topC1 G; xB redG.@Z/ ! K top G; C 0 .EG/ Š K .C 0 .EG/ Ì G/ (19)is an isomorphism. This in turn implies that the connecting mapG; credG.EG/ ! Ktop G; C 0 .EG/ K topC1

Coarse and equivariant co-assembly maps 89is surjective. Thus we can lift 1 2 RKK G 0 .EGI C; C/ Š K 0.C 0 .EG/ Ì G/ to˛ 2 K top1G; c redG.EG/ Š K top1G; c redG .jGj/ :Then ‰ .˛/ 2 KK G 0 .C; P/ is the desired dual-Dirac morphism.The group Ktop G; xB redG.@Z/ that appears in the above argument is a reducedtopological G-equivariant K-theory for @Z and hence differs from K top G; C.@Z/ .The relationship between these two groups is analysed in [6].References[1] H. Abels, Parallelizability of proper actions, global K-slices and maximal compact subgroups,Math. Ann. 212 (1974/75), 1–19.[2] J. Chabert, S. Echterhoff, H. Oyono-Oyono, Going-down functors, the Künneth Formula,and the Baum-Connes conjecture, Geom. Funct. Anal. 14 (2004), 491–528.[3] A. Connes, A., M. Gromov, H. Moscovici, H. Group cohomology with Lipschitz controland higher signatures, Geom. Funct. Anal. 3 (1993), 1–78.[4] A. N. Dranishnikov, Lipschitz cohomology, Novikov’s conjecture, and expanders, Tr. Mat.Inst. Steklova 247 (2004), 59–73. translation: Proc. Steklov Inst. Math. 4 (247) (2004),50–63.[5] H. Emerson, R. Meyer, Dualizing the coarse assembly map, J. Inst. Math. Jussieu 5 (2006),161–186.[6] H. Emerson, R. Meyer, Euler characteristics and Gysin sequences for group actions onboundaries, Math. Ann. 334 (2006), 853–904.[7] H. Emerson, R. Meyer, A descent principle for the Dirac–dual-Dirac method, Topology 46(2007), 185–209.[8] S. C. Ferry, A. Ranicki, J. Rosenberg, Novikov conjectures, index theorems and rigidity.Vol. 1, London Math. Soc. Lecture Notes Ser. 226, Cambridge University Press, Cambridge1995.[9] N. Higson, Bivariant K-theory and the Novikov conjecture, Geom. Funct. Anal. 10 (2000),563–581.[10] G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91(1988), 147–201.[11] R. Meyer, R. Nest, The Baum–Connes conjecture via localisation of categories, Topology45 (2006), 209–259.

On K 1 of a Waldhausen categoryFernando Muro and Andrew Tonks IntroductionA general notion of K-theory, for a category W with cofibrations and weak equivalences,was defined by Waldhausen [10] as the homotopy groups of the loop space of a certainsimplicial category wS.W,K n W D n jwS.Wj Š nC1 jwS.Wj ; n 0:Waldhausen K-theory generalizes the K-theory of an exact category E, defined as thehomotopy groups of jQEj, where QE is a category defined by Quillen.Gillet and Grayson defined in [2] a simplicial set G.E which is a model for jQEj.This allows one to compute K 1 E as a fundamental group, K 1 E D 1 jG.Ej. Usingthe standard techniques for computing 1 Gillet and Grayson produced algebraic representativesfor arbitrary elements in K 1 E. These representatives were simplified bySherman [8], [9] and further simplified by Nenashev [5]. Nenashev’s representativesare pairs of short exact sequences on the same objects,A B C:Such a pair gives a loop in jG.Ej which corresponds to a 2-sphere in jwS.Ej, obtainedby pasting the 2-simplices associated to each short exact sequence along their commonboundary. BA C Figure 1. Nenashev’s representative of an element in K 1 E.Nenashev showed in [6] certain relations which are satisfied by these pairs of shortexact sequences, and he later proved in [7] that pairs of short exact sequences togetherwith these relations yield a presentation of K 1 E. The first author was partially supported by the MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the MEC postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research cantract; and thesecond author by the MEC-FEDER grant MTM2004-03629.

92 F. Muro and A. TonksIn this paper we produce representatives for all elements in K 1 of an arbitraryWaldhausen category W which are as close as possible to Nenashev’s representativesfor exact categories. They are given by pairs of cofiber sequences where the cofibrationshave the same source and target. The cofibers, however, may be non-isomorphic, butthey are weakly equivalent via a length 2 zig-zag.A C 1 B C: C 2This diagram is called a pair of weak cofiber sequences. It corresponds to a 2-sphere injwS.Wj obtained by pasting the 2-simplices associated to the cofiber sequences alongthe common part of the boundary. The two edges which remain free after this operationare filled with a disk made of two pieces corresponding to the weak equivalences. BA C 1 C C 2Figure 2. Our representative of an element in K 1 W.In addition we show in Section 3 that pairs of weak cofiber sequences satisfy acertain generalization of Nenashev’s relations.Let K1 wcs W be the group generated by the pairs of weak cofiber sequences in Wmodulo the relations in Section 3. The results of this paper show in particular that thereis a natural surjectionK1 wcs W K 1 Wwhich, by [7], is an isomorphism in case W D E is an exact category. See Section 6for a comparison of K1 wcs E with Nenashev’s presentation. We do not know whetherthis natural surjection is an isomorphism for an arbitrary Waldhausen category W,butwe prove here further evidence which supports the conjecture: the homomorphism W K 0 W ˝ Z=2 ! K 1 Winduced by the action of the Hopf map 2 s in the stable homotopy groups of spheresfactors asK 0 W ˝ Z=2 ! K1 wcs W K 1 W:We finish this introduction with some comments about the method of proof. We donot rely on any previous similar result because we do not know of any generalization of

On K 1 of a Waldhausen category 93the Gillet–Grayson construction for an arbitrary Waldhausen category W. The largestknown class over which the Gillet–Grayson construction works is the class of pseudoadditivecategories, which extends the class of exact categories but still does not coverall Waldhausen categories, see [3]. We use instead the algebraic model D W definedin [4] for the 1-type of the Waldhausen K-theory spectrum KW. The algebraic objectD W is a chain complex of non-abelian groups concentrated in dimensions i D 0; 1whose homology is K i W,K 1 W .D 0 W/ ab ˝ .D 0 W/ abh;i D 1 W@ D 0 W K 0 W;i.e. the lower row is exact. This non-abelian chain complex is equipped with a bilinearmap h; i which determines the commutators. This makes D W a stable quadraticmodule in the sense of [1]. This stable quadratic module is defined in [4] in terms ofgenerators and relations. Generators correspond simply to objects, weak equivalences,and cofiber sequences in W.Acknowledgement. The authors are very grateful to Grigory Garkusha for askingabout the relation between the algebraic model for Waldhausen K-theory defined in [4]and Nenashev’s presentation of K 1 of an exact category in [7].(A)1 The algebraic model for K 0 and K 1Definition 1.1. A stable quadratic module C is a diagram of group homomorphismsC0 ab ˝ C 0absuch that given c i ;d i 2 C i , i D 0; 1,1. @hc 0 ;d 0 iDŒd 0 ;c 0 ,2. h@.c 1 /; @.d 1 /iDŒd 1 ;c 1 ,h;i! C1@! C03. hc 0 ;d 0 iChd 0 ;c 0 iD0.Here Œx; y D x y C x C y is the commutator of two elements x;y 2 K in anygroup K, and K ab is the abelianization of K. It follows from the axioms that the imageof h; i and Ker @ are central in C 1 , the groups C 0 and C 1 have nilpotency class 2, and@.C 1 / is a normal subgroup of C 0 .A morphism f W C ! D of stable quadratic modules is given by group homomorphismsf i W C i ! D i , i D 0; 1, compatible with the structure homomorphisms ofC and D , i.e. f 0 @ D @f 1 and f 1 h; iDhf 0 ;f 0 i.Stable quadratic modules were introduced in [1, Definition IV.C.1]. Notice, however,that we adopt the opposite convention for the homomorphism h; i, i.e. we composewith the symmetry isomorphism on the tensor square.

94 F. Muro and A. TonksRemark 1.2. There is a natural right action of C 0 on C 1 defined byc c 01 D c 1 Chc 0 ; @.c 1 /i:The axioms of a stable quadratic module imply that commutators in C 0 act trivially onC 1 , and that C 0 acts trivially on the image of h; i and on Ker @.The action gives @W C 1 ! C 0 the structure of a crossed module. Indeed a stablequadratic module is the same as a commutative monoid in the category of crossedmodules such that the monoid product of two elements in C 0 vanishes when one ofthem is a commutator, see [4, Lemma 4.18].A stable quadratic module can be defined by generators and relations in degree 0and 1. In the appendix we give details about the construction of a stable quadraticmodule defined by generators and relations. This is useful to understand Definition 1.3below from a purely group-theoretic perspective.We assume the reader has certain familiarity with Waldhausen categories and relatedconcepts. We refer to [12] for the basics, see also [11]. The following definition wasintroduced in [4].Definition 1.3. Let C be a Waldhausen category with distinguished zero object , andcofibrations and weak equivalences denoted by and !, respectively. A genericcofiber sequence is denoted byA B B=A:We define D C as the stable quadratic module generated in dimension zero by thesymbols• ŒA for any object in C,and in dimension one by• ŒA ! A 0 for any weak equivalence,• ŒA B B=A for any cofiber sequence,such that the following relations hold.(R1) @ŒA ! A 0 DŒA 0 C ŒA.(R2) @ŒA B B=A DŒB C ŒB=A C ŒA.(R3) Œ D 0.(R4) ŒA 1 A! A D 0.(R5) ŒA 1 A! A D 0, Œ A 1 A! A D 0.

On K 1 of a Waldhausen category 95(R6) For any pair of composable weak equivalences A ! B ! C ,ŒA ! C D ŒB ! CC ŒA ! B:(R7) For any commutative diagram in C as followsA BB=A A 0 B0 B 0 =A 0we haveŒA ! A 0 C ŒB=A ! B 0 =A 0 ŒA D ŒA 0 B 0 B 0 =A 0 C ŒB ! B 0 C ŒA B B=A:(R8) For any commutative diagram consisting of four cofiber sequences in C as followsC=BB=A C=AA B Cwe haveŒB C C=BC ŒA B B=AD ŒA C C=AC ŒB=A C=A C=B ŒA :(R9) For any pair of objects A; B in CHerehŒA; ŒBi D ŒB i 2 A _ B p 1 A C ŒA i 1 A _ B p 2 B:A i 1 A _ Bp 1are the inclusions and projections of a coproduct in C.Remark 1.4. These relations are quite natural, and some illustration of their meaningis given in [4, Figure 2]. They are not however minimal: relation (R3) follows from(R2) and (R5), and (R4) follows from (R6). Also (R5) is equivalent toi 2p 2B

96 F. Muro and A. Tonks(R5 0 ) Œ D 0.This equivalence follows from (R8) on considering the diagrams A 1 A A 1 AA,A1 A A1 A A.We now introduce the basic object of study of this paper.Definition 1.5. A weak cofiber sequence in a Waldhausen category C is a diagramA B C 1Cgiven by a cofiber sequence followed by a weak equivalence in the opposite direction.We associate the elementŒA B C 1C D ŒA B C1 C ŒC ! C 1 ŒA 2 D C (1.1)to any weak cofiber sequence. By (R1) and (R2) we have@ŒA B C 1C D ŒB C ŒC C ŒA:The following fundamental identity for weak cofiber sequences is a slightly lesstrivial consequence of the relations (R1)–(R9).Proposition 1.6. Consider a commutative diagramA0 j AAr AA 000w AA 00j 0 B 0 j BjBr Bj 000 B 000w Bj 00B 00 r 00r 0 D 0 w 0 C 0 rr 000 w 000j D rD D D000 D 00j Cw Cr CC 000 w Dw 00 C 00w C

On K 1 of a Waldhausen category 97such that the induced map j [ A 0 j B W A [ A 0 B 0 B is a cofibration. Denote thehorizontal weak cofiber sequences by l A , l B , l D and l C , and the vertical ones by l 0 , l,l 000 and l 00 . Then the following equation holds in D C.Œl A Œl C ŒA Œl C Œl B C Œl 00 ŒB0 C Œl 0 DhŒC 0 ; ŒA 00 i:Proof. Applying (R7) to the two obvious equivalences of cofiber sequences givesŒD 0 D D 000 D Œw C Œl C Œw C ŒC 0 Œw 000 ŒC 0 Œw 0 ;(a)ŒA 000 B 000 D 000 D Œw B C Œl 00 Œw 00 ŒA00 Œw D ŒA00 Œw A ;(b)using the notation of (1.1).Now consider the following commutative diagrams:A 0 j AA A 0 j AA j 0 B 0 push X jj 0 B 0 push X j BB,i 2 r 0i 1 r A A 000 _ D 0 .We now have four commutative diagrams of cofiber sequences as in (R8).c) D0d) A000A 000 A 000 _ D0A 0 A Xe) D 000D 0 DA X BD0 A 000 _ D0A 0 B0 Xf) D000A 000 B000B 0 X B

98 F. Muro and A. TonksTherefore we obtain the corresponding relationsŒA X D 0 C Œl A Œw A ŒA0 D ŒA 0 X A 000 _ D 0 C ŒA 000 A 000 _ D 0 D 0 ŒA0 ;ŒB 0 X A 000 C Œl 0 Œw 0 ŒA0 D ŒA 0 X A 000 _ D 0 C ŒD 0 A 000 _ D 0 A 000 ŒA0 ;ŒX B D 000 C ŒA X D 0 D Œl Œw ŒA C ŒD 0 D D 000 ŒAD Œl C .Œl C Œw C ŒC 0 Œw 000 ŒC 0 Œw 0 / ŒA ;ŒX B D 000 C ŒB 0 X A 000 D Œl B Œw B ŒB0 C ŒA 000 B 000 D 000 ŒB0 D Œl B C .Œl 00 Œw 00 ŒA00 Œw D ŒA00 Œw A / ŒB0 :(c)(d)(e)(f)Here we have used equations (a) and (b). Now .c/ C .d/ and .e/ C .f/ yieldŒw A ŒA0 Œl A ŒA X D 0 C ŒB 0 X A 000 C Œl 0 Œw 0 ŒA0 D ŒA 000 A 000 _ D 0 D 0 ŒA0 C ŒD 0 A 000 _ D 0 A 000 ŒA0 DhŒD 0 ; ŒA 000 i;(g)ŒA X D 0 C ŒB 0 X A 000 D Œw 0 ŒA C a ŒC 0 CŒAŒl C ŒA Œl C Œl B C Œl 00 ŒB0 a ŒA00 CŒB 0 (h)Œw A ŒB0 ;where we use (R9) and Remark 1.2, and we write a D Œw 000 C Œw C D Œw D C Œw 00 ,by (R6). In fact it is easy to check by using Remark 1.2 and the laws of stable quadraticmodules that the terms a ŒC 0 CŒA and a ŒA00 CŒB 0 cancel in this expression, since@ Œl C ŒA Œl C Œl B C Œl 00 ŒB0 D .ŒC 0 C ŒA/ C .ŒA 00 C ŒB 0 /:Substituting (h) into the left hand side of (g) we obtainŒw A ŒA0 Œl A C Œw 0 ŒA Œl C ŒA Œl C Œl B C Œl 00 ŒB0 DhŒD 0 ; ŒA 000 i;Œw A ŒB0 C Œl 0 Œw 0 ŒA0 which, using again Definition 1.1 and Remark 1.2, may be rewritten asŒl A Œl C ŒA Œl C Œl B C Œl 00 ŒB0 C Œl 0 D Œw 0 ŒA00 CŒA 0 Œw A ŒA0 hŒA 000 ; ŒD 0 iCŒw 0 ŒA0 C Œw A ŒC 0 CŒA 0 :The result then follows from the identityh@Œw A ; @Œw 0 ihŒA 00 ; @Œw 0 iChŒC 0 ; @Œw A iDhŒC 0 ; ŒA 00 iChŒA 000 ; ŒD 0 i:

On K 1 of a Waldhausen category 992 Pairs of weak cofiber sequences and K 1Let C be a Waldhausen category. A pair of weak cofiber sequences is a diagram givenby two weak cofiber sequences with the first, second, and fourth objects in common:A C 1 B C C 2 . We associate to any such pair the following element89r C ˆ= B1 w Cj ˆ: 2 2D ŒA j 1 B r 1 Cr 2 1 CC ŒA j 2 B r 2 C 2 Cw 1 w 2 >;C 2D ŒA j 2 B r 2 C 2 C ŒA j 1 B r 1 C 1 C;w 2 w 1(2.1)which lies in K 1 C D Ker @, see diagram (A) in the introduction. For the secondequality in (2.1) we use that Ker @ is central, see Remark 1.2, so we can permute theterms cyclically.The following theorem is one of the main results of this paper.Theorem 2.1. Any element in K 1 C is represented by a pair of weak cofiber sequences.This theorem will be proved later in this paper. We first give a set of useful relationsbetween pairs of weak cofiber sequences and develop a sum-normalized version of themodel D C.3 Relations between pairs of weak cofiber sequencesSuppose we have six pairs of weak cofiber sequences in a Waldhausen category CA 0A 0A 001 w1Ajr A 1A 1 A A00 ; j A 2r A 2A 002w A 2C1 0 w10 j r 0 10 1 B 0 Cj20 0 ; r 0 2C 0 2w 0 2B 0j B 1j B 2B1 00 w1Br1B B B00 ; r B 2B 002w B 2r 1C 1 w 1A j 1 B C;j 2r 2 wC22C 0j C 1j C 2LC1 00 w1C r1C C C 00 ; r C 2LC200OC 00w C 21 w 100r jA 00 00 1 001 B 00 Cj200 00 ; r 002OC 002w 002

100 F. Muro and A. Tonksthat we denote for the sake of simplicity as A , B , C , 0 , , and 00 , respectively.Moreover, assume that for i D 1; 2 there exists a commutative diagramA0 j A iA r A iA 00 iwiAA 00j 0 iB 0 j B ij iBr B iL| iBi00w B ij 00ir 0 iCi 0 r i Lr i O| i Or iC iCi00 Ow iB 00 r 00iOCi00w 0 iw iLw iw 00iC 0 j C iCr C iLC 00iw C iC 00such that the induced map j i [ A 0 jiB W A[ A 0 B 0 B is a cofibration. Notice that six ofthe eight weak cofiber sequences in this diagram are already in the six diagrams above.The other two weak cofiber sequences are just assumed to exist with the property ofmaking the diagram commutative.Theorem 3.1. In the situation above the following equation holds˚A ˚ B C ˚ C D ˚ 0 fg C ˚ 00 : (S1)Proof. For i D 1; 2 let A i , B i , C i , 0 i , i, and 00ibe the upper and the lower weakcofiber sequences of the pairs above, respectively. By Proposition 1.6Œ A 1 ŒC 1 ŒA Œ 1 C Œ B 1 C Œ00 1 ŒB0 C Œ 0 1 DhŒC 0 ; ŒA 00 iD Œ A 2 ŒC 2 ŒA Œ 2 C Œ B 2 C Œ00 2 ŒB0 C Œ 0 2 :Since @. C 1 / D @.C 2/ and @.001 / D @.00 2/ the actions cancel,Œ A 1 ŒC 1 Œ 1CŒ B 1 CŒ00 1 CŒ0 1 D ŒA 2 ŒC 2 Œ 2CŒ B 2 CŒ00 2 CŒ0 2 :Now one can rearrange the terms in this equation, by using that pairs of weak cofibersequences are central in D 1 C, obtaining the equation in the statement.Theorem 3.1 encodes the most relevant relation satisfied by pairs of weak cofibersequences in K 1 C. They satisfy a further relation which follows from the very definitionin (2.1), but which we would like to record as a proposition by analogy with [6].

On K 1 of a Waldhausen category 101Proposition 3.2. A pair of weak cofiber sequences given by two times the same weakcofiber sequence is trivial.89C ˆ= B jˆ: C D 0:(S2)wr >;C 1We establish a further useful relation in the following proposition.Proposition 3.3. Relations (S1) and (S2) imply that the sum of two pairs of weakcofiber sequences coincides with their coproduct.89 89C 1 _ NC 1ˆ< CA _ AN >= ˆ= B Cˆ: >;ˆ: >;C 2 _ NC 2C 289ˆ< NC 1 C AN >= NB NC :ˆ: >; NC 2Proof. Apply relations (S1) and (S2) to the following pairs of weak cofiber sequencesA BC 1 C; C 2AN NB NC 1 NC 2NC;A _ ANC 1 _ NC 1 B _ NB C _ NC; C 2 _ NC 2p 2 AN 1 A i 1A _ ANA; Ni1p 2AN 1p 2 B i 1B _ NB i 1 p 2NB 1 NB 1NB;p 2 NC 1 C i 1C _ NCNC:i 1p 2 NC 14 Waldhausen categories with functorial coproductsWaldhausen categories admit finite coproducts A _ B. The operation _ need not bestrictly associative and unital. However, as we check below, any Waldhausen categorycan be replaced by an equivalent one with strict coproducts.

102 F. Muro and A. TonksWe now give an explicit definition of what we understand by a strict coproductfunctor. Let C be a Waldhausen category endowed with a symmetric monoidal structureC which is strictly associative.A C B/ C C D A C .B C C/;strictly unital with unit object , the distinguished zero object,CA D A D A C;but not necessarily strictly commutative, such thatA D A C !A C B CB D Bis always a coproduct diagram. Such a category will be called a Waldhausen categorywith a functorial coproduct. Then we define the sum-normalized stable quadraticmodule D CC as the quotient of D C by the extra relation(R10) ŒB i 2 A C B p 1 A D 0.Remark 4.1. In D C C, the relations (R2) and (R10) imply thatŒA C B D ŒA C ŒB:Furthermore, relation (R9) becomes equivalent to(R9 0 ) hŒA; ŒBi DŒ B;A W B C A Š ! A C B.Here B;A is the symmetry isomorphism of the symmetric monoidal structure. Thisequivalence follows from the commutative diagramA A i 2 p 1B C A B B;AŠi 1 A C B p2 Btogether with (R4), (R7) and (R10).Theorem 4.2. Let C be a Waldhausen category with a functorial coproduct such thatthere exists a set S which freely generates the monoid of objects of C under the operationC. Then the projectionp W D C D C Cis a weak equivalence. Indeed, it is part of a strong deformation retraction.We prove this theorem later in this section.Waldhausen categories satisfying the hypothesis of Theorem 4.2 are general enoughas the following proposition shows.

On K 1 of a Waldhausen category 103Proposition 4.3. For anyWaldhausen category C there is anotherWaldhausen categorySum.C/ with a functorial coproduct whose monoid of objects is freely generated bythe objects of C except from . Moreover, there are natural mutually inverse exactequivalences of categories'Sum.C/ C .Proof. The statement already says which are the objects of Sum.C/. The functor 'sends an object Y in Sum.C/, which can be uniquely written as a formal sum of nonzeroobjects in C, Y D X 1 CCX n , to an arbitrarily chosen coproduct of theseobjects in C'.Y / D X 1 __X n :For n D 0; 1 we make special choices, namely for n D 0, '.Y / D, and for n D 1we set '.Y / D X 1 . Morphisms in Sum.C/ are defined in the unique possible way sothat ' is fully faithful. Then the formal sum defines a functorial coproduct on Sum.C/.The functor sends to the zero object of the free monoid and any other object in Cto the corresponding object with a single summand in Sum.C/, so that ' D 1. Theinverse natural isomorphism 1 Š ',X 1 CCX n Š .X 1 __X n /;is the unique isomorphism in Sum.C/ which ' maps to the identity. Cofibrationsand weak equivalences in Sum.C/ are the morphisms which ' maps to cofibrationsand weak equivalences, respectively. This Waldhausen category structure in Sum.C/makes the functors ' and exact.Let us recall the notion of homotopy in the category of stable quadratic modules.Definition 4.4. Given morphisms f; g W C ! C 0 of stable quadratic modules, ahomotopy f ' ˛g from f to g is a function ˛ W C 0 ! C1 0 satisfying1. ˛.c 0 C d 0 / D ˛.c 0 / f 0.d 0 / C ˛.d 0 /,2. g 0 .c 0 / D f 0 .c 0 / C @˛.c 0 /,3. g 1 .c 1 / D f 1 .c 1 / C ˛@.c 1 /.The following lemma then follows from the laws of stable quadratic modules.Lemma 4.5. A homotopy ˛ as in Definition 4.4 satisfies˛.Œc 0 ;d 0 / D hf 0 .d 0 /; f 0 .c 0 /iChg 0 .d 0 /; g 0 .c 0 /i; (a)˛.c 0 / C g 1 .c 1 / D f 1 .c 1 / C ˛.c 0 C @.c 1 //:(b)Now we are ready to prove Theorem 4.2.

104 F. Muro and A. TonksProof of Theorem 4.2. In order to define a strong deformation retraction,˛ D C pj DC C,1 ˛' jp; pj D 1;the crucial step will be to define the homotopy ˛ W D 0 C ! D 1 C on the generators ŒAof D 0 C. Then one can use the equations1. ˛.c 0 C d 0 / D ˛.c 0 / d 0C ˛.d 0 /,2. jp.c 0 / D c 0 C @˛.c 0 /,3. jp.c 1 / D c 1 C ˛@.c 1 /,for c i ;d i 2 D i C to define the composite jp and the homotopy ˛ on all of D C.Itisa straightforward calculation to check that the map jp so defined is a homomorphismand that ˛ is well defined. In particular, Lemma 4.5 (a) implies that ˛ vanishes oncommutators of length 3. In order to define j we must show that jp factors throughthe projection p, that is,jp.ŒB i 2 A C B p 1 A/ D 0:If we also show thatp˛ D 0then equations (2) and (3) above say pjp.c i / D p.c i / C 0, and since p is surjective itfollows that the composite pj is the identity.The set of objects of C is the free monoid generated by objects S 2 S. Thereforewe can define inductively the homotopy ˛ by˛.ŒS C B/ D ŒB i 2 S C B p 1 SC ˛.ŒB/for B 2 C and S 2 S, with ˛.ŒS/ D 0. We claim that a similar relation then holds forall objects A, B of C,˛.ŒA C B/ D ŒB i 2 A C B p 1 A C ˛.ŒA C ŒB/ (4.1)D ŒB i 2 A C B p 1 A C ˛.ŒA/ ŒB C ˛.ŒB/:If A Dor A 2 S then (4.1) holds by definition, so we assume inductively it holds forgiven A, B and show it holds for S C A, B also:˛.ŒS C A C B/ D ŒA C B S C A C B SC ˛.ŒA C B/D ŒA C B S C A C B SC ŒB A C B A C ˛.ŒA/ ŒB C ˛.ŒB/D ŒB S C A C B S C A C ŒA S C A S ŒB C ˛.ŒA/ ŒB C ˛.ŒB/D ŒB S C A C B S C A C ˛.ŒS C A/ ŒB C ˛.ŒB/:

On K 1 of a Waldhausen category 105Here we have used (R8) for the composable cofibrationsB A C B S C A C B:It is clear from the definition of ˛ that the relation p˛ D 0 holds. It remains to see thatjp factors through D C C, that is,jp.c 1 / D 0if c 1 D ŒB A C B A:Let c 0 D ŒA C B so that c 0 C @.c 1 / D ŒA C ŒB. Then by Lemma 4.5 (b) we havejp.c 1 / D ˛.c 0 / C c 1 C ˛.c 0 C @.c 1 //D ˛.ŒA C B/ C ŒB A C B A C ˛.ŒA C ŒB/:This is zero by (4.1).We finish this section with four lemmas which show useful relations in D CC.Lemma 4.6. The following equality holds in D CC.ŒA C A 0 B C B 0 B=A C B 0 =A 0 C C C 0 D ŒA B B=A C ŒB0 C ŒA 0 B 0 B 0 =A 0C 0 ChŒA; ŒC 0 i:Proof. Use (R4), (R10) and Proposition 1.6 applied to the commutative diagramA0 B 0 B 0 =A0 C 0A C A 0 B C B 0B=A C B 0 =A 0 C C C 0A B B=A CAs special cases we haveA B B=A C .Lemma 4.7. The following equality holds in D C C.ŒA C A 0 B C B 0 B=A C B 0 =A 0 D ŒA B B=A ŒB0 C ŒA 0 B 0 B 0 =A 0 ChŒA; ŒB 0 =A 0 i:Lemma 4.8. The following equality holds in D C C.ŒA C B ! A 0 C B 0 D ŒA ! A 0 ŒB0 C ŒB ! B 0 :

106 F. Muro and A. TonksWe generalize (R9 0 ) in the following lemma.Lemma 4.9. Let C be a Waldhausen category with a functorial coproduct and letA 1 ;:::;A n be objects in C. Given a permutation 2 Sym.n/ of n elements we denotebyŠ A1 ;:::;A nW A 1 CCA n ! A1 CCA nthe isomorphism permuting the factors of the coproduct. Then the following formulaholds in D C C:Œ A1 ;:::;A n D X i>j i < jhŒA i ; ŒA j i:Proof. The result holds for n D 1 by (R4). Suppose 2 Sym.n/ for n 2, and notethat the isomorphism A1 ;:::;A nfactors naturally as. 0 A 1 ;:::;A n 1C 1/.1 C An ;B/W A 1 CCA n ! A 01CCA 0n 1C A n! A 1 CCA n 1 C A nwhere . 0 1 ;:::;0 n 1 ;n/ D . 1;:::; b k ;:::; n ; k / and B D A kC1 C C A n .Therefore by (R4), (R6) and Lemma 4.8,Œ A1 ;:::;A n D ŒA 0 1 ;:::;A n 1C 1 C Œ1 C An ;BD .ŒA 0 1 ;:::;A n 1 ŒAn C 0/ C .0 C Œ An ;B/:By induction, (R9 0 ) and Remark 4.1 this is equal toXhŒA 0 p; ŒA 0 qiChŒB; ŒA n iDX hŒA i ; ŒA j iCi>j i < j q 0 p < 0 qas required.5 Proof of Theorem 2.1X i>j i < j DnhŒA i ; ŒA j iThe morphism of stable quadratic modules D F W D C ! D D induced by an exactfunctor F W C ! D, see [4], takes pairs of weak cofiber sequences to pairs of weakcofiber sequences,8ˆ= ˆ;C 2ˆ:F .A/ F.B/9F.C 1 / >=F.C/ : F.C 2 />;By [4, Theorem 3.2] exact equivalences of Waldhausen categories induce homotopyequivalences of stable quadratic modules, and hence isomorphisms in K 1 . Therefore

On K 1 of a Waldhausen category 107if the theorem holds for a certain Waldhausen category then it holds for all equivalentones. In particular by Proposition 4.3 it is enough to prove the theorem for any Waldhausencategory C with a functorial coproduct such that the monoid of objects is freelygenerated by a set S. By Theorem 4.2 we can work in the sum-normalized constructionD C C in this case.Any element x 2 D C 1C is a sum of weak equivalences and cofiber sequences withcoefficients C1 or 1. By Lemma 4.8 modulo the image of h; i we can collect onthe one hand all weak equivalences with coefficient C1 and on the other all weakequivalences with coefficient 1. Moreover, by Lemma 4.7 we can do the same forcofiber sequences. Therefore the following equation holds modulo the image of h; i.x D ŒV 1! V2 ŒX 1 X 2 X 3 C ŒY 1 Y 2 Y 3 C ŒW 1! W2 (R4), (R10), 4.8, 4.7 DŒV 1 C W 1 ! V2 C W 1 ŒX 1 C Y 1 X 2 C Y 3 C Y 1 X 3 C Y 3 C ŒX 1 C Y 1 X 3 C X 1 C Y 2 X 3 C Y 3 C ŒV 1 C W 1 ! V1 C W 2 renaming D ŒL ! L 1 ŒA E 1 DC ŒA E 2 D C ŒL ! L 2 mod h; i:Suppose that @.x/ D 0 modulo commutators. Then modulo Œ; 0 D ŒL C ŒL 1 ŒA ŒD C ŒE 1 ŒE 2 C ŒD C ŒA ŒL 2 C ŒLD ŒL 1 C ŒE 1 ŒE 2 ŒL 2 mod Œ; ;i.e. ŒL 1 C E 1 D ŒL 2 C E 2 mod Œ; :(5.1)The quotient of D C 0C by the commutator subgroup is the free abelian group on S,hence by (5.1) there are objects S 1 ;:::;S n 2 S and a permutation 2 Sym.n/ suchthatL 1 C E 1 D S 1 CCS n ;L 2 C E 2 D S 1 CCS n ;so there is an isomorphism S1 ;:::;S nW L 2 C E 2 ! L1 C E 1 :Again by Lemmas 4.8 and 4.7 modulo the image of h; ix D ŒL C D ! L 1 C D ŒA L 1 C E 1 L 1 C DC ŒA L 2 C E 2 L 2 C D C ŒL C D ! L 2 C D4.9 D ŒL C D ! L 1 C D ŒA L 1 C E 1 L 1 C D C

108 F. Muro and A. TonksC Œ S1 ;:::;S nW L 2 C E 2! L1 C E 1 C ŒA L 2 C E 2 L 2 C D C ŒL C D ! L 2 C D(R7) D ŒL C D ! L 1 C D ŒA L 1 C E 1 L 1 C DC ŒA L 1 C E 1 L 2 C D C ŒL C D ! L 2 C Drenaming D ŒC ! C 1 ŒA B C 1 C ŒA B C 2 C ŒC ! C 2 D ŒC ! C 1 ŒA ŒA B C 1 C ŒA B C 2 C ŒC ! C 2 ŒAD ŒA B C 1CC ŒA B C2C mod h; i;i.e. there is y 2 D C 1C in the image of h; i such that8C ˆ=C y:>;Now assume that @.x/ D 0. Since the pair of weak cofiber sequences is also inthe kernel of @ we have @.y/ D 0. In order to give the next step we need a technicallemma.Lemma 5.1. Let C be a stable quadratic module such that C 0 is a free group ofnilpotency class 2. Then any element y 2 Ker @ \ Imageh; iis of the form y Dha; aifor some a 2 C 0 .Proof. For any abelian group A let y˝2Abe the quotient of the tensor square A ˝ A bythe relations a ˝ b C b ˝ a D 0, a; b 2 A, and let ^2A be the quotient of A ˝ A bythe relations a ˝ a D 0, a 2 A, which is also a quotient of y˝2A.The projection ofa ˝ b 2 A ˝ A to y˝2Aand ^2A is denoted by a y˝b and a ^ b, respectively.There is a commutative diagram of group homomorphismsC0 ab ˝ C 0ab 0 ˝ Z=2 N y˝2C ab q^2C 00abh;i c 1c 0 @C 1 C 0C abwhere N.a ˝ 1/ D a y˝ a, the factorization c 1 of h; i is given by c 1 .a y˝ b/ Dha; bi,which is well defined by Definition 1.1 (3), and c 0 .a ^ b/ D Œb; a is well known to beinjective in the case C 0 is free of nilpotency class 2.

On K 1 of a Waldhausen category 109Moreover, C0 ab is a free abelian group, hence the middle row is a short exact sequence,see [1, I.4]. Therefore any element y 2 C 1 which is both in the image of c 1and the kernel of @ is in the image of c 1 N as required.The group D C 0C is free of nilpotency class 2, therefore by the previous lemmay Dha; ai for some a 2 D C 0C. By the laws of a stable quadratic module the elementha; ai only depends on a mod 2, compare [4, Definition 1.8], and so we can supposethat a is a sum of basis elements with coefficient C1. Hence by Remark 4.1 we cantake a D ŒM for some object M in C,y DhŒM ; ŒM i:The element hŒM ; ŒM i is itself a pair of weak cofiber sequences89p ˆ=M C M1 M ;i ˆ: 1p 2 M >;therefore x is also a pair of weak cofiber sequences, by Proposition 3.3, and the proofof Theorem 2.1 is complete.6 Comparison with Nenashev’s approach for exact categoriesFor any Waldhausen category C we denote by K wcs1C the abelian group generated bypairs of weak cofiber sequences8ˆ=B C >;C 2modulo the relations (S1) and (S2) in Section 3. Theorem 3.1 and Proposition 3.2 showthe existence of a natural homomorphismK1 wcs C K 1 C (6.1)which is surjective by Theorem 2.1.Given an exact category E Nenashev defines in [6] an abelian group D.E/ bygenerators and relations which surjects naturally to K 1 E. Moreover, he shows in [7]that this natural surjection is indeed a natural isomorphismGenerators of D.E/ are pairs of short exact sequences´μA j 1 r 1 B C :j 2 r 2D.E/ Š K 1 E: (6.2)

110 F. Muro and A. TonksThey satisfy two kind of relations. The first, analogous to (S2), says that a generatorvanishes provided j 1 D j 2 and r 1 D r 2 . The second is a simplification of (S1): givensix pairs of short exact sequencesA 0A 0j A 1 rA 1 A A 00 ; B 0 j A 2j 0 2r A 2 j 0 1 r0 1B 0 C 0 ;r 0 2j B 1j B 2A j 1 Bj 2rB 1B B 00 ; C 0 r B 2j C 1j C 2rC 1C C 00 ;r C 2r 1j C; A 00 001 r001 B 00 C 00 ;r 2denoted for simplicity as A , B , C , 0 , , and 00 , such that the diagramj 002r 002A 0 j iAj 0 iB 0 j B ir 0i C 0 j C iA j ir A i A00j 00iB rB iB 0 ir i C r C C00ir 00icommutes for i D 1; 2 then˚A ˚ B C ˚ C D ˚ 0 fg C ˚ 00 :Proposition 6.1. For any exact category E there is a natural isomorphismD.E/ Š K1 wcs E:Proof. There is a clear homomorphism D.E/ ! K wcs´A j 1 Bj 2μr 1Cr 21E defined by89r ˆ= B Cj ˆ: 2r 2 C 1 : >;Exact categories regarded as Waldhausen categories have the particular feature thatall weak equivalences are isomorphisms, hence one can also define a homomorphismK wcs1E ! D.E/ as8ˆ=w 1 w C 2r 2 Š>;C 27!´A j 1 Bj 2w1 1r1w 12 r 2μC :We leave the reader to check the compatibility with the relations and the fact that thesetwo homomorphisms are inverse of each other.

On K 1 of a Waldhausen category 111Remark 6.2. The composite of the isomorphism in Proposition 6.1 with the naturalepimorphism (6.1) for C D E coincides with Nenashev’s isomorphism (6.2), thereforethe natural epimorphism (6.1) is an isomorphism when C D E is an exact category.7 Weak cofiber sequences and the stable Hopf mapAfter the previous section one could conjecture that the natural epimorphismK wcs1 C K 1 Cin (6.1) is an isomorphism not only for exact categories but for any Waldhausen categoryC. In order to support this conjecture we show the following result.Theorem 7.1. For any Waldhausen category C there is a natural homomorphism W K 0 C ˝ Z=2! K wcs1 Cwhich composed with (6.1) yields the homomorphism W K 0 C ˝ Z=2 ! K 1 C determinedby the action of the stable Hopf map 2 s in the stable homotopy groups ofspheres.For the proof we use the following lemma.Lemma 7.2. The following relation holds in K wcs8ˆ< ˆ:1C9 89A 00 w1 0 w 11 A >= ˆ= A 0 1A 00 w>; ˆ: A2 0 w 21A 0 w 2>;8A ˆ=:>; ; A 00 w1 0 w 11 A 00 A;1A 00 w2 0 w 2A 0 w 1 1 A 0 A;1A 0 w 2 ; A 00 w101 A 00 A0 ;1A 00 w 0 2A 1 1 A A: 1A 1

112 F. Muro and A. TonksNow we are ready to prove Theorem 7.1.Proof of Theorem 7.1. We define the homomorphism by89 8p ˆ= ˆ; ˆ: A _ A1A _ A A;AHere the second equality follows from (S1) and (S2) applied to the following pairs ofweak cofiber sequences9>=:>; A 11 A A;1A 1 ; A _ A 1 1 A _ A A _ A;1A _ A A;Ap 1 A 1 A i 2 A _ A A;i 1 p 2A 1A 1 1 A A;1A 1p 1 A 1 A i 2 A _ A A:i 2 p 1A 1Given a cofiber sequence A j B r C the equation ŒCCŒA D ŒB followsfrom (S1) and (S2) applied to the following pairs of weak cofiber sequencesp 1 A 1 A i 2 A _ A A;i 1 p 2A 1r C 1 A j B j C;rC 1A _ A p 1 B 1 B i 2 B _ B B;i 1 j _j j _jp 2B 1r_r C _ C 1 B _ B C _ C;r_rC _ C 1p 1 C 1 C i 2 C _ C C;i 1 p 2C 1r C 1 A j B j C:rC 1Given a weak equivalence w W A ! A 0 the equation ŒA D ŒA 0 follows from(S1) and (S2) applied to the following pairs of weak cofiber sequences ; A 0 w 1 A 0 A;1A 0 w A 0A 0 1 i p 1 2 A i 0 _ A 0 A10 ; p 2A 0 1A 0 _ A 01 w_w A 0 _ A 0 A _ A;1A 0 _ A 0 w_wp 1 A 1 A i 2 A _ A A;i 1 p 2A 1A 0 w 1 A 0 A:1A 0 w

On K 1 of a Waldhausen category 113The equation 2ŒA D 0 follows from (S2) and Lemma 7.2 by using the secondformula for ŒA above and the fact that 2 A;A D 1.We have already shown that is a well-defined homomorphism. The composite of and (6.1) coincides with the action of the stable Hopf map by the main result of [4].Appendix. Free stable quadratic modules and presentationsFree stable quadratic modules, and also stable quadratic modules defined by generatorsand relations, can be characterized up to isomorphism by obvious universal properties.In this appendix we give more explicit constructions of these notions.Let squad be the category of stable quadratic modules and letU W squad! Set Setbe the forgetful functor, U.C / D .C 0 ;C 1 /. The functor U has a left adjoint F , and astable quadratic module F.E 0 ;E 1 / is called a free stable quadratic module on the setsE 0 and E 1 . In order to give an explicit description of F.E 0 ;E 1 / we fix some notation.Given a set E we denote by hEi the free group with basis E, and by hEi ab the freeabelian group with basis E. The free group of nilpotency class 2 with basis E, denotedby hEi nil , is the quotient of hEi by triple commutators. Given a pair of sets E 0 andE 1 , we write E 0 [ @E 1 for the set whose elements are the symbols e 0 and @e 1 for eache 0 2 E 0 , e 1 2 E 1 .To define F.E 0 ;E 1 /, consider the groupsF.E 0 ;E 1 / 0 DhE 0 [ @E 1 i nil ;F.E 0 ;E 1 / 1 D .hE 0 i ab ˝ Z=2/ Ker ı:Here ı W F.E 0 ;E 1 / 0 !hE 0 i ab is the homomorphism given by ıe 0 D e 0 and ı@e 1 D 0.In the notation of the proof of lemma 5.1 there are isomorphismsKer ı Š^2hE0 i ab hE 0 E 1 i ab hE 1 i nil ;.hE 0 i ab ˝ Z=2/ Ker ı Š Ő 2 hE 0 i ab hE 0 E 1 i ab hE 1 i nil ;and intuitively we think of F.E 0 ;E 1 / 1 as a group generated by symbols he 0 ;e0 0 i,he 0 ;@e 1 i and e 1 . The symbol h@e 1 ;@e1 0 i is unnecessary since it will be given by thecommutator Œe1 0 ;e 1.We define structure homomorphisms on F.E 0 ;E 1 / as follows. The boundary@W F.E 0 ;E 1 / 1 ! F.E 0 ;E 1 / 0is the projection onto Ker ı followed by the inclusion of the kernel. The bracketh; iW F.E 0 ;E 1 / ab0 ˝ F.E 0;E 1 / ab0 ! F.E 0 ;E 1 / 1

114 F. Muro and A. Tonksis given by the product of the following two homomorphisms,c 0 W F.E 0 ;E 1 / ab0 ˝ F.E 0;E 1 / ab0 ! hE 0 i ab ˝ Z=2;defined on the generators x;y 2 E 0 [ @E 1 by c 0 .x; y/ D x ˝ 1 if x D y 2 E 0 andc 0 .x; y/ D 0 otherwise, andc W F.E 0 ;E 1 / ab0 ˝ F.E 0;E 1 / ab0 ! Ker ıinduced by the commutator bracket, c.a;b/ D Œb; a.It is now straightforward to define explicitly the stable quadratic module C presentedby generators E i and relations R i F.E 0 ;E 1 / i in degrees i D 0; 1, byC 0 D F.E 0 ;E 1 / 0 =N 0 ;C 1 D F.E 0 ;E 1 / 1 =N 1 :Here N 0 F.E 0 ;E 1 / 0 is the normal subgroup generated by the elements of R 0 and@R 1 , and N 1 F.E 0 ;E 1 / 1 is the normal subgroup generated by the elements ofR 1 and hF.E 0 ;E 1 / 0 ;N 0 i. The boundary and bracket on F.E 0 ;E 1 / induce a stablequadratic module structure on C which satisfies the following universal property:given a stable quadratic module C 0 , any pair of functions E i ! Ci 0 .i D 0; 1/ suchthat the induced morphism F.E 0 ;E 1 / ! C 0 annihilates R 0 and R 1 induces a uniquemorphism C ! C 0 .In [1] Baues considers the totally free stable quadratic module C with basis givenby a function g W E 1 !hE 0 i nil . In the language of this paper C is the stable quadraticmodule with generators E i in degree i D 0; 1 and degree 0 relations @.e 1 / D g.e 1 / forall e 1 2 E 1 .References[1] H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Exp.Math. 2, Walter de Gruyter, Berlin 1991.[2] H. Gillet, D. R. Grayson, The loop space of the Q-construction, Illinois J. Math. 31 (1987),574–597.[3] T. Gunnarsson, R. Schwänzl, R. M. Vogt, F. Waldhausen, An un-delooped version of algebraicK-theory, J. Pure Appl. Algebra 79 (1992), 255–270.[4] F. Muro, A. Tonks, The 1-type of a Waldhausen K-theory spectrum, Adv. Math. 216 (2007),178–211, .[5] A. Nenashev, Double short exact sequences produce all elements of Quillen’s K 1 ,inAlgebraicK-theory (Poznań, 1995), Contemp. Math. 199, Amer. Math. Soc., Providence, R.I.,1996, 151–160.[6] A. Nenashev, Double short exact sequences and K 1 of an exact category, K-Theory 14(1998), 23–41.[7] A. Nenashev, K 1 by generators and relations, J. Pure Appl. Algebra 131 (1998), 195–212.

On K 1 of a Waldhausen category 115[8] C. Sherman, On K 1 of an abelian category, J. Algebra 163 (1994), 568–582.[9] C. Sherman, On K 1 of an exact category, K-Theory 14 (1998), 1–22.[10] F. Waldhausen, Algebraic K-theory of topological spaces. I, in Algebraic and geometrictopology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc.Sympos. Pure Math. 23, Amer. Math. Soc., Providence, R.I., 1978, 35–60.[11] F. Waldhausen, Algebraic K-theory of spaces, In Algebraic and geometric topology (NewBrunswick, N.J., 1983), Lecture Notes in Math. 1126, Springer-Verlag, Berlin 1985,318–419.[12] C. Weibel, An introduction to algebraic K-theory, book in progress available at http://math.rutgers.edu/weibel/Kbook.html.

Twisted K-theory – old and newby Max KaroubiSome history and motivation about this paperThe subject “K-theory with local coefficients”, now called “twisted K-theory”, wasintroduced by P. Donovan and the author in [19] almost forty years ago. 1 It associatesto a compact space X and a “local coefficient system”˛ 2 GBr.X/ D Z=2 H 1 .XI Z=2/ Tors.H 3 .XI Z//an abelian group K˛.X/ which generalizes the usual Grothendieck–Atiyah–HirzebruchK-theory of X when we restrict ˛ being in Z=2 (cf. [5]). This “graded Brauer group”GBr.X/ has the following group structure: if ˛ D ."; w 1 ;W 3 / and ˛0 D ." 0 ;w1 0 ;W0 3 /are two elements, one defines the sum ˛C˛0 as ."C" 0 ;w 1 Cw1 0 ;W 3CW3 0Cˇ.w1w1 0 //,where ˇ W H 2 .XI Z=2/ ! H 3 .XI Z/ is the Bockstein homomorphism. With thisdefinition, one has a generalized cup-product 2 :K˛.X/ K˛0.X/ ! K˛C˛0.X/:The motivation for this definition is to give in K-theory a satisfactory Thom isomorphismand Poincaré duality pairing which are analogous to the usual ones in cohomologywith local coefficients. More precisely, as proved in [28], if V is a real vector bundleon a compact space X with a positive metric, the K-theory of the Thom space of V isisomorphic to a certain “algebraic” group K C.V / .X/ associated to the Clifford bundleC.V /, viewed as a bundle of Z=2 graded algebras. A careful analysis of this groupshows that it depends only on the class of C.V / in GBr.X/, the three invariants beingrespectively the rank of V mod. 2, w 1 .V / and ˇ.w 2 .V // D W 3 .V /, where w 1 .V /,w 2 .V / are the first two Stiefel–Whitney classes of V . In particular, if V is a c spinorialbundle of even rank, one recovers a well-known theorem of Atiyah and Hirzebruch.On the other hand, if X is a compact manifold, it is well known that such a Thomisomorphism theorem induces a pairing between K-groupsK˛.X/ K˛0.X/ ! Zif ˛ C ˛0 is the class of C.V / in GBr.X/, where V is the tangent bundle of X.The necessity to revisit these ideas comes from a new interest in the subject becauseof its relations with Physics [43], as shown by the number of recent publications.However, for these applications, the first definition recalled above is not completesince the coefficient system is restricted to the torsion elements of H 3 .XI Z/. Asit was pointed out by J. Rosenberg [38] and later on by C. Laurent, J.-L. Tu, P. Xu1 See the appendix for a short history of the subject.2 Strictly speaking, this product is defined up to non canonical isomorphism ; see 2.1 for more details.

118 M. Karoubi[35], M. F. Atiyah and G. Segal [8], this restriction is in fact not necessary. In orderto avoid it, one may use for instance the Atiyah–Jänich theorem [2], [27] about therepresentability of K-theory by the space of Fredholm operators (already quoted in[19] for the cup-product which cannot be defined otherwise).In the present paper, we would like to make a synthesis between different viewpointson the subject: [19], [38], [35] and [8] (partially of course) 3 . We hope to have been“pedagogical” in some sense to the non experts.However, this paper is not just historical. It presents the theory with another pointof view and contains some new results. We extend the Thom isomorphism to this moregeneral setting (see also [16]), which is important in order to relate the “ungraded”and “graded” twisted K-theories. We compute many interesting equivariant twistedK-groups, complementing the basic papers [35], [8] and [9]. For this purpose, we usethe “Chern character” for finite group actions, as defined by Baum, Connes, Kuhn andSłomińska [11], [33], [41], together with our generalized Thom isomorphism. Theselast computations are related to some previous ones [31] and to the work of manyauthors. Finally, we introduce new cohomology operations which are complementaryto those defined in [19] and [9].We don’t pretend to be exhaustive in a subject which has already many ramifications.In an appendix to this paper we try to give a short historical survey and a list of interestingcontributions of many authors related to the results quoted here.General plan of the paperLet us first recall the point of view developed in [19], in order to describe the backgroundmaterial. We consider a locally trivial bundle of Z=2-graded central simple complexalgebras A, i.e. modelled on M 2n .C/ or M n .C/ M n .C/, with the obvious gradings 4 .Then A has a well-defined class ˛ in the group GBr.X/ (as introduced above). Onthe other hand, one may consider the category of “A-bundles” , whose objects arevector bundles provided with an A-module structure (fibrewise). We call this categoryE A .X/; the graded objects of this category are vector bundles which are modules overA y˝ C 0;1 , where C 0;1 is the Clifford algebra C C D CŒx=.x 2 1/. The groupK˛.X/ is now defined as the “Grothendieck group” of the forgetful functorE A y˝C 0;1 .X/ ! E A .X/:We refer the reader to [28], p. 191, for this definition which generalizes the usualGrothendieck group of a category. For our purpose, we make it quite explicit at the endof §1, using the concept of “grading”.Despite its algebraic simplicity, this definition of K˛.X/ is not quite satisfactoryfor various reasons. For instance, it is not clear how to define in a simple way a3 As it was pointed out to me by J. Rosenberg, one should also add the following reference, in the spirit of[18]: Ellen Maycock Parker, The Brauer group of graded continuous trace C*-algebras, Trans. Amer. Math.Soc. 308 (1988), Nr. 1, 115–132.4 up to graded Morita equivalence.

Twisted K-theory – old and new 119“cup-product”K˛.X/ K˛0.X/ ! K˛C˛0.X/as mentioned earlier (even if ˛ and ˛0 are in the much smaller group Z=2).To correct this defect, a second definition may be given in terms of Fredholmoperators in a Hilbert space. More precisely, we consider graded Hilbert bundles Ewhich are also graded A-modules in an obvious sense, together with a continuousfamily of Fredholm operatorsD W E ! Ewith the following properties:1. D is self-adjoint of degree 1,2. D commutes with the action of A (in the graded sense).One gets an abelian semi-group from the homotopy classes of such pairs .E; D/, withthe addition rule.E; D/ C .E 0 ;D 0 / D .E ˚ E 0 ;D˚ D 0 /:The associated group gives the second definition of K˛.X/ which is equivalent to thefirst one (see [19], p. 18, and [29], p. 88). We use also the notation K A .X/ instead ofK˛.X/ where ˛ is the class of A in GBr.X/ when we want to be more explicit.With this new viewpoint, the cup-product alluded to above becomes obvious. It isdefined by the following formula 5 :.E; D/ Y .E 0 ;D 0 / D .E y˝ E 0 ;D y˝ 1 C 1 y˝ D 0 /where the symbol y˝ denotes the graded tensor product of bundles or morphisms. It is amap from K A .X/ K A0 .X/ to K A y˝A 0 .X/ and therefore it induces a (non canonical)map from K˛.X/ K˛0.X/ to K˛C˛0.X/.For simplicity’s sake, we have only considered complex K-theory. We could aswell study the real case: one has to replace GBr.X/ byGBrO.X/ D Z=8 H 1 .XI Z=2/ H 2 .XI Z=2/:If we take for the coefficient system ˛ D n to be in Z=8, we get the usual groupsKO n .X/ as defined using Clifford algebras in [28] and [29], p. 88 (these groups beingwritten xK n in the later reference).In this paper, we essentially follow the same pattern, but with bundles of infinitedimension in the spirit of [38]. As a matter of fact, all the technical tools are alreadypresent in [19], [29] and [38], for instance the Fredholm operator machinery which isnecessary to define the cup product. However, this paper is not a rewriting of thesepapers, since we take a more synthetic view point and have other applications in mind.For instance, the K-theory of Banach algebras K n .A/ and its graded version, denoted5 See the appendix about the origin of such a formula.

120 M. Karoubihere by GrK n .A/, are more systematically used. On the other hand, since the equivariantK-theory and its relation with cohomology have been studied carefully in [17],[35] and [9], we limit ourselves to the applications in this case. One of them is thedefinition of operations in twisted K-theory in the graded and ungraded situations.Here is the contents of the paper:1. K-theory of Z=2-graded Banach algebras2. Ungraded twisted K-theory in the finite and infinite-dimensional cases3. Graded twisted K-theory in the finite and infinite-dimensional cases4. The Thom isomorphism5. General equivariant K-theory6. Some computations in the equivariant case7. Operations in twisted K-theoryAppendix. A short historical survey of twisted K-theoryAcknowledgments. I would like to thank A. Adem, A. Carey, P. Hu, I. Kriz, C. Leruste,V. Mathai, J. Rosenberg, J.-L. Tu, P. Xu and the referee for their remarks and suggestionsafter various drafts of this paper.1 K-theory of Z=2-graded Banach algebrasHigher K-theory of real or complex Banach algebras A is well known (cf. [28] or [12]for instance). Starting from the usual Grothendieck group K.A/ D K 0 .A/, there aremany equivalent ways to define “derived functors” K n .A/, for n 2 Z, such that anyexact sequence of Banach algebras0 A0 A A00 0induces an exact sequence of abelian groups::: K nC1 .A/ K nC1 .A"/ K n .A 0 / K n .A/ K n .A"/ ::::Moreover, by Bott periodicity, these groups are periodic of period 2 in the complexcase and 8 in the real case.The K-theory of Z=2-graded Banach algebras A (in the real or complex case) isless well known 6 and for the purpose of this paper we shall recall its definition whichis already present but not systematically used in [28] and [19]. We first introduce C p;qas the Clifford algebra of R pCq with the quadratic form.x 1 / 2 .x p / 2 C .x pC1 / 2 CC.x pCq / 2 :6 This is of course included in the general KK-theory of Kasparov which was introduced later than ourbasic references, at least for C*-algebras.

Twisted K-theory – old and new 121It is naturally Z=2-graded. If A is an arbitrary Z=2-graded Banach algebra, A p;q is thegraded tensor product A y˝ C p;q .ForA unital, we now define the graded K-theory ofA (denoted by GrK.A/)astheK-theory of the forgetful functor: W P .A 0;1 / ! P .A/(see [28] or 1.4 for a concrete definition). Here P .A/ denotes in general the categoryof finitely generated projective left A-modules. One may remark that the objectsof P .A 0;1 / are graded objects of the category P .A/. The functor simply “forgets”the grading. One should also notice that GrK.A p;q / is naturally isomorphic toGrK.A pC1;qC1 /, since A p;q is Morita equivalent to A pC1;qC1 (in the graded sense).If A is not unital, we define GrK.A/ by the usual method, as the kernel of theaugmentation mapGrK.A C / ! GrK.k/where A C is the k-algebra A with a unit added (k D R or C, according to the theory,with the trivial grading).There is a “suspension functor” on the category of graded algebras, associating toA the graded tensor product A 0;1 D A y˝ C 0;1 . One of the fundamental results 7 in[28], p. 210, is the fact that this suspension functor is compatible with the usual one:in other words, we have a well-defined isomorphismGrK.A 0;1 / GrK.A.R//where A.T / denotes in general the algebra of continuous maps f.t/ on the locallycompact space T with values in A, such that f.t/ ! 0 when t goes to infinity. As aconsequence, we have an isomorphism between the following groups (for n 0):GrK.A 0;n / GrK.A.R n //which we call GrK n .A/. More generally, we put GrK n .A/ D GrK.A p;q / for q p Dn 2 Z. These groups GrK n .A/ satisfy the same exactness property as the groupsK n .A/ above, from which they are naturally derived. They are of course linked withthem by the following exact sequence (for all n 2 Z):K nC1 .A y˝ C 0;1 / K nC1 .A/ GrK n .A/ Kn .A y˝ C 0;1 / K n .A/:In particular, if we start with an ungraded Banach algebra A, we see that Bott periodicityfollows from these previous considerations, thanks to the periodicity of Clifford algebrasup to graded Morita equivalence: this was the main theme developed in [6], [44] and[28], in order to give a more conceptual proof of the periodicity theorems.When A is unital, it is technically important to describe the group GrK.A/ in amore concrete way. If E is an object of P .A/,agrading of E is given by an involution" which commutes (resp. anticommutes) with the action of the elements of degree 07 Strictly speaking, one has to replace the category C p;q with an arbitrary graded category. However, theproof of Theorem 2.2.2 in [28] easily extends to this case.

122 M. Karoubi(resp. 1) in A. In this way, E with a grading " may be viewed as a module over thealgebra A y˝C 0;1 . We now consider triples .E; " 1 ;" 2 /, where " 1 and " 2 are two gradingsof E. The homotopy classes of such triples obviously form a semi-group. Its quotientby the semi-group of “elementary” triples (i.e. such that " 1 D " 2 ) is isomorphic toGrK.A/.2 Ungraded twisted K-theory in the finite andinfinite-dimensional casesLet X be a compact space and let us consider bundles of algebras A with fiber M n .C/.As it was shown by Serre [25], such bundles are classified by Čech cocycles (up to Čechcoboundaries):g ji W U i \ U j ! PU.n/where PU.n/ is U.n/=S 1 , the projective unitary group. In other words, the bundleof algebras A may be obtained by gluing together the bundles of C*-algebras .U i M n .C//, using the transition functions g ji [Note that U.n/ acts on M n .C/ by innerautomorphisms and therefore induces an action of PU.n/ on the algebra M n .C/]. TheBrauer group of X denoted by Br.X/ is the quotient of this semi-group of bundles(via the tensor product) by the following equivalence relation: A is equivalent to A 0iff there exist vector bundles V and V 0 such that the bundles of algebras A y˝ End.V /and A 0 y˝ End.V 0 / are isomorphic. It was proved by Serre [25] that Br.X/ is naturallyisomorphic to the torsion subgroup of H 3 .XI Z/.The Serre–Swan theorem (cf. [28] for instance) may be easily translated in thissituation to show that the category of finitely generated projective A-module bundles E(as in [19]) is equivalent to the category P .A/ of finitely generated projective modulesover A D .X;A/, the algebra of continuous sections of the bundle A. The keyobservation for the proof is that E is a direct factor of a “trivial” A-bundle; this iseasily seen with finite partitions of unity, since X is compact. One should notice that ifA is equivalent to A 0 the associated categories P .A/ and P .A 0 / are equivalent. Notehowever that this equivalence is non canonical since A ˝ End.V / and A 0 ˝ End.V 0 /are not canonically isomorphic.Definition 2.1. The ungraded 8 twisted K-theory K .A/ .X/ is by definition the K-theoryof the ring A (which is the same as the K-theory of the category E A .X/ mentioned inthe introduction). By abuse of notation, we shall simply call it K.A/. We also defineK .A/n.X/ as the K n -group of the Banach algebra .X;A/. It only depends on the classof A in Br.X/ D Tors.H 3 .XI Z//.The key observation made by J. Rosenberg [38] is the following: we can “stabilize”the situation (in the C*-algebra sense), i.e. embed M n .C/ into the algebra of compactoperators K in a separable Hilbert space H , thanks to the split inclusion of C n in l 2 .N/.8 We use the notation K .A/ .X/, not to be confused with the graded twisted K-group K A .X/ whichwill be defined in the next section.

Twisted K-theory – old and new 123Now, a bigger group PU.H / D U.H /=S 1 is acting on K by inner automorphisms. Ifwe take a Čech cocycleg ji W U i \ U j ! PU.H /we may use it to construct a bundle A of (non unital) C*-algebras with fiber K.Let us now consider the commutative diagramS 1 S 1 U.n/U.H/PU.n/ PU.H /.Thanks to Kuiper’s theorem [34], we remark that the classifying space of U.H/ is contractible.Therefore, the classifying space BPU.H / of the topological group PU.H /,isa nice model of the Eilenberg–Mac Lane space K.Z;3/(compare with the well-knownpaper of Dixmier and Douady [18]). Moreover, if we start with a finite-dimensionalalgebra bundle A over X with fiber M n .C/, the diagram above shows how to associateto A another bundle of algebras A 0 with fiber K, together with a C*-inclusionfrom A to A 0 . We note that the invariant W 3 .A/ in Br.X/ D Tors.H 3 .XI Z// definedin [25] is simply induced by the classifying map from X to BPU.H / (which factorsthrough BPU.n/). In this finite example, it is an n-torsion class since one has anothercommutative diagram n S1SU.n/PU.n/ U.n/PU.n/.Theorem 2.2. The inclusion from A to A 0 induces an isomorphismK r .A/ D K r ..X; A// ! K r ..X; A 0 // D K r .A 0 /where the K r define the classical topological K-theory of C*-algebras.Proof. The proof is classical for a trivial algebra bundle, since C is Morita equivalentto K (in the C*-algebra sense). It extends to the general case by a no less classicalMayer–Vietoris argument.Definition 2.3. Let now A be an algebra bundle with fiber K on a compact space Xwith structural group PU.H /. We define K .A/ .X/ (also denoted by K.A/) astheK-theory of the (non unital) Banach algebra .X;A/. This K-theory only depends of

124 M. Karoubithe class ˛ of A in H 3 .XI Z/ and we shall also call it K .˛/ .X/ [Due to 2.4, this is ageneralization of definition 2.2].Before treating the graded case in the next section, we would like to give an equivalentdefinition of K.A/ in terms of Fredholm operators, as was done in [19] for thetorsion elements in H 3 .XI Z/ and in [8] for the general case. The basic idea is to remarkthat PU.H / acts not only on the C*-algebra of compact operators in H , but alsoon the ring of bounded operators End.H / and on the Calkin algebra End.H /=K (withthe norm topology 9 )). Let us call B the algebra bundle with fiber End.H / associatedto the cocycle defined in 2.1 and B=A the quotient algebra bundle. Therefore, we havean exact sequence of C*-algebras bundles0 A B B=A 0.which induces an exact sequence for the associated rings of sections (thanks to a partitionof unity again)0 .X;A/ .X;B/ .X;B=A// 0.If B is trivial, it is well known that the algebra of continuous maps from X to End.H /has trivial K n -groups because this algebra is flabby 10 . By a Mayer–Vietoris argument,it follows that K n ..X; B// is also trivial. Therefore the connecting homomorphismK 1 .B=A/ D K 1 ..X; B=A// ! K 0 ..X; A// K .A/ .X/ D K.A/is an isomorphism, a well-known observation in index theory.Let us now consider the elements of B which map onto .B=A/ via the map .These elements form a bundle of Fredholm operators on H (the twist comes from theaction of PU.H /). This subbundle of B will be denoted by Fredh.H /. Therefore, wehave a principal fibration.X;A/ .X;Fredh.H // .X;.B=A/ /with contractible fiber the Banach space .X;A/ (this fibration admits a local sectionthanks to Michael’s theorem [37]). Therefore, the space of sections of Fredh.H /has the homotopy type of .X;.B=A/ /. In particular, the path components are inbijective correspondence via the map . The following theorem is a generalization ofa well-known theorem of Atiyah and Jänich [2], [27]:Theorem 2.4 ([8]). The set of homotopy classes of continuous sections of the fibrationFredh.H / Xis naturally isomorphic to K .A/ .X/.9 other topologies may be also considered, see [8].10 A unital Banach algebra ƒ is called flabby if there exists a continuous functor from P .ƒ/ to itselfsuch that C Id is isomorphic to . For instance, ƒ D End.H / is flabby since P .A/ is equivalent tothe category of Hilbert spaces which are isomorphic to direct factors in H ; is then defined by the infiniteHilbert sum .E/ D E ˚˚E ˚ :

Twisted K-theory – old and new 125Proof. As we have seen above, the two spaces .X;Fredh.H // and .X;.B=A/ /have the same homotopy type. On the other hand, it is a well known consequence ofKuiper’s theorem [34] than the (non unital) ring map .X;B=A/ ! .X;M r .B=A//induces a bijection between the path components of the associated groups of invertibleelements (see for instance [32], p. 93). Therefore, 0 ..X; Fredh.H /// may beidentified withlim 0 ..X; GL r .B=A/// D K 1 .B=A/!rand therefore with K.A/, as we already mentioned in 2.6.Remark 2.5. We may also consider the following “stabilized” bundleFredh s .H / D lim! nFredh.H n /and, without Kuiper’s theorem, prove in the same way that the set of connected componentsof the space of sections of this bundle is isomorphic to K .A/ .X/.There is an obvious ring homomorphism (since the Hilbert tensor product H ˝ His isomorphic to H )K ˝ K KIf A and A 0 are bundles of algebras on X modelled on K, we may use this homomorphismto get a new algebra bundle on X, which we denote by A ˝ A 0 . From thecocycle point of view, we have a commutative diagram, where the top arrow is inducedby complex multiplicationS 1 S 1U.H/ U.H/PU.H / PU.H / S1U.H ˝ H/ PU.H ˝ H/.It follows that W 3 .A ˝ A 0 / D W 3 .A/ C W 3 .A 0 / in H 3 .XI Z/ and one gets a “cupproduct”K .˛/ .X/ K .˛0/ .X/ K .˛C˛0/ .X/.(well defined up to non canonical isomorphism: see 2.1). This is a particular case of a“graded cup-product” which will be introduced in the next section.3 Graded twisted K-theory in the finite and infinite-dimensionalcasesWe are going to change our point of view and now consider Z=2-graded finite-dimensionalalgebras which are central and simple (in the graded sense). We are only inter-

126 M. Karoubiested in the complex case. The real case is treated with great details in [19] and doesnot seem to generalize in the infinite-dimensional framework 11 .In the complex case, there are just two “types” of such graded algebras (up to Moritaequivalence 12 ) which are R D C and C C D CŒx=.x 2 1/. For a type R of algebra,the graded inner automorphisms of A D R y˝ End.V 0 ˚ V 1 / may be given by either anelement of degree 0 or an element of degree 1 in A . This gives us an augmentation(whose kernel is denoted by Aut 0 .A//:Aut.A/ Z=2.Therefore, for bundles of Z=2-graded algebras modelled on A, we already get aninvariant in H 1 .XI Z=2/, called the “orientation” of A and which may be representedby a real line bundle. A typical example is the (complexified) Clifford bundle C.V /,associated to a real vector bundle V of rank n. Its orientation invariant is the firstStiefel–Whitney class associated to V (cf. [19]). Note that the type R of C.V / is Cif n is even and CŒx=.x 2 1/ Š C C if n is odd 13 .For the second invariant, let us start with M 2n .C/ as a basic graded algebra to fixthe ideas and let us put a C*-algebra metric on A. We have an exact sequence of groupsas in the ungraded case (where PU 0 .2n/ D PU.2n/ \ Aut 0 .A//1 S1 U.n/ U.n/ PU 0 .2n/ 1.Therefore, a bundle with structural group PU 0 .2n/ also has a class in H 2 .XI S 1 / DH 3 .XI Z/ which is easily seen to have order n as in the ungraded case. A similarargument holds if the graded algebra is M n .C/ M n .C/. It follows that the “gradedBrauer group” GBr.X/ isGBr.X/ D Z=2 H 1 .XI Z=2/ Tors.H 3 .XI Z//as already quoted in the introduction (see [19] for the explicit group law on GBr.X/). IfA is a bundle of Z=2-graded finite-dimensional algebras, we define the graded twistedK-theory K A .X/ as the graded K-theory of the graded algebra .X;A/ as recalled inSection 1. This definition only depends on the class of A in GBr.X/. We recover thedefinition in [19] by using again the Serre–Swan theorem as in 2.1. For instance, if weconsider the bundle of (complex) Clifford algebras C.V /, associated to a real vectorbundle V , the invariants we get are w 1 .V / and W 3 .V /, as quoted in the introduction.If these invariants are trivial, the bundle V is c spinorial of even rank and C.V / may be11 This is not quite true if we work in the context of “Real” K-theory in the sense of Atiyah [2]. We shall notconsider this generalization here, although it looks interesting in the light of “equivariant twisted K-theory”as we shall show in §6.12 This means that we are allowed to take the graded tensor product with End.V 0 ˚ V 1 / with the obviousgrading.13 If V is oriented and even dimensional for instance, this does not imply that C.V / is a bundle of gradedalgebras of type M 2 .ƒ/ for a certain bundle of ungraded algebras ƒ. However, for suitable vector bundlesV 0 and V 1 , this is the case for the graded tensor product C.V / y˝ End.V 0 ˚ V 1 /.

Twisted K-theory – old and new 127identified with the bundle of endomorphisms End.S C ˚ S /, where S C and S arethe even and odd “spinors” associated to the Spin c -structure.In order to define graded twisted K-theory in the infinite-dimensional case, wefollow the same pattern as in §2. For instance, let us take a graded bundle of algebras Amodelled on M 2 .K/: it has 2 invariants, one in H 1 .XI Z=2/, the other in H 3 .XI Z/(and not just in the torsion part of this group). We then define K A .X/ as the gradedK-theory of the graded algebra .X;A/, according to §1. The same definition holdsfor bundles of graded algebras modelled on K K D KŒx=.x 2 1/ D K y˝ Kx.If C 0;1 is the Clifford algebra C C with its usual graded structure, the generalresults of §1 show that the group K A .X/ fits into an exact sequence:K .A y˝C 0;1 /1.X/ K.A/1.X/ K A .X/ K.A y˝C 0;1/ .X/ K .A/ .X/where K .A/i.X/ denotes in general the K i -group of the Banach algebra .X;A/.Let us now assume that A is oriented modelled on M 2 .K/ (which means that thestructural group of A may be reduced to PU 0 .H ˚ H/; see below or 3.2 in the finitedimensionalsituation). We are going to show that A may be written as M 2 .A 0 /, withthe obvious grading, A 0 being an ungraded bundle of algebras modelled on K. For thispurpose, we write the commutative diagram S1S 1 U.H/U.H/ U.H/PU.H / PU 0 .H ˚ H/where the first horizontal map is the identity and the others are induced by the diagonal.This shows that H 1 .XI PU.H // H 1 .XI PU 0 .H ˚ H//, which is equivalent tosaying that A may be written as M 2 .A 0 / for a certain bundle of algebras A 0 .Therefore, A y˝C 0;1 is Morita equivalent to A 0 A 0 and K A .X/ is the K-theory ofthe ring homomorphism (more precisely the functor defined by the associated extensionof the scalars, as we shall consider in other situations)A 0 A 0 M 2 .A 0 /defined by .a; b/ 7!0b a0 (no grading). We should also note that A 0 is Morita equivalentto A as an ungraded bundle of algebras. Since the usual K-theory (resp. gradedK-theory) is invariant under Morita equivalence (resp. graded Morita equivalence), theprevious considerations lead to the following theorem:Theorem 3.1. Let A be an oriented bundle of graded algebras modelled on M 2 .K/.Then K A .X/ is isomorphic to K .A/ .X/ via the identification above.

128 M. KaroubiThe same method may be applied when A is an oriented bundle of graded algebrasmodelled on K K D KŒx=.x 2 1/. The graded oriented automorphisms of K Kinduced by PU.H ˚ H/are diagonal matrices of the type a 00 aThis shows that A is isomorphic to A 0 A 0 and A y˝ C 0;1 is isomorphic to M 2 .A 0 /.Therefore, K A .X/ is the Grothendieck group of the ring homomorphism A 0 !M 2 .A 0 /, defined bya 7!: a 00 a:Hence we have the following theorem, analogous to 3.5:Theorem 3.2. Let A be an oriented bundle of graded algebras modelled on K K.Then A is isomorphic to A 0 A 0 and the group K A .X/ is isomorphic to K 1 .A 0 / viathe identification above.Remark 3.3. One may notice that if A is a bundle of oriented graded algebras modelledon M 2 .K/, the associated bundle with fiber M 2 .K C / 14 has a section " of degree 0and of square 1, which commutes (resp. anticommutes) with the elements of degree 0(resp. 1); it is simply defined by the matrix 1 00 1Finally, we would like to give an equivalent definition of K A .X/ in terms of Fredholmoperators as in §2. This was done in [19] if the class of A belongs to the torsiongroup of H 3 .XI Z/ and in [8] for the general case. We shall give another treatmenthere, using again the K-theory of graded Banach algebras.Following the general notations of §2, we have an exact sequence of bundles ofgraded Banach algebras:0 A B B=A 0.Since the graded K-groups of B are trivial, we see as in 2.6 that GrK..X; A// isisomorphic to GrK 1 ..X; B=A//.In order to shorten the notations, we denote by B the graded Banach algebra.X;B/, byƒ the graded Banach algebra .X;B=A/ and by f the surjective mapB ! ƒ. The following lemma 15 may be proved in the same way as in [29], p. 78:14 K C is the ring K which a unit added15 It might be helpful for a better understanding to notice that the category of finitely generated freeEnd.H /-modules is equivalent to the category of Hilbert spaces H n for n 2 N. This “local” situation istwisted by the group PU.H /.

Twisted K-theory – old and new 129Lemma 3.4. Any element of GrK 1 .ƒ/ may be written as the homotopy class of apair .E; "/ where E is a free graded B-module and " is a grading of degree 1 of theassociated ƒ-module (see 1.4 for the definition of a grading).By the well-known dictionary between modules and bundle theory, we may view " asa grading of a suitable bundle of free End.H /=K-modules. By spectral theory, we mayalso assume that " is self-adjoint. Finally, following [29], we define a quasi-grading 16of E as a family of Fredholm endomorphisms D such that1. D D D,2. D is of degree 1.The following theorem is the analogue in the graded case of Theorem 2.8 (cf [29],p. 78/79).Theorem 3.5. The (graded) twisted K-group K A .X/ is the Grothendieck group associatedto the semi-group of homotopy classes of pairs .E; D/ where E isafreeZ=2-graded B-module and D is a family of Fredholm endomorphisms of E which areself-adjoint and of degree 1.Remark 3.6. Let us assume that A is oriented modelled on M 2 .K/. The descriptionabove gives a Fredholm description of GrK 1 .A/ D K 1 .A/: we just take the homotopyclasses of sections of the associated bundle of self-adjoint Fredholm operatorsFredh .B/ whose essential spectrum is divided into two non empty parts 17 in R C andR .This Fredholm description of K A .X/ enables us to define a cup-productK A .X/ K A0 .X/ KA y˝A 0 .X/where A y˝ A 0 denotes the graded tensor product of A and A 0 . This cup-product isgiven by the same formula as in [19], p. 19, and generalizes it:.E; D/ Y .E 0 ;D 0 / D .E y˝ E 0 ;D y˝ 1 C 1 y˝ D 0 /(see the appendix about the origin of this formula in usual topological K-theory).To conclude this section, let us consider a locally compact space X and a bundle ofgraded algebras A on X. For technical reasons, we assume the existence of a compactspace Z containing X as an open subset, such that A extends to a bundle (also calledA) onZ. There is an obvious definition of K A .X/ as a relative term in the followingexact sequence (where T D Z X and A 0 D A y˝ C 0;1 ):K A0 .Z/ KA 0 .T / K A .X/ K A .Z/ K A .T /.16 “quasi-graduation” in French.17 See the appendix about the role of self-adjoint Fredholm operators in K-theory.

130 M. KaroubiBy the usual excision theorem in topological K-theory, one may prove that this definitionof K A .X/ is independent from the choice of Z.The method described before and also in [29], §3, shows how to generalize thedefinition of K A .X/ in this case: one takes homotopy classes of pairs .E; D/ as in 3.12,with the added assumption that the family D is acyclic at infinity. In other words, thereis a compact set S X, such that D x is an isomorphism when x … S (see [29], p. 89–97for the technical details of this approach). This Fredholm description of K A .X/ willbe important in the next section for the description of the Thom isomorphism.4 The Thom isomorphism in twisted K-theory 18Let V be a finite-dimensional real vector bundle on a locally compact space X whichextends over a compactification of X as was assumed in 3.15. Then the complexifiedClifford bundle C.V / has a well-defined class in the graded Brauer group of X. IfAis another twist on X, we can consider the graded tensor product A y˝ C.V / and theassociated group K A y˝C.V / .X/. As was described in 3.15, it is the group 19 associatedto pairs .E; D/ where E is a graded bundle of free B-modules and D is a family ofFredholm endomorphisms which are of degree 1, self-adjoint and acyclic at 1. Letus now consider the projection W V ! X. For simplicity’s sake, we shall often callE;A; B;::: the respective pull-backs of E;A; B;::: via this projection. Since Band C.V / are subbundles of B y˝ C.V /, E may be provided with the induced B andC.V /-modules structures.Theorem 4.1. Let d.E;D/be an element of K A y˝C.V / .X/ with the notations above.We define an element t.d.E;D//in the group K A .V / as d. .E/; D 0 / where D 0 is thefamily of Fredholm operators on .E/, defined over the point v in V (with projectionx on X) asD 0 .x;v/ D D x C .v/where .v/ denotes the action of the element v of the vector bundle V considered as asubbundle of C.V /. The homomorphismt W K A y˝C.V / .X/ ! K A .V /(t for “Thom”) defined above is then an isomorphism.Proof. 20 We should first notice that V may be identified with the open unit ball bundle ofthe vector bundle V and is therefore an open subset of the closed unit ball bundle of V .Moreover, since V and A extends to a compactification of X, the required conditionsin 3.15 for the definition of the twisted K-theory of X and V are fulfilled.18 See also [16].19 We should note that E is a B-module, not an A-module. Nevertheless, we shall keep the notation K A .20 According to a suggestion of J. Rosenberg, it should be possible to give a proof with the KK-theoryof Kasparov by describing an explicit inverse to the homomorphism t. However, KK-theory is out of thescope of this paper.

Twisted K-theory – old and new 131We shall now provide two different proofs of the theorem.The first one, more elementary in spirit, consists in using a Mayer–Vietoris argumentwhich we can apply here since the two sides of the formula above behave as cohomologytheories 21 with respect to the base X. Therefore, we may assume that A and V aretrivial: this is a special case of the theorem stated in [30], pp. 211/212.The second one is more subtle and may be generalized to the equivariant case. Letus first describe the Thom isomorphism for complex V . This is a slight modificationof Atiyah’s argument using the elliptic Dolbeault complex [3]. More precisely, weconsider the composite map W K A .X/ ! K A y˝C.V / .X/ t ! K A .V /:The first map is the cup-product with the algebraic “Thom class” which is ƒV providedwith the classical Clifford graded module structure. This map is an isomorphismfrom well-known algebraic considerations (Morita equivalence). Therefore t is anisomorphism if and only if is an isomorphism. On the other hand, is just thecup-product with the topological Thom class T V which belongs to the usual topologicalK-theory K.V / of Atiyah and Hirzebruch [5], [6]. In order to prove that is anisomorphism, we may now use the exact sequence in 3.3 to reduce ourselves to theungraded twisted case. In other words, it is enough to show that the cup-product withT V induces an isomorphism‰ W K .A/ .X/ ! K .A/ .V /:In order to prove this last point, Atiyah defines a reverse map 22‰ 0 W K .A/ .V / ! K .A/ .X/:He shows that ‰ 0 ‰ D Id and, by an ingenious argument, deduces that ‰‰ 0 D Id aswell.Now, as soon as Theorem 4.2 is proved for complex V , the general case followsfrom a trick already used in [30], p. 241: we consider the following three Thom homomorphismswhich behave “transitively”:K A y˝C.V /y˝C.V /y˝C.V / .X/ KA y˝C.V /y˝C.V / .V / KA y˝C.V / .V ˚ V/ K A .V ˚ V ˚ V/.We know that the composites of two consecutive arrows are isomorphisms since V ˚Vcarries a complex structure. It follows that the first arrow is an isomorphism, which isessentially the theorem stated (using Morita equivalence again).21 Strictly speaking, one has to “derive” the two members of the formula, which can be done easily sincethey are Grothendieck groups of graded Banach categories.22 More precisely, one has to define an index map parametrized by a Banach bundle, which is also classical[20].

132 M. KaroubiLet A be a graded twist. As we have seen before, it has two invariants in H 1 .XI Z=2/and in H 3 .XI Z/. The first one provides a line bundle L in such a way that the gradedtensor product A 1 D A y˝ C.L/ is oriented. From Thom isomorphism and the considerationsin 3.5/7, we deduce the following theorem which gives the relation betweenthe ungraded and graded twisted K-groups.Theorem 4.2. Let A be a graded twist of type M 2 .K/ and A 1 D A y˝ C.L/, whereL is the orientation bundle of A. Then A 1 may be written as A 2 A 2 where A 2 isungraded of type K. Therefore, we have the following isomorphismsK A .X/ Š K A y˝C.L/ y˝C.L/ .X/ Š K A 1.L/ Š K .A 2/1.L/:Let A be a graded twist of type K K. With the same notations, we have the followingisomorphismsK A .X/ Š K A y˝C.L/ y˝C.L/ .X/ Š K A 1.L/ Š K .A 1/ .L/:5 General equivariant K-theoryNote: this section is inserted here for the convenience of the reader as an introductionto §6. It is mainly a summary of results found in [39], [12], [35] and [8].Let A be a Banach algebra and G a compact Lie group acting on A via a continuousgroup homomorphismG Aut.A/where Aut.A/ is provided with the norm topology. We are interested in the categorywhose objects are finitely generated projective A-modules E together with a continuousleft action of G on E such that we have the following identity holds (g 2 G, a 2 A,e 2 E)g:.a:e/ D .g:a/:.g:e/with an obvious definition of the dots. We define K G .A/ as the Grothendieck groupof this category C D P G .A/. By the well-known dictionary between bundles andmodules, we recover the usual equivariant K-theory defined by Atiyah and Segal [39]if A is the ring of continuous maps on a compact space X (thanks to the lemma below).It may also be defined as a suitable semi-direct product of G by A (cf. [12]).More generally, if A is a Z=2-graded algebra (where G acts by degree 0 automorphisms),we define the graded equivariant K-theory of A, denoted by GrK G .A/, as theGrothendieck group of the forgetful functorC 0;1 Cwhere C 0;1 is the category of “graded objects” in P G .A/ which is defined asP G .A y˝ C 0;1 /.In the same spirit as in §1, we define “derived” groups K p;qG .A/ as GrK G.A y˝C p;q ).They satisfy the same formal properties as the usual groups K n .A/ (also denoted by

Twisted K-theory – old and new 133K n .A/ with n D q p), for instance Bott periodicity. The following key lemmaenables us to translate many general theorems of K-theory into the equivariant framework:Lemma 5.1. Let E be an object of the category P G .A/. Then E is a direct summandof an object of the type A ˝C M where M is a finite-dimensional G-module.Proof. ([39], p. 134). Let us consider the union of all finite-dimensional invariantsubspaces of the G-Banach space E. According to a version of the Peter-Weyl theoremquoted in [39], this union is dense in A. We now consider a set e 1 ;:::;e n of generatorsof E as an A-module. Since is dense in A and E is projective, one may choosethese generators in the subspace . Let M 1 ;:::;M n be finite-dimensional invariantsubspaces of E containing e 1 ;:::;e n respectively and let M be the following directsum:M D M 1 ˚˚M nWe define an equivariant surjection W A ˝C M Š .A ˝C M 1 / ˚˚.A ˝C M n / ! E:between projective left A-modules by the formula. 1 ˝ m1;:::; n ˝ m n / D 1 m 1 CC n m n :This surjection admits a section, which we can average out thanks to a Haar measurein order to make it equivariant. Therefore, E is a direct summand in A ˝C M as statedin the lemma.As for usual K-theory, one may also define equivariant K-theory for non unitalrings and, using Lemma 5.2 above, prove that any equivariant exact sequence of ringson which G acts0 A0 A A00 0induces an exact sequence of equivariant K-groupsK n 1G .A/ K n 1G .A00 / K n G .A0 / K n G .A/ K n G .A00 /and an analogous exact sequence in the graded equivariant framework.After these generalities, let us assume that G acts on a compact space X and let Abe a bundle of algebras modelled on K. We define the (ungraded) equivariant twistedK-group K .A/G.X/ as K G.A/, where A is the Banach algebra of sections of the bundleA. Similarly, if A is a bundle of graded algebras modelled on K K or M 2 .K/, wedefine the (graded) equivariant twisted K-group KG A.X/ as GrK G.A/, where A is thegraded Banach algebra of sections of A.As seen in §3, it is also natural to consider free graded B-modules together with acontinuous action of G (compatible with the action on X) and a family of Fredholmoperators D which are self-adjoint of degree 1, commuting with the action of G. With

134 M. Karoubithe same ideas as in §3, we can show that the Grothendieck group of this category isisomorphic to KG A .X/. This is essentially the definition proposed in [8].An example was in fact given at the beginning of the history of twisted K-theory.In [19], §8, we defined a “power operation”P W K A .X/ ! K A y˝nS n.X/where S n denotes the symmetric group on n letters acting on A y˝n (in the spirit ofAtiyah’s paper on power operations [2]). According to the general philosophy ofAdamsand Atiyah, one can deduce from this n th power operation new “Adams operations”(n being odd and the group law in GBr.X/ being written multiplicatively):‰ n W K˛.X/ ! K˛n.X/ ˝Z nwhere ˛ belongs to Z=2H 1 .XI Z=2/H 3 .XI Z/ and n denotes the free Z-modulegenerated by the n th roots of unity in C (the ring of cyclotomic integers 23 ). One canprove, following [19], that these additive maps ‰ n satisfy all the required propertiesproved by Adams. We shall come back to them in §7, making the link with operationsrecently defined by Atiyah and Segal [9].In order to fix ideas, let us consider the graded version of equivariant twisted K-theory KG A .X/, where A is modelled on C C with the obvious grading. We haveagain a “Thom isomorphism”:t W K A y˝C.V /G.X/ ! K A G .V /whose proof is the same as in the non equivariant case (using elliptic operators). Onthe other hand, if L is the orientation line bundle of A, the group G acts on L inawaycompatible with the action on X. Therefore, A 1 D A y˝C.L/ is an algebra bundle withtrivial orientation. and A 1 y˝C 0;1 is naturally isomorphic to A 1 A 1 D A 1 Œx=.x 2 1/.As in 4.4, this shows that the graded twisted group KG A .X/ is naturally isomorphic tothe ungraded twisted K-group K A 1G.L/. The same method shows that we can reducegraded twisted K-groups to ungraded ones if A is a bundle of graded algebras modelledon M 2 .K/.Let us mention finally one of the main contributions of Atiyah and Segal to thesubject ([8], §6, see also [17]), which we interpret in our language. One is interested inalgebra bundles A on X modelled on K, provided with a left G-action. The isomorphismclasses of such bundles are in bijective correspondence with principal bundlesP over PU.H / (acting on the right) together with a left action of G.Such an algebra bundle T is called “trivial” if T may be written as the bundle ofalgebras of compact operators in End.V /, where V is a G-Hilbert bundle. Equivalently,this means that the structure group of T can be lifted equivariantly to U.H/ (in a waycompatible with the G-action).23 The introduction of this ring is absolutely necessary in the graded case.

Twisted K-theory – old and new 135We now say that two such algebra bundles A and A 0 are equivalent if there existtwo trivial algebra bundles T and T 00 such that A ˝ T and A 0 ˝ T 00 are isomorphicas G-bundles of algebras. The quotient is a group since the dual of A is its inverse viathe tensor product of principal PU.H /-bundles. This is the “equivariant Brauer group”Br G .X/.A closely related definition (in the framework of C*-algebras and for a locallycompact group G) is given in [17]. It is very likely that it coincides with this onefor compact Lie groups, in the light of an interesting filtration described in this paper,probably associated to a spectral sequence.Theorem 5.2 (cf. [8], Proposition 6.3). Let X Gassociated to X. Then the natural mapD EG G X be the Borel spaceBr G .X/ ! Br.X G /is an isomorphism.The interest of this theorem lies in the fact that the equivariant K-theoriesK G ..X; A// and K G ..X; A 0 // are isomorphic if A and A 0 are equivalent. Thisfollows from the well-known Morita invariance in operator K-theory. We shall studyconcrete applications of this principle in the next section.6 Some computations of twisted equivariant K-groupsLet us look at the particular case of the ungraded twisted K-groups K .A/G.X/ where Gis a finite group acting on the trivial bundle of algebras A D X M n .C/ via a grouphomomorphism G ! PU.n/. 24 We define zG as the pull-back diagramzGGSU.n/ PU.n/Therefore, zG is a central covering of G with fiber n (whose elements are denoted byGreek letters such as ). The following definition is already present in [31], §2.5 (forn D 2):Definition 6.1. A finite-dimensional representation of zG is of “linear type” if .u/ D.u/ for any 2 n . We now consider the category E A QG .X/ whose objects arelzG-A-modules as before, except that we request that the zG-action be of linear type andcommutes with the action of A. By Morita invariance, E A QG .X/ is equivalent to thelcategory EG Q .X/ l of finite-dimensional zG-bundles on X, the action of zG on the fibersbeing of linear type.24 However, we don’t assume that G acts trivially on X in general.

136 M. KaroubiTheorem 6.2. 25 The (ungraded) twisted K-theory K .A/G.X/ is canonically isomorphicto the Grothendieck group of the category EG z .X/ l.Proof. One just repeats the argument in the proof of Theorem 2.6 in [31], where A is aClifford algebra C.V / and Z=2 plays the role of n . We simply “untwist” the actionof zG thanks to the formula (F) written explicitly in the proof of 2.6 (loc. cit.).For A D X A with A D M n .C/, the previous argument shows that K .A/G.X/ isa subgroup of the usual equivariant K-theory KG z .X/. From now on, we shall writeKG A .X/ instead of K.A/G.X/. Similarly, in the graded case (A D M n.C/ M n .C/ orM 2n .C/), we shall write KG A.X/ instead of KA G.X/. If X is a point and G is finite,K .A/G.X/ is just the K-theory of the semi-direct product G Ë A.Theorem 6.3. Let G be a finite group acting on the algebra of matrices A D M n .C/and let zG be the central extension G by n described in 6.1. Then, for X reduced to apoint, the group K .A/G.X/ D K.G Ë A/ is a free abelian group of rank the number ofconjugacy classes in G which split into n conjugacy classes in zG.Proof. We can apply the same techniques as the ones detailed in [31], §2.6/12 (forn D 2). By the theory of characters on zG, one is looking for functions f on zG (whichwe call of “linear type”) such that1. f.hgh 1 / D f.g/,2. f .x/ D f .x/ if is an n th root of the unity.The C-vector space of such functions is in bijective correspondence with the space offunctions on the set of conjugacy classes of G which split into n conjugacy classesof zG.Like the Brauer group of a space X, one may define in a similar way the Brauergroup Br.G/ of a finite group G by considering algebras A D M n .C/ as above with aG-action (see [24] for a broader perspective ; this is also a special case of the generaltheory of Atiyah and Segal mentioned at the end of §5). From the diagram written in6.1, one deduces a cohomology invariantw 2 .A/ 2 H 2 .GI n /and (via the Bockstein homomorphism) a second invariant W 3 .A/ 2 H 2 .GI S 1 / DH 2 .GI Q=Z/ D H 3 .GI Z/. It is easy to show that this correspondence induces awell-defined mapW 3 W Br.G/ ! H 3 .GI Z/:The following theorem is a special case of 5.9 in a more algebraic situation.25 There is an obvious generalization when A is infinite-dimensional. However, for our computations, werestrict ourselves to the finite-dimensional case.

Twisted K-theory – old and new 137Theorem 6.4. Let G be a finite group. Then the previous homomorphismis bijective.W 3 W Br.G/ ! H 3 .GI Z/Proof. First of all, we remark that H 3 .GI Z/ Š H 2 .GI Q=Z/ is the direct limit ofthe groups H 2 .GI m / through the maps H 2 .GI m / ! H 2 .GI p / when m dividesp. This stabilization process corresponds on the level of algebras to the tensor productA 7! A ˝ End.V /, where V is a G-vector space of dimension p=m. Therefore, themap W 3 is injective.The proof of the surjectivity is a little bit more delicate (see also 5.9). We can sayfirst that it is a particular case of a much more general result proved by A. Fröhlich andC. T. C. Wall [24] about the equivariant Brauer group of an arbitrary field k: there is asplit exact sequence (with their notations)0 Br.k/ BM.k; G/ H 2 .GI U.k// 0where Br.k/ is the usual Brauer group of k, U.k/ is the group of invertible elements ink and BM.k; G/ is a group built out of central simple algebras over k with a G-action.Since Br.C/ D 0 and H 2 .GI U.k// D H 2 .GI Q=Z/, the theorem is an immediateconsequence.Here is an elementary proof suggested by the referee (for k D C). If we start witha central extension1 n zG G 1we consider the finite dimensional vector spaceH Dff 2 L 2 . zG/ such that f ./ D 1 f./with .; / 2 n zGg:Then zG acts by left translation on H in such a way that this action is of linear typeas described in 6.2. Therefore, we have a projective representation of G on H andEnd.H / is a matrix algebra where G acts.Remark 6.5. Let us consider an arbitrary central extension zG of G by n Š Z=nassociated to a cohomology class c 2 H 2 .GI Z=n/. We are interested in the set ofelements g of G such that the conjugacy class hgi splits into n conjugacy classes in zG 1 .This set only depends on the image of c in H 3 .GI Z/ via the Bockstein homomorphismH 2 .GI Z=n/ ! H 3 .GI Z/ (cf. 6.6). In other words, two central extensions of G byZ=n with the same associated image by the Bockstein homomorphism have the sameset of n-split conjugacy classes.

138 M. KaroubiIn order to show this fact, let us consider the following diagram: n nmzG 1 zG G G,and an element g 1 of zG 1 . The conjugacy class of g D .g 1 / splits into n conjugacyclasses in zG 1 if and only if there is a trace function f on zG 1 with values in C such thatf.g 1 c/ D f.g 1 /c when c 2 n . Such a trace function extends obviously to zG, whichyields to the result, since the direct limit of the groups H 2 .GI Z=nm/ is preciselyH 2 .GI Q=Z/ Š H 3 .GI Z/.This remark may be generalized as follows according to a suggestion of J.-P. Serre:let zG 1 and zG be two group extensions (not necessary central) of G by abelian groupsC 1 and C of orders m 1 and m respectively, such that the following diagram commutes(with ˛ injective):˛C 1CzG 1 zG G G.The previous argument shows that if an element g of G splits into m conjugacy classesin zG, it splits into m 1 conjugacy classes in zG 1 : take trace functions f on zG with valuesin C such that f.gc/ D f.g/c (we write multiplicatively the abelian group C ). Theconverse is true if the extension of G by C is central.Let us now assume that X is not reduced to a point. We can use the Baum–Connes–Kuhn–Słomińska Chern character [11], [33], [41] which is defined on K .X/ (for anyfinite group ), with values in the direct sum L hi H even .X g / C./ . In this formula,hi runs through all the conjugacy classes of G, C./ being the centralizer of (thecohomology is taken with complex coefficients). One of the main features of this“Chern character”K .X/ L hi H even .X / C./is the isomorphism it induces between K .X/˝Z C and the cohomology with complexcoefficients on the right-hand side. If E is a -vector bundle, the map is definedexplicitly by Formula 1.13, p. 170 in [11].Let us now take for the group zG previously considered and let us analyze theformula in this case. We shall view the right-hand side not just as a function on the

Twisted K-theory – old and new 139set of conjugacy classes hi, but as a function f on the full group with certain extraproperties which we shall explain now.If we replace by 0 such that 0 D 1 , we have a canonical isomorphism W H .X 0 / C.0/ ! H .X / C./ induced by x 7! x. This map exchanges f./and f. 0 / and we have the relation f./D .f . 0 //, which says that f is essentiallya function on the set of conjugacy classes.On the other hand, if the action of zG is of linear type, we have an extra relation, aneasy consequence of the formula in [11], which is f ./ D f ./ when is an n throot of unity. To summarize, we get the following theorem.Theorem 6.6. Let G be a finite group and A D M n .C/ with a G-action. Thenthe ungraded twisted equivariant K-theory K .A/G.X/ is a subgroup of the equivariantK-theory KG z .X/, where zG is the pull-back diagramzGGSU.n/ PU.n/.More precisely, K .A/G.X/ ˝Z C may be identified with the C-vector space of functionsf on D zG with f./ in H even .X g / C./ , ./ D g, such that the following twoidentities hold:1. If 0 D 1 , one has f./D .f . 0 //, according to the formula above.2. f ./ D f ./ if is an n th root of unity.In particular, if X is reduced to a point, we have D Id and K .A/G.X/ is free withrank the number of conjugacy classes of G which split into n conjugacy classes in zG(as seen in 6.5.)Remark 6.7. This theorem is not really new. In a closely related context, one findssimilar results in [1] and [42]. We should also notice that the same ideas have beenused in [31] for representations of “linear type”. Finally, the theorem easily extendsto locally compact spaces if we consider cohomology with compact supports on theright-hand side.Theorem 6.8. Let A be any finite-dimensional graded semi-simple complex algebrawith a graded action of a finite group G. Then the graded K-theory GrK 0 .A 0 / ˚GrK 1 .A 0 / of the semi-direct product A 0 D G Ë A is a non trivial free Z-module. Inparticular, if V is a real finite-dimensional vector space with a G action, the groupKG A.V / ˚ KA G.V ˚ 1/ is free non trivial thanks to the Thom isomorphism.Proof. The algebra G ËA is graded semi-simple over the complex numbers. Therefore,it is a direct sum of graded algebras Morita equivalent to M n .C/ M n .C/ or M 2n .C/.In both cases, the graded K-theory is non trivial. The last part of the theorem followsfrom 4.2.

140 M. KaroubiThe following theorem is a direct consequence of the previous considerations:Theorem 6.9. Let us now assume that A D M 2n .C/ is G-oriented as a graded algebra:in other words, there is an involutive element " of A of degree 0 which commutes withthe action of G and commutes (resp. anticommutes) with the elements of A of degree 0(resp. 1). Then, the graded K-theory GrK .A 0 /, with A 0 D GËA, is a finitely generatedfree module concentrated in degree 0. More precisely, one has GrK 0 .A 0 / D K.A 0 /and GrK 1 .A 0 / D 0. In particular, if V is an even-dimensional real vector space and ifA y˝ C.V / is G-oriented, we have (via the Thom isomorphism)KG A .V / D KA y˝C.V /G.P / D K.G Ë A y˝ C.V // and KG A .V ˚ 1/ D 0where P is a point. If we write A y˝ C.V / as an algebra of matrices M r .C/ with arepresentation of G and call zG the associated central extension by r , the rank ofKG A .V / is the number of conjugacy classes of G which split into r conjugacy classesin zG.In the abelian case, the following two theorems are related to results obtained byP. Hu and I. Kriz [26], using different methods.Theorem 6.10. Let us consider the algebra A D M n .C/ provided with an action ofan abelian group G and P a point. Then the ungraded twisted K-theory K .A/G.P / DK .A 0 /, with A 0 D G Ë A, is concentrated in degree 0 and is a free Z-module. If wetensor this group with the rationals and if we look at it as an R.G/ ˝ Q D QŒGmodule,it may be identified with R.G 0 / ˝ Q for a suitable subgroup G 0 of G. Inparticular, the rank of K 0 .A 0 / divides the order of G.Proof. The first part of the theorem is a consequence of the previous more generalconsiderations. As we have shown before, the algebra A 0 gives rise to the followingcommutative diagram:zG nSU.n/G PU.n/,the fibers of the vertical maps being n . The subset of elements Qg in zG n such that .Qg/splits into n conjugacy classes is just the center Z. zG n / of zG n (since G is abelian).Let us put n D .Z. zG n //. Then K.A 0 / may be written as KG A .P / where P is apoint. According to Theorem 6.10, this is the subgroup of the representation ring ofzG n generated by representations of linear type. At this stage, it is convenient to maken D1by extension of the roots of unity, so that we have an extension of G by Q=ZQ=Z zG G(the “linear type” finite-dimensional representations of zG are the same as the originallinear type finite-dimensional representations of zG n ). We call R. zG/ l / the associated

Twisted K-theory – old and new 141Grothendieck group. By the theory of characters, we see that R. zG/ l ˝Q is isomorphicto R.Z. zG// l ˝ Q, since the characters of such linear type representations of zG vanishoutside Z. zG/. On the other hand, if we denote by G 0 the image of Z. zG/ in G, theextension of abelian groupsQ=Z Z. zG/ G0splits (non canonically). This means that we can identify R. zG/ l ˝ Q with the representationring R.G 0 / ˝ Q as an R.G/ ˝ Q-module. This proves the last part of thetheorem.Theorem 6.11. Let us consider a graded algebra A D M 2n .C/ provided with a nonoriented action of an abelian group G (with respect to the grading). Then the gradedtwisted K-theory K A G .P / D GrK .A 0 /, with A 0 D G Ë A, is concentrated in a singledegree (0 or 1) and is a free Z-module. If we tensor this group with the complexnumbers and if we look at it as an R.G/ ˝ C D CŒG-module, it may be identifiedwith R.G 0 / ˝ Q for a suitable subgroup G 0 of G. In particular, the rank of K A G .P / divides the order of G.Proof. Let L be the orientation bundle of A (with respect to the action of G). If wechange A into A y˝ C.L/ and if we apply the Thom isomorphism theorem, we have tocompute K A G .L/ where G acts on A (resp. L) in an oriented way (resp. non orientedway).Let us apply Theorem 6.10 in this situation: since G is abelian, the function f ofthe theorem must be equal to 0 on the elements of zG which are not in Z. zG/. Therefore,the relevant group K A G .L/ is reduced to K G 0.L/ (after tensoring with C and whereG 0 is the image of Z.G/ in G). We now consider two cases:1. the action of G 0 on L is oriented, in which case we only find R.G 0 / (with ashift of dimension). Therefore, the dimension of KG A .P / ˝ C is the order of G0which divides the order of G.2. the action of G 0 on L is not oriented. We then find a direct sum of copies of C,each one corresponding to an element of G 0 such that .g 0 / D 1. The dimensionof KG A.L/ ˝ C is therefore half the order of G0 , hence divides jGj=2.Remark 6.12. For an oriented action of G on M 2n .C/, Theorem 3.5 enables us tosolve the analogous problem in ungraded twisted K-theory, which is done in 6.14.On the other hand, as we already mentioned, the two last theorems are related toresults of P. Hu and I. Kriz [26].In [31] we also perform analogous types of computations when A is a Cliffordalgebra and G is any finite group, again using the Thom isomorphism. For instance,if G is the symmetric group S n acting on the Clifford algebra of R n via the canonicalrepresentation of S n on R n , there is a nice relation between K-theory and the pentagonalidentity of Euler (cf. [31], p. 532).

142 M. KaroubiWe would like to point out also that the theory K˙.X/ introduced recently byAtiyahand Hopkins [7] is a particular case of twisted equivariant K-theory. As a matter offact, it was explicitly present in [28], §3, 40 years ago, before the formal introductionof twisted K-theory. Here is a detailed explanation of this identification.According to [7], the definition of K˙.X/ (in the complex or real case) is thegroup K Z=2 .X R 8 /, where Z=2 acts on X and also on R 8 D R R 7 by .; / 7!. ;/. According to the Thom isomorphism in equivariant K-theory (proved inthe non spinorial case in [30]), it coincides with an explicit graded twisted K-groupK A Z=2 .X/, as defined in [28]. Here A is the Clifford algebra C.R2 / D C 1;1 of R 2provided with the quadratic form x 2 y 2 and where Z=2 acts via the involution.; / ! . ;/ on R R (this is also mentioned briefly in [7], p. 2, footnote 1).This identification is valid as well in the real framework, where we have 8-periodicity.These groups K A .X/ were considered in [28], §3.3, in a broader context: A mayZ=2be any Clifford algebra bundle C.V / (where V is a real vector bundle provided witha non degenerate quadratic form) and Z=2 may be replaced by any compact Lie groupacting in a coherent way on X and V . The paper [31] gives a method to computethese equivariant twisted K-groups. As quoted in the appendix, the real and complexself-adjoint Fredholm descriptions (for the non twisted case) which play an importantrole in [7] were considered independently in [10] and [32].7 Operations on twisted K-groupsThis section is a partial synthesis of [19] and [9].Let us start with the simple case of bundles of (ungraded) infinite C*-algebrasmodelled on K, like in [9]. As it was shown in [2] and [19], we have a n th power mapP W K A .X/ ! K .A˝n /S n.X/where the symmetric group S n acts on A˝n by permutation of the factors.Lemma 7.1. 26 The group K .A˝n /S n.X/ is isomorphic to the group K .A˝n / 0S n.X/ wherethe symbol 0 means that S n is acting trivially on A˝nProof. As we have shown many times in §6, this “untwisting” of the action of thesymmetric group on A˝n is due to the following fact: the standard representationS n ! PU.H ˝n /can be lifted into a representation W S n ! U.H˝n / in a way compatible with thediagonal action of elements of PU.H /, a fact which is obvious to check.26 We should note that this lemma is not true for K Sn .A˝n / for a general noncommutative ring A.Therefore, it is not possible to define -operations in this case. Twisted K-theory is somehow intermediarybetween the commutative case and the noncommutative one.

Twisted K-theory – old and new 143Remark 7.2. If we take a bundle of finite dimensional algebras modelled on A DEnd.E/ where E D C r , there is another way to check (functorially) this untwisting ofthe action of S n on A˝n : we identify A˝n with End.E˝n / and .A˝n / with Aut.E˝n /.We have the following commutative diagram:S nEnd.E˝n / D Aut.E˝n / End.E˝n / D End.E/˝n .The vertical map sends the invertible element ˛ to the automorphism .u 7! ˛u˛ 1/.If is a permutation, the horizontal map sends to the automorphismu 1 ˝˝u n 7! u .1/ ˝˝u .n/while the map sends to the automorphism of E˝n defined byx 1 ˝˝x n 7! x .1/ ˝˝x .n/ :Finally, the composition , computed on a decomposable tensor of E˝n , gives therequired result:x 1 ˝˝x n x .1/ ˝˝x .n/ u u 1 .x .1/ / ˝˝u n .x .n/ /As was shown in [2], a Z-module map 1 u .1/ .x 1 / ˝˝u .n/ .x n /.R.S n / Zdefines an operation in twisted K-theory by taking the composite of the following maps:K .A/ .X/ K .A˝n /S n.X/ Š K .A˝n / 0S n.X/ Š K .A/˝n .X/ ˝ R.S n / K .A/˝n .X/.This is essentially 27 what was done in [9], §10, in order to define the n operation ofGrothendieck in this context for instance.Let us call .F; r/ or even F (for short) a representative of the image of .E; D/ bythe composite of the maps (we now use the Fredholm description of twisted K-theory)K .A/ .X/ K .A˝n /S n.X/ K .A˝n / 0.X/.We can define the Adams operations ‰ n with the method described in [19] (which weintend to generalize later on). For this, we restrict the action of S n to the cyclic group27 The second homomorphism was not explicitly given however.A n

144 M. KaroubiZ=n identified with the group of n th roots of the unity. Let us call F r the subbundle ofF where the action of Z=n is given by ! r , ! being a fixed primitive root of the unity.Then ‰ n .E; D/ is defined by the following sum:Xn 1‰ n .E; D/ D F r ! r :It belongs formally to K .A/˝n .X/˝Z n where n is the ring of n-cyclotomic integers.However, if n is prime, using the action of the symmetric group S n , it is easy to checkthat F r is isomorphic to F 1 if r ¤ 1. Therefore we end up in K .A/˝n .X/, consideredas a subgroup of K .A/˝n .X/ ˝ n , as it was expected. It is essentially proved in [2]Zthat this definition of ‰ n agrees with the classical one.There is another operation in twisted K-theory which is “complex conjugation”,classically denoted by ‰ 1 , which maps K .A/ .X/ to K .A/ 1 .X/ (if we write multiplicativelythe group law in Br.X/). It is shown in [9], §10, how we can combinethis operation with the previous ones in order to get “internal” operations, i.e. mappingK .A/ .X/ to itself.It is more tricky to define operations in graded twisted K-theory. If ƒ is a Z=2-graded algebra, it is no longer true in general that a graded involution of ƒ is induced byan inner automorphism with an element of degree 0 and of order 2. A typical exampleis the Clifford algebra.C ˚ C/ y˝2 D .C 0;1 / y˝2which may be identified with the graded algebra M 2 .C/. Ifweput 0 10 ie 1 D and e1 02 Di 0we see that there is no inner automorphism by an element of order 2 and degree 0permuting e 1 and e 2 .In order to define such operations in the graded case, we may proceed in at least twoways.We first remark that if A 0 is an oriented bundle modelled on M 2 .K/, the groupsK .A0/ .X/ and K A0 .X/ are isomorphic (see 3.5). Moreover, the ungraded tensor productA 0˝n is isomorphic to the graded one A 0 y˝n (since M 2 .C/ y˝ M 2 .C/ is isomorphic toM 4 .C/ D M 2 .C/ ˝ M 2 .C//. Let us now assume that the bundle of algebras A ismodelled on K K but not necessarily oriented and let L be the orientation real linebundle of A. Then A 0 D A y˝ C.L/ is oriented modelled on M 2 .K/ (C.V / denotesin general the Clifford bundle associated to V ). We may apply the previous methodto define operations from K .A0/ .Y / D K A0 .Y / to Ky˝n.Y A0 /. If we apply the Thomisomorphism to the vector bundle Y D L with basis X (see §4), we deduce (for nodd) operations from K A .X/ to K A y˝n.X/. However, if A is oriented, we loose someinformation since what we get are essentially the previous operations applied to thesuspension of X. Note also that the same method may be applied to K A .X/, when Ais modelled on M 2 .K/.0

Twisted K-theory – old and new 145Another way to proceed is to use a variation of the method developed in [19], p. 21,for any A. We consider the following diagram (with X connected):F.X;S 1 /F.X;S 1 /u nU 0 .A y˝n/A n Aut 0 .A y˝n /vU 0 .A y˝ A/ y˝n Aut 0 .A y˝ A/ y˝n .In this diagram A n is the alternating group (for n 3), n is the cartesian product ofA n and U 0 .A y˝n / over Aut 0 .A y˝n / and the horizontal maps between the U ’s and theAut’s are essentially given by g 7! g ˝ g. Note that these maps induce a map fromF.X;S 1 / to itself which is f.z/7! f.z 2 /.By the general theory developed in 7.2, we know that there exists a canonicalmap from A n to U 0 .A y˝ A/ y˝n such that its composite with the projection fromU 0 .A y˝A/ y˝n to Aut 0 .A y˝A/ y˝n is the composite of the last horizontal maps. Let usnow define the subgroup C n of n which elements x are defined by '..x// D v.u.x//.It is clear that C n is a double cover of A n which is either the product of A n by Z=2 orthe Schur group C n [40] (since H 2 .A n I Z=2/ is isomorphic to Z=2 when n>3).Using the methods of §6, we have therefore the following composite maps:K A .X/ KA y˝nA n.X/ KA y˝n.C n / .X/ KA y˝n.X/ ˝ R.C n /,where the notation K A y˝n.C n / .X/ means that the group C n acts trivially on A y˝n . Followingagain Atiyah [2], we see then that any group homomorphism R.C n / ! Z gives rise toan operation K A .X/ ! K A y˝n.X/ Therefore, in the graded case, the Schur group C nreplaces the symmetric group S n , which we have used in the ungraded case.In order to define more computable operations, we may replace the alternatinggroup A n by the cyclic group Z=n (when n is odd) as it was already done in [19].The reduction of the central extension C n becomes trivial and we have a commutativediagramU 0 .A y˝n / Z=n Aut.A y˝n /.The Adams operation ‰ n is then given by the same formula as in 7.4 and [19]‰ n W K A .X/ ! K A˝n .X/ ˝ nZ

146 M. Karoubiwhere n is the ring of n-cyclotomic integers. We can show that this operation isadditive and multiplicative up to canonical isomorphisms (see Theorem 30, p. 23 in[19]).We should also notice, following [9], that we can define the Adams operation ‰ 1in graded twisted K-theory as well and combine it with the ‰ n ’s in order to define“internal” operations from K A .X/ to K A y˝n.X/ ˝ n .ZThe simplest non-trivial example is‰ n W Z Š K 1 .S 1 / ! K n .S 1 / ˝Z n Š K 1 .S 1 / ˝ nZwhere n is a product of different odd primes. Since the operation ‰ n on K 2 .S 2 / is themultiplication by n, we deduce that D p . 1/ .n 1/=2 n belongs to n (a well-knownresult due to Gauss) and that ‰ n on K 1 .S 1 / is essentially the inclusion of Z in ndefined by 1 7! .As a concluding remark, we should notice that the image of ‰ n as defined in 7.8 isnot arbitrary. If k and n are coprimes, the multiplication by k on the group Z=n definesan element of the symmetric group S n . The signature of this permutation is called theLegendre symbol . k n /. Moreover, this permutation conjugates the elements r and rk inthe group Z=n. If the Legendre symbol is 1, this permutation can be lifted to the Schurgroup C n . Let us denote now by F r (as in 7.4) the element of the twisted K-groupassociated to the eigenvalue e 2ir . Then we see that F r and F rk are isomorphic ifthe Legendre symbol . k n / is equal to 1 since r and rk are conjugate by an element ofthe Schur group. If n is prime for instance, ‰ n .E/ may therefore be written in thefollowing way:‰ n .E/ D F 0 C XU! k C XV! k ;. k n/D1. k n/D 1where U (resp. V )isanyF k with Legendre’s symbol equal to 1 (resp. 1). Thisshows in particular that the element D p . 1/ .n 1/=2 n in 7.9 is a “Gauss sum”, awell-known result.8 Appendix: A short historical survey of twisted K-theoryTopological K-theory was of course invented by Atiyah and Hirzebruch in 1961 [4]after the fundamental work of Grothendieck [14] and Bott [15]. However, twistedK-theory which is an elaboration of it took some time to emerge. One should quotefirst the work of Atiyah, Bott and Shapiro [6] where Clifford modules and Cliffordbundles (in relation with the Dirac operator) where used to reinterpret Bott periodicityand Thom isomorphism in the presence of Spin or Spin c structures. We then started tounderstand in [28], as quoted in the introduction, that K-theory of the Thom space maybe defined almost algebraically as K-theory of a functor associated to Clifford bundles.Finally, it was realized in [19] that we can go a step further and consider general gradedalgebra bundles instead of Clifford bundles associated to vector bundles (with a suitablemetric).

Twisted K-theory – old and new 147On the other hand, the theorem of Atiyah and Jänich [2], [27] showed that Fredholmoperators play a crucial role in K-theory. It was discovered independently in [10] and[29], [32] that Clifford modules may also be used as a variation for the spectrum of(real and complex) K-theory. In particular, the cup-product from K n .X/ K p .Y / toK nCp .X Y/may be reinterpreted in a much simpler way with this variation. Whatappeared as a convenience for usual K-theory in these previous references became anecessity for the definition of the product in the twisted case, as it was shown in [19].The next step was taken by J. Rosenberg [38], some twenty years later, in redefiningGBr.X/ for any class in H 3 .XI Z/ (not just torsion classes), thanks to the new roleplayed by C*-algebras in K-theory. The need for such a generalization was not clearin the 60’s (although all the tools were there) since we were just interested in usualPoincaré pairing. In passing, we should mention that one can define more generaltwistings in any cohomology theory and therefore in K-theory (see for instance [8],p. 18). However, the geometric ones are the most interesting for the purpose of ourpaper.We already mentioned the paper of E. Witten [43] in the introduction which madeuse of this more general twisting of J. Rosenberg, together with [35], [8], [9]. Butwe should also quote many other papers: [13] for the relations with the theory ofgerbes and D-branes in Physics; [1] for the relation with orbifolds; [36] who used theChern character defined byA. Connes and M. Karoubi in order to prove an isomorphismbetween twisted K-theory and a computable “twisted cohomology” (at least rationally);[22], [21] about the relation between loop groups and twisted K-theory; [23] (again) forthe relation with equivariant cohomology and many other works which are mentionedinside the paper and in the references. In particular, M. F. Atiyah and M. Hopkins [7]introduced another type of K-theory, denoted by them K˙X/, linked to deep physicalproblems, which is a particular case of twisted equivariant K-theory. This definitionof K˙.X/ was already given in [28], §3.3, in terms of Clifford bundles with a groupaction (see 6.16 and also [31] for the details).We invite the interested reader to use the classical research tools on the Web formany other references in mathematics and physics.References[1] A. Adem and Y. Ruan, Twisted Orbifold K-theory, Commun. Math. Phys. 237 (2003),533–556.[2] M. F. Atiyah, K-theory, Notes by D. W. Anderson, 2nd ed., Adv. Book Classics, Addison-Wesley Publishing Company, Redwood City, CA, 1989.[3] M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. OxfordSer. 19 (1968), 113–140.[4] M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull.Amer. Math. Soc. 65 (1959), 276–281.[5] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, in Proc. Sympos.Pure Math. 3, Amer. Math. Soc., Providence, R.I., 1961, 7–38.

148 M. Karoubi[6] M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), 3–38.[7] M. F. Atiyah and M. Hopkins, A variant of K-theory, in Topology, geometry and quantumfield theory, London Math. Soc. Lecture Note Ser. 308, Cambridge University Press,Cambridge 2004, 5–17.[8] M. F. Atiyah and G. Segal, Twisted K-theory. Ukr. Mat. Visn. 1 (2004), 287–330; Englishtransl. Ukrain. Math. Bull. 1 (2004), 291–334 .[9] M. F. Atiyah and G. Segal, Twisted K-theory and cohomology, Nankai Tracts Math. 11(2006), 5–43.[10] M. F. Atiyah and I. M. Singer, Index theory for skew adjoint Fredholm operators, Inst.Hautes Études Sci. Publ. Math. 37 (1969), 5–26 (compare with [32]).[11] P. Baum andA. Connes, Chern character for discrete groups, in A fête of Topology,AcademicPress, Boston, MA, 1988, 163–232.[12] B. Blackadar, K-theory for operator algebras, 2nd ed., Math. Sci. Res. Inst. Publ. 5, CambridgeUniversity Press, Cambridge 1998.[13] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray and D. Stevenson, Twisted K-theoryand K-theory of Bundle Gerbes, Commun. Math. Phys. 228 (2002), 17–45.[14] A. Borel et J.-P. Serre, Le théorème de Riemann-Roch (d’après Grothendieck), Bull. Soc.Math. France 86 (1958), 97–136.[15] R. Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313–337.[16] A. L. Carey and Bai-Ling Wang, Thom isomorphism and push-forward maps in twistedK-theory, Preprint 2006, math. KT/0507414.[17] D. Crocker, A. Kumjian, I. Raeburn, D. P. Williams, An equivariant Brauer group and actionof groups on C*-algebras, J. Funct. Anal. 146 (1997), 151–184.[18] J. Dixmier et A. Douady, Champs continus d’espaces hilbertiens et de C*-algèbres, Bull.Soc. Math. France 91 (1963), 227–284.[19] P. Donovan and M. Karoubi, Graded Brauer groups and K-theory with local coefficients,Inst. Hautes Études Sci. Publ. Math. 38 (1970), 5–25; French summary in: Groupe de Braueret coefficients locaux en K-théorie, Comptes Rendus Acad. Sci. Paris 269 (1969), 387–389.[20] A. T. Fomenko and A. S. Miscenko, The index of elliptic operators over C*-algebras, Izv.Akad. Nauk. SSSR Ser. Mat. 43 (1979), 831–859.[21] D. S. Freed, Twisted K-theory and loop groups, Preprint 2002, arXiv:math/0206237.[22] D. S. Freed, M. Hopkins and C. Teleman, Twisted K-theory and loop group representations,Preprint 2005, arXiv:math/0312155; Loop Groups and Twisted K-Theory II, Preprint 2007,arXiv:math/0511232.[23] D. S. Freed, M. Hopkins and C. Teleman, Twisted equivariant K-theory with complexcoefficients, J. Topol. 1 (2008), 16–44.[24] A. Fröhlich and C. T. C. Wall, Equivariant Brauer groups, in Quadratic forms and theirapplications (Dublin, 1999), Contemp. Math. 272, Amer. Math. Soc., Providence, RI, 2000,57–71.[25] A. Grothendieck, Le groupe de Brauer, Séminaire Bourbaki Vol. 9, Exp. No. 290, Soc.Math. France, Paris 1995, 199–219.

Twisted K-theory – old and new 149[26] P. Hu and I. Kriz, The RO.G/-graded coefficients of .Z=2/ n -equivariant K-theory, Preprint2006, ArXiv:math.KT/0609067.[27] K. Jänich, Vektorraumbündel und der Raum der Fredholm-operatoren. Math. Ann. 161(1965), 129–142.[28] M. Karoubi, Algèbres de Clifford et K-théorie, Ann. Sci. Ecole Norm. Sup. (4) 1 (1968),161–270.[29] M. Karoubi, Algèbres de Clifford et opérateurs de Fredholm, in Séminaire Heidelberg-Saarbrücken-Strasbourg sur la K-théorie Lecture Notes in Math. 136, Springer-Verlag,Berlin 1970, 66–106; Summary in Comptes Rendus Acad. Sci. Paris 267 (1968), 305–308.[30] M. Karoubi, Sur la K-théorie équivariante, in Séminaire Heidelberg-Saarbrücken-Strasbourgsur la K-théorie Lecture Notes in Math. 136, Springer-Verlag, Berlin 1970, 187–253.[31] M. Karoubi, Equivariant K-theory of real vector spaces and real projective spaces, TopologyAppl. 122 (2002), 531–546.[32] M. Karoubi, Espaces classifiants en K-théorie, Trans. Amer. Math. Soc. 147 (1970), 75–115(compare with [10]); Summary in Comptes Rendus Acad. Sci. Paris 268 (1969), 710–713.[33] N. Kuhn, Character rings in algebraic topology, in Advances in Homotopy Theory (Cortona1988), London Math. Soc. Lecture Note Ser. 139, Cambridge University Press, Cambridge1989, 111–126.[34] N. Kuiper, The homotopy type of the unitary group in Hilbert space, Topology 3 (1965),19–30.[35] C. Laurent, J.-L. Tu and P. Xu, Twisted K-theory of differential stacks, Ann. Sci. ÉcoleNorm. Sup. 37 (2004), 841–910.[36] V. Mathai and D. Stevenson, On a generalized Connes-Hochschild-Kostant-Rosenberg theorem,Adv. Math. 200 (2006), 303–335.[37] E. A. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959),556–575.[38] J. Rosenberg, Continuous-trace algebras from the bundle theoretic point of view, J. Austral.Math. Soc. A 47 (1989), 368–381.[39] G. Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151.[40] I. Schur, Darstellungen der symmetrischen und alternierenden Gruppe durch gebroche lineareSubstitutionen, J. Reine Angew. Math. 139 (1911), 155–250.[41] J. Słomińska, On the equivariant Chern homomorphism, Bull. Acad. Polon. Sci. Sér. Sci.Math. Astronom. Phys. 24 (1976), 909–913.[42] J.-L. Tu and Ping Xu, The Chern character for twisted K-theory of orbifolds, Adv. Math.207 (2006), 455–483.[43] E. Witten, D-branes and K-theory, J. High Energy Phys. 12 (1998), paper 19.[44] R. Wood, Banach algebras and Bott periodicity, Topology 4 (1966), 371–389.

Equivariant cyclic homology for quantum groupsChristian Voigt1 IntroductionEquivariant cyclic homology can be viewed as a noncommutative generalization ofequivariant de Rham cohomology. For actions of finite groups or compact Lie groups,different aspects of the theory have been studied by various authors [4], [5], [6], [17],[18]. In order to treat noncompact groups as well, a general framework for equivariantcyclic homology following the Cuntz–Quillen formalism [8], [9], [10] has been introducedin [25]. For instance, in the setting of discrete groups or totally disconnectedgroups this yields a new approach to classical constructions in algebraic topology [26].However, in contrast to the previous work mentioned above, a crucial feature of theconstruction in [25] is the fact that the basic ingredient in the theory is not a complexin the usual sense of homological algebra. In particular, the theory does not fitinto the traditional scheme of defining cyclic homology using cyclic modules or mixedcomplexes.In this note we define equivariant periodic cyclic homology for quantum groups.This generalizes the constructions in the group case developed in [25]. Again we workin the setting of bornological vector spaces. Correspondingly, the appropriate notionof a quantum group in this context is the concept of a bornological quantum groupintroduced in [27]. This class of quantum groups includes all locally compact groupsand their duals as well as all algebraic quantum groups in the sense of van Daele [24].As in the theory of van Daele, an important ingredient in the definition of a bornologicalquantum group is the Haar measure. It is crucial for the duality theory and also explicitlyused at several points in the construction of the homology theory presented in this paper.However, with some modifications our definition of equivariant cyclic homology couldalso be adapted to a completely algebraic setting using Hopf algebras with invertibleantipodes instead.From a conceptual point of view, equivariant cyclic homology should be viewedas a homological analogon to equivariant KK-theory [15], [16]. The latter has beenextended by Baaj and Skandalis to coactions of Hopf-C -algebras [3]. However, inour situation it is more convenient to work with actions instead of coactions.An important ingredient in equivariant cyclic homology is the concept of a covariantmodule [25]. In the present paper we will follow the terminology introduced in [12]and call these objects anti-Yetter–Drinfeld modules instead. In order to construct thenatural symmetry operator on these modules in the general quantum group setting weprove a formula relating the fourth power of the antipode with the modular functions ofa bornological quantum group and its dual. In the context of finite dimensional Hopfalgebras this formula is a classical result due to Radford [23].

152 C. VoigtAlthough anti-Yetter–Drinfeld modules occur naturally in the constructions oneshould point out that our theory does not fit into the framework of Hopf-cyclic cohomology[13]. Still, there are relations to previous constructions for Hopf algebrasby Akbarpour and Khalkhali [1], [2] as well as Neshveyev and Tuset [22]. Remarkin particular that cosemisimple Hopf algebras or finite dimensional Hopf algebras canbe viewed as bornological quantum groups. However, basic examples show that thehomology groups defined in [1], [2], [22] only reflect a small part of the informationcontained in the theory described below.Let us now describe how the paper is organized. In Section 2 we recall the definitionof a bornological quantum group. We explain some basic features of the theoryincluding the definition of the dual quantum group and the Pontrjagin duality theorem.This is continued in Section 3 where we discuss essential modules and comodulesover bornological quantum groups as well as actions on algebras and their associatedcrossed products. We prove an analogue of the Takesaki–Takai duality theorem in thissetting. Section 4 contains the discussion of Radford’s formula relating the antipodewith the modular functions of a quantum group and its dual. In Section 5 we studyanti-Yetter–Drinfeld modules over bornological quantum groups and introduce the notionof a paracomplex. Section 6 contains a discussion of equivariant differential formsin the quantum group setting. After these preparations we define equivariant periodiccyclic homology in Section 7. In Section 8 we show that our theory is homotopy invariant,stable and satisfies excision in both variables. Finally, Section 9 contains a briefcomparison of our theory with the previous approaches mentioned above.Throughout the paper we work over the complex numbers. For simplicity we haveavoided the use of pro-categories in connection with the Cuntz–Quillen formalism to alarge extent.2 Bornological quantum groupsThe notion of a bornological quantum group was introduced in [27]. We will workwith this concept in our approach to equivariant cyclic homology. For information onbornological vector spaces and more details we refer to [14], [20], [27]. All bornologicalvector spaces are assumed to be convex and complete.A bornological algebra H is called essential if the multiplication map induces anisomorphism H y˝H H Š H . The multiplier algebra M.H/ of a bornological algebraH consists of all two-sided multipliers of H , the latter being defined by the usual algebraicconditions. There exists a canonical bounded homomorphism W H ! M.H/.A bounded linear functional W H ! C on a bornological algebra is called faithfulif .xy/ D 0 for all y 2 H implies x D 0 and .xy/ D 0 for all x implies y D 0.If there exists such a functional the map W H ! M.H/ is injective. In this case onemay view H as a subset of the multiplier algebra M.H/.In the sequel H will be an essential bornological algebra with a faithful bounded linearfunctional. For technical reasons we assume moreover that the underlying bornologicalvector space of H satisfies the approximation property.

Equivariant cyclic homology for quantum groups 153A module M over H is called essential if the module action induces an isomorphismH y˝H M Š M . Moreover an algebra homomorphism f W H ! M.K/ is essentialif f turns K into an essential left and right module over H . Assume that W H !M.H y˝ H/ is an essential homomorphism. The map is called a comultiplicationif it is coassociative, that is, if . y˝ id/ D .id y˝ / holds. Moreover the Galoismaps l ; r ; l ; r W H y˝ H ! M.H y˝ H/for are defined by l .x ˝ y/ D .x/.y ˝ 1/; l .x ˝ y/ D .x ˝ 1/.y/; r .x ˝ y/ D .x/.1 ˝ y/; r .x ˝ y/ D .1 ˝ x/.y/:Let W H ! M.H y˝ H/ be a comultiplication such that all Galois maps associatedto define bounded linear maps from H y˝ H into itself. If ! is a bounded linearfunctional on H we define for every x 2 H a multiplier .id y˝ !/.x/ 2 M.H/ by.id y˝ !/.x/ y D .id y˝ !/ l .x ˝ y/;y .id y˝ !/.x/ D .id y˝ !/ l .y ˝ x/:In a similar way one defines .! y˝ id/.x/ 2 M.H/. A bounded linear functional W H ! C is called left invariant if.id y˝ /.x/ D .x/1for all x 2 H . Analogously one defines right invariant functionals.Let us now recall the definition of a bornological quantum group.Definition 2.1. A bornological quantum group is an essential bornological algebra Hsatisfying the approximation property with a comultiplication W H ! M.H y˝ H/such that all Galois maps associated to are isomorphisms together with a faithful leftinvariant functional W H ! C.The definition of a bornological quantum group is equivalent to the definition of analgebraic quantum group in the sense of van Daele [24] in the case that the underlyingbornological vector space carries the fine bornology. The functional is unique up toa scalar and referred to as the left Haar functional of H .Theorem 2.2. Let H be a bornological quantum group. Then there exists an essentialalgebra homomorphism W H ! C and a linear isomorphism S W H ! H which isboth an algebra antihomomorphism and a coalgebra antihomomorphism such that. y˝ id/ D id D .id y˝ /and.S y˝ id/ r D y˝ id; .id y˝ S/ l D id y˝ :Moreover the maps and S are uniquely determined.

154 C. VoigtUsing the antipode S one obtains that every bornological quantum group is equippedwith a faithful right invariant functional as well. Again, such a functional is uniquelydetermined up to a scalar. There are injective bounded linear maps F l ; F r ; G l ; G r W H !H 0 D Hom.H; C/ defined by the formulasF l .x/.h/ D .hx/;G l .x/.h/ D .hx/;F r .x/.h/ D .xh/;G r .x/.h/ D .xh/:The images of these maps coincide and determine a vector space yH . Moreover, thereexists a unique bornology on yH such that these maps are bornological isomorphisms.The bornological vector space yH is equipped with a multiplication which is inducedfrom the comultiplication of H . In this way yH becomes an essential bornologicalalgebra and the multiplication of H determines a comultiplication on yH .Theorem 2.3. Let H be a bornological quantum group. Then yH with the structuremaps described above is again a bornological quantum group. The dual quantum groupof yH is canonically isomorphic to H .Explicitly, the duality isomorphism P W H ! yH is given by P D yG l F l S orequivalently P D yF r G r S. Here we write yG l and yF r for the maps defined aboveassociated to the dual Haar functionals on yH . The second statement of the previoustheorem should be viewed as an analogue of the Pontrjagin duality theorem.In [27] all calculations were written down explicitly in terms of the Galois mapsand their inverses. However, in this way many arguments tend to become lengthy andnot particularly transparent. To avoid this we shall use the Sweedler notation in thesequel. That is, we write.x/ D x .1/ ˝ x .2/for the coproduct of an element x, and accordingly for higher coproducts. Of coursethis has to be handled with care since expressions like the previous one only have aformal meaning. Firstly, the element .x/ is a multiplier and not contained in an actualtensor product. Secondly, we work with completed tensor products which means thateven a generic element in H y˝ H cannot be written as a finite sum of elementarytensors as in the algebraic case.3 Actions, coactions and crossed productsIn this section we review the definition of essential comodules over a bornologicalquantum group and their relation to essential modules over the dual. Moreover weconsider actions on algebras and their associated crossed products and prove an analogueof the Takesaki–Takai duality theorem.Let H be a bornological quantum group. Recall from Section 2 that a module Vover H is called essential if the module action induces an isomorphism H y˝H V Š V .A bounded linear map f W V ! W between essential H -modules is called H -linear or

Equivariant cyclic homology for quantum groups 155H -equivariant if it commutes with the module actions. We denote the category of essentialH -modules and equivariant linear maps by H -Mod. Using the comultiplicationof H one obtains a natural H -module structure on the tensor product of two H -modulesand H -Mod becomes a monoidal category in this way.We will frequently use the regular actions associated to a bornological quantumgroup H .Fort 2 H and f 2 yH one definest*f D f .1/ f .2/ .t/; f ( t D f .1/ .t/f .2/and this yields essential left and right H -module structures on yH , respectively.Dually to the concept of an essential module one has the notion of an essentialcomodule. Let H be a bornological quantum group and let V be a bornological vectorspace. A coaction of H on V is a right H -linear bornological isomorphism W V y˝H ! V y˝ H such that the relationholds..id ˝ r / 12 .id ˝ 1r / D 12 13Definition 3.1. Let H be a bornological quantum group. An essential H -comodule isa bornological vector space V together with a coaction W V y˝ H ! V y˝ H .A bounded linear map f W V ! W between essential comodules is called H -colinear if it is compatible with the coactions in the obvious sense. We write Comod- Hfor the category of essential comodules over H with H -colinear maps as morphisms.The category Comod- H is a monoidal category as well.If the quantum group H is unital, a coaction is the same thing as a bounded linearmap W V ! V y˝ H such that . y˝ id/ D .id y˝ / and .id y˝ / D id.Modules and comodules over bornological quantum groups are related in the sameway as modules and comodules over finite dimensional Hopf algebras.Theorem 3.2. Let H be a bornological quantum group. Every essential left H -moduleis an essential right yH -comodule in a natural way and vice versa. This yields inverseisomorphisms between the category of essential H -modules and the category of essentialyH -comodules. These isomorphisms are compatible with tensor products.Since it is more convenient to work with essential modules instead of comoduleswe will usually prefer to consider modules in the sequel.An essential H -module is called projective if it has the lifting property with respectto surjections of essential H -modules with bounded linear splitting. It is shown in [27]that a bornological quantum group H is projective as a left module over itself. Thiscan be generalized as follows.Lemma 3.3. Let H be a bornological quantum group and let V be any essentialH -module. Then the essential H -modules H y˝ V and V y˝ H are projective.

156 C. VoigtProof. Let V be the space V equipped with the trivial H -action induced by the counit.We have a natural H -linear isomorphism ˛l W H y˝ V ! H y˝ V given by ˛l.x ˝v/ D x .1/ ˝ S.x .2/ / v. Similarly, the map ˛r W V y˝ H ! V y˝ H given by˛r.v ˝ x/ D S 1 .x .1/ / v ˝ x .2/ is an H -linear isomorphism. Since H is projectivethis yields the claim.Using category language an H -algebra is by definition an algebra in the categoryH -Mod. We formulate this more explicitly in the following definition.Definition 3.4. Let H be a bornological quantum group. An H -algebra is a bornologicalalgebra A which is at the same time an essential H -module such that the multiplicationmap A y˝ A ! A is H -linear.If A is an H -algebra we will also speak of an action of H on A. Remark thatwe do not assume that an algebra has an identity element. The unitarization A C ofan H -algebra A becomes an H -algebra by considering the trivial action on the extracopy C.According to Theorem 3.2 we can equivalently describe an H -algebra as a bornologicalalgebra A which is at the same time an essential yH -comodule such that the multiplicationis yH -colinear.Under additional assumptions there is another possibility to describe this structurewhich resembles the definition of a coaction in the setting of C -algebras. Let us callan essential bornological algebra A regular if it is equipped with a faithful boundedlinear functional and satisfies the approximation property. If A is regular it followsfrom [27] that the natural bounded linear map A y˝ H ! M.A y˝ H/is injective.Definition 3.5. Let H be a bornological quantum group. An algebra coaction of Hon a regular bornological algebra A is an essential algebra homomorphism ˛ W A !M.A y˝ H/such that the coassociativity condition.˛ y˝ id/˛ D .id y˝ /˛holds and the maps ˛l and ˛r from A y˝ H to M.A y˝ H/given by˛l.a ˝ x/ D .1 ˝ x/˛.a/; ˛r.a ˝ x/ D ˛.a/.1 ˝ x/induce bornological automorphisms of A y˝ H .It can be shown that an algebra coaction ˛ W A ! M.A y˝ H/on a regular bornologicalalgebra A satisfies .id y˝ /˛ D id. In particular, the map ˛ is always injective.Proposition 3.6. Let H be a bornological quantum group and let A be a regularbornological algebra. Then every algebra coaction of yH on A corresponds to a uniqueH -algebra structure on A and vice versa.Proof. Assume that ˛ is an algebra coaction of yH on A and define D ˛r. Bydefinition is a right yH -linear automorphism of A y˝ yH . Moreover we have for a 2 A

Equivariant cyclic homology for quantum groups 157and f; g 2 yH the relation.id y˝ r / 12 .id y˝ r 1 /.a ˝ f ˝ g/ D .id y˝ r /..˛.a/ ˝ 1/.1 ˝ r1 .f ˝ g///D .id y˝ /.˛.a//.1 ˝ r r1 .f ˝ g//D .˛ y˝ id/.˛.a//.1 ˝ f ˝ g/D 12 13 .a ˝ f ˝ g/in M.A y˝ yH y˝ yH/. Using that A is regular we deduce that.id y˝ r / 12 .id y˝ 1r /.a ˝ f ˝ g/ D 12 13 .a ˝ f ˝ g/in A y˝ yH y˝ yH and hence defines a right yH -comodule structure on A. In additionwe have. y˝ id/ 13 23 .a ˝ b ˝ f/D . y˝ id/ 13 .a ˝ ˛.b/.1 ˝ f//D ˛.a/˛.b/.1 ˝ f/D ˛.ab/.1 ˝ f/D . y˝ id/.a ˝ b ˝ f/and it follows that A becomes an H -algebra using this coaction.Conversely, assume that A is an H -algebra implemented by the coactionW A y˝ yH ! A y˝ yH . Define bornological automorphisms l and r of A y˝ yH by l D .id y˝ S 1 / 1 .id y˝ S/; r D :The map l is left yH -linear for the action of yH on the second tensor factor and r isright yH -linear. Since is a compatible with the multiplication we have r . y˝ id/ D . y˝ id/ 13r 23 rand l . y˝ id/ D . y˝ id/ 23l 13l:In addition one has the equation.id y˝ / 12lD .id y˝ / 13rrelating l and r . These properties of the maps l and r imply that˛.a/.b ˝ f/D r .a ˝ f /.b ˝ 1/; .b ˝ f/˛.a/D .b ˝ 1/ l .a ˝ f/defines an algebra homomorphism ˛ from A to M.A y˝ yH/. As in the proof of Proposition7.3 in [27] one shows that ˛ is essential. Observe that we may identify the naturalmap A y˝A .A y˝ yH/ ! A y˝ yH induced by ˛ with 13r W A y˝A .A y˝ yH y˝ y yHH/ !.A y˝A A/ y˝ . yH y˝ y yHH/since A is essential.The maps ˛l and ˛r associated to the homomorphism ˛ can be identified with land r , respectively. Finally, the coaction identity .id y˝ r / 12 .id y˝ r 1 / D 12 13implies .˛ y˝ id/˛ D .id y˝ /˛. Hence ˛ defines an algebra coaction of yH on A.It follows immediately from the constructions that the two procedures describedabove are inverse to each other.

158 C. VoigtTo every H -algebra A one may form the associated crossed product A Ì H . Theunderlying bornological vector space of A Ì H is A y˝ H and the multiplication isdefined by the chain of mapsA y˝ H y˝ A y˝ H 24r A y˝ H y˝ A y˝ H id y˝ y˝id A y˝ A y˝ H y˝id A y˝ Hwhere denotes the action of H on A. Explicitly, the multiplication in A Ì H is givenby the formula.a Ì x/.b Ì y/ D ax .1/ b ˝ x .2/ yfor a; b 2 A and x;y 2 H . On the crossed product A Ì H one has the dual action ofyH defined byf .a Ì x/ D a Ì .f * x/for all f 2 yH . In this way AÌH becomes an yH -algebra. Consequently one may formthe double crossed product AÌH Ì yH . In the remaining part of this section we discussthe Takesaki–Takai duality isomorphism which clarifies the structure of this algebra.First we describe a general construction which will also be needed later in connectionwith stability of equivariant cyclic homology. Assume that V is an essential H -moduleand that A is an H -algebra. Moreover let b W V V ! C be an equivariant boundedlinear map. We define an H -algebra l.bI A/ by equipping the space V y˝ A y˝ V withthe multiplication.v 1 ˝ a 1 ˝ w 1 /.v 2 ˝ a 2 ˝ w 2 / D b.w 1 ;v 2 /v 1 ˝ a 1 a 2 ˝ w 2and the diagonal H -action.As a particular case of this construction consider the space V D yH with the regularaction of H given by .t * f /.x/ D f.xt/and the pairingˇ.f; g/ D O .fg/:We write K H for the algebra l.ˇI C/ and A y˝ K H for l.ˇI A/. Remark that the actionon A y˝ K H is not the diagonal action in general. We denote an element f ˝ a ˝ g inthis algebra by jf i˝a ˝hgj in the sequel. Using the isomorphism yF r S 1 W yH ! Hwe identify the above pairing with a pairing H H ! C. The corresponding actionof H on itself is given by left multiplication and using the normalization D S. /we obtain the formulaˇ.x; y/ D ˇ.SG r S.x/;SG r S.y// D ˇ.F l .x/; F l .y// D .S 1 .y/x/ D .S.x/y/for the above pairing expressed in terms of H .Let H be a bornological quantum group and let A be an H -algebra. We define abounded linear map A W A Ì H Ì yH ! A y˝ K H by A .a Ì x Ì F l .y// Djy .1/ S.x .2/ /i˝y .2/ S.x .1/ / a ˝hy .3/ jand it is easily verified that A is an equivariant bornological isomorphism. In addition, astraightforward computation shows that A is an algebra homomorphism. Consequentlywe obtain the following analogue of the Takesaki–Takai duality theorem.

Equivariant cyclic homology for quantum groups 159Proposition 3.7. Let H be a bornological quantum group and let A be an H -algebra.Then the map A W A Ì H Ì yH ! A y˝ K H is an equivariant algebra isomorphism.For algebraic quantum groups a discussion of Takesaki–Takai duality is containedin [11]. More information on similar duality results in the context of Hopf algebras canbe found in [21].If H D D.G/ is the smooth convolution algebra of a locally compact group G thenan H -algebra is the same thing as a G-algebra. As a special case of Proposition 3.7 oneobtains that for every G-algebra A the double crossed product AÌH Ì yH is isomorphicto the G-algebra A y˝ K G used in [25].4 Radford’s formulaIn this section we prove a formula for the fourth power of the antipode in terms of themodular elements of a bornological quantum group and its dual. This formula wasobtained by Radford in the setting of finite dimensional Hopf algebras [23].Let H be a bornological quantum group. If is a left Haar functional on H thereexists a unique multiplier ı 2 M.H/ such that. y˝ id/.x/ D .x/ıfor all x 2 H . The multiplier ı is called the modular element of H and measuresthe failure of from being right invariant. It is shown in [27] that ı is invertible withinverse S.ı/ D S 1 .ı/ D ı 1 and that one has .ı/ D ı ˝ ı as well as .ı/ D 1.In terms of the dual quantum group the modular element ı defines a character, that is,an essential homomorphism from yH to C. Similarly, there exists a unique modularelement O ı 2 M. yH/for the dual quantum group which satisfies. O y˝ id/ O.f / D O.f / O ıfor all f 2 yH .The Haar functionals of a bornological quantum group are uniquely determined upto a scalar multiple. In many situations it is convenient to fix a normalization at somepoint. However, in the discussion below it is not necessary to keep track of the scalingof the Haar functionals. If ! and are linear functionals we shall write ! if thereexists a nonzero scalar such that ! D . We use the same notation for elements in abornological quantum group or linear maps that differ by some nonzero scalar multiple.Moreover we shall identify H with its double dual using Pontrjagin duality.To begin with observe that the bounded linear functional ı*on H defined byis faithful and satisfies.ı * /.x/ D .xı/..ı * / y˝ id/.x/ D . y˝ id/..xı/.1 ˝ ı 1 // D .xı/ıı 1 D .ı * /.x/:

160 C. VoigtIt follows that ı*is a right Haar functional on H . In a similar way we obtain aright Haar functional (ıon H . Henceı* (ıby uniqueness of the right Haar functional which yields in particular the relationsF l .xı/ G l .x/;F r .ıx/ G r .x/for the Fourier transform.According to Pontrjagin duality we have x D yG l F l S.x/ for all x 2 H . SinceyG l .f / yF l .f O ı/ for every f 2 yH as well as.F l .S.x// ı/.h/ O D .h .1/ S.x// ı.h O .2/ / D .hS.x .2/ // ı.ıS O 2 .x .1/ // F l .S.x .2/ //.h/ ı.x O .1/ / D F l .S.x ( ı//.h/ Owe obtain x yF l F l S.x ( O ı/ or equivalentlyS 1 . O ı*x/ yF l F l .x/: (4.1)Using equation (4.1) and the formula x S 1 yF l F r .x/ obtained from Pontrjaginduality we computewhich impliesyF l F l . O ı 1 *S 2 .x// S 1 S 2 .x/ D S.x/ yF l F r .x/F l .S 2 .x// F r . O ı*x/ (4.2)since yF l is an isomorphism. Similarly, we have yF l SG l id and using yF l .f / yG l .f ı O 1 / together with.SG l .x/ ı O 1 /.h/ D .S.h .1/ /x/ ı O 1 .h .2/ / .S.h/x .2/ / ı O 1 .x .1/ / D SG l .x ( ı O 1 /.h/we obtain yG l SG l .x/ x( O ı. According to the relation F l .xı/ G l .x/ this may berewritten as S 1 yF r F l .xı/ x( O ı which in turn yieldsyF r F l .x/ S..xı 1 /( O ı/: (4.3)Due to Pontrjagin duality we have x D yF r G r S.x/ for all x 2 H and using G r .x/ F r .ıx/ we obtainyF r F r .S.x// yF r G r .ı 1 S.x// D xı: (4.4)According to equation (4.3) and equation (4.4) we haveyF r F l .S 2 .ı 1 .x ( ı O 1 /ı// S..S 2 .ı 1 .x ( ı O 1 /ı/ı 1 /( ı/ O S.S 2 .ı 1 x// D S 1 .ı 1 x/ yF r F r .x/

Equivariant cyclic homology for quantum groups 161and since yF r is an isomorphism this impliesF r .S 2 .x// F l .ı 1 .x ( O ı 1 /ı/: (4.5)Assembling these relations we obtain the following result.Proposition 4.1. Let H be a bornological quantum group and let ı and O ı be the modularelements of H and yH , respectively. ThenS 4 .x/ D ı 1 . O ı*x( O ı 1 /ıfor all x 2 H .Proof. Using equation (4.2) and equation (4.5) we computeF l .S 4 .x// F r . O ı*S 2 .x// D F r .S 2 . O ı*x// F l .ı 1 . O ı*x( O ı 1 /ı/which impliesS 4 .x/ ı 1 . ı*x( O ı O 1 /ıfor all x 2 H since F l is an isomorphism. The claim follows from the observation thatboth sides of the previous equation define algebra automorphisms of H .5 Anti-Yetter–Drinfeld modulesIn this section we introduce the notion of an anti-Yetter–Drinfeld module over a bornologicalquantum group. Moreover we discuss the concept of a paracomplex.We begin with the definition of an anti-Yetter–Drinfeld module. In the context ofHopf algebras this notion was introduced in [12].Definition 5.1. Let H be a bornological quantum group. An H -anti-Yetter–Drinfeldmodule is an essential left H -module M which is also an essential left yH -module suchthatt .f m/ D .S 2 .t .1/ /*f (S 1 .t .3/ // .t .2/ m/:for all t 2 H; f 2 yH and m 2 M . A homomorphism W M ! N between anti-Yetter–Drinfeld modules is a bounded linear map which is both H -linear and yH -linear.We will not always mention explicitly the underlying bornological quantum groupwhen dealing with anti-Yetter–Drinfeld modules. Moreover we shall use the abbreviationsAYD-module and AYD-map for anti-Yetter–Drinfeld modules and their homomorphisms.According to Theorem 3.2 a left yH -module structure corresponds to a right H -comodule structure. Hence an AYD-module can be described equivalently as a bornologicalvector space M equipped with an essential H -module structure and an H -comodulestructure satisfying a certain compatibility condition. Formally, this compatibility conditioncan be written down asfor all t 2 H and m 2 M ..t m/ .0/ ˝ .t m/ .1/ D t .2/ m .0/ ˝ t .3/ m .1/ S.t .1/ /

162 C. VoigtWe want to show that AYD-modules can be interpreted as essential modules overa certain bornological algebra. Following the notation in [12] this algebra will bedenoted by A.H /. As a bornological vector space we have A.H / D yH y˝ H and themultiplication is defined by the formula.f ˝ x/ .g ˝ y/ D f.S 2 .x .1/ /*g(S 1 .x .3/ // ˝ x .2/ y:There exists an algebra homomorphism H W H ! M.A.H // given by H .x/ .g ˝ y/ D S 2 .x .1/ /*g(S 1 .x .3/ / ˝ x .2/ yand.g ˝ y/ H .x/ D g ˝ yx:It is easily seen that H is injective. Similarly, there is an injective algebra homomorphismH y W yH ! M.A.H // given byas well asH y .f / .g ˝ y/ D fg ˝ y.g ˝ y/ y H .f / D g.S2 .y .1/ /*f (S 1 .y .3/ // ˝ y .2/and we have the following result.Proposition 5.2. For every bornological quantum group H the bornological algebraA.H / is essential.Proof. The homomorphism H y induces on A.H / the structure of an essential left yH -module. Similarly, the space A.H / becomes an essential left H -module using thehomomorphism H . In fact, if we write yH for the space yH equipped with the trivialH -action the map c W A.H / ! yH y˝ H given byc.f ˝ x/ D S.x .1/ /*f (x .3/ ˝ x .2/is an H -linear isomorphism. Actually, the actions of yH and H on A.H / are definedin such a way that A.H / becomes an AYD-module. Using the canonical isomorphismH y˝H A.H / Š A.H / one obtains an essential yH -module structure on H y˝H A.H /given explicitly by the formulaf .x ˝ g ˝ y/ D x .2/ ˝ .S.x .1/ /*f (x .3/ /g ˝ y:Correspondingly, we obtain a natural isomorphism yH y˝ yH .H y˝H A.H // Š A.H /.It is straightforward to verify that the identity map yH y˝ .H y˝ A.H // Š yH y˝ H y˝yH y˝ H ! A.H / y˝ A.H / induces an isomorphism yH y˝ yH .H y˝H A.H // !A.H / y˝A.H / A.H /. This yields the claim.We shall now characterize AYD-modules as essential modules over A.H /.

Equivariant cyclic homology for quantum groups 163Proposition 5.3. Let H be a bornological quantum group. Then the category ofAYD-modules over H is isomorphic to the category of essential left A.H /-modules.Proof. Let M Š A.H / y˝A.H / M be an essential A.H /-module. Then we obtaina left H -module structure and a left yH -module structure on M using the canonicalhomomorphisms H W H ! M.A.H // and y H W yH ! M.A.H //. Since the action ofH on A.H / is essential we have natural isomorphismsH y˝H M Š H y˝H A.H / y˝A.H / M Š A.H / y˝A.H / M Š Mand hence M is an essential H -module. Similarly we haveyH y˝ yH M Š yH y˝ yH A.H / y˝A.H /y˝ M Š A.H / y˝A.H / M Š Msince A.H / is an essential yH -module. These module actions yield the structure of anAYD-module on M .Conversely, assume that M is an H -AYD-module. Then we obtain an A.H /-modulestructure on M by setting.f ˝ t/ m D f .t m/for f 2 yH and t 2 H . Since M is an essential H -module we have a naturalisomorphism H y˝H M Š M . As in the proof of Proposition 5.2 we obtain aninduced essential yH -module structure on H y˝H M and canonical isomorphismsA.H / y˝A.H / M Š yH y˝ yH .H y˝H M/ Š M . It follows that M is an essentialA.H /-module.The previous constructions are compatible with morphisms and it is easy to checkthat they are inverse to each other. This yields the assertion.There is a canonical operator T on every AYD-module which plays a crucial rolein equivariant cyclic homology. In order to define this operator it is convenient topass from yH to H in the first tensor factor of A.H /. More precisely, consider thebornological isomorphism W A.H / ! H y˝ H given by.f ˝ y/ D yF l .f / ˝ y( O ı 1where O ı 2 M. yH/is the modular function of yH . The inverse map is given by 1 .x ˝ y/ D SG l .x/ ˝ y( O ı:It is straightforward to check that the left H -action on A.H / corresponds toand the left yH -action becomest .x ˝ y/ D t .3/ xS.t .1/ / ˝ t .2/ yf .x ˝ y/ D .f * x/ ˝ y

164 C. Voigtunder this isomorphism. The right H -action is identified withand the right yH -action corresponds to.x ˝ y/ t D x ˝ y.t ( O ı 1 /.x ˝ y/ g D x .2/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 .x .1/ /ı/ ˝ Oı.y .1/ /y .3/ ( O ı 1where ı is the modular function of H . Using this description of A.H / we obtain thefollowing result.Proposition 5.4. The bounded linear map T W A.H / ! A.H / defined byis an isomorphism of A.H /-bimodules.T.x˝ y/ D x .2/ ˝ S 1 .x .1/ /yProof. It is evident that T is a bornological isomorphism with inverse given byT 1 .x ˝ y/ D x .2/ ˝ x .1/ y. We computeT.t .x ˝ y// D T.t .3/ xS.t .1/ / ˝ t .2/ y/D t .5/ x .2/ S.t .1/ / ˝ S 1 .t .4/ x .1/ S.t .2/ //t .3/ yD t .3/ x .2/ S.t .1/ / ˝ t .2/ S 1 .x .1/ /yD t T.x˝ y/and it is clear that T is left yH -linear. Consequently T is a left A.H /-linear map.Similarly, we haveT ..x ˝ y/ t/ D T.x˝ y.t ( O ı 1 // D x .2/ ˝ S 1 .x .1/ /y.t ( O ı 1 / D T.x˝ y/ tand hence T is right H -linear. In order to prove that T is right yH -linear we computeT 1 ..x ˝ y/ g/D T 1 .x .2/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 .x .1/ /ı/ ˝ Oı.y .1/ /y .3/ ( ı O 1 /D x .3/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 .x .1/ /ı/ ˝ Oı.y .1/ /x .2/ .y .3/ ( ı O 1 /D x .3/ .S 2 .y .2/ /*g(S 1 .y .4/ //.S 2 . ı*x O .1/ /ı/ ˝ Oı.y .1/ /.x .2/ y .3/ /( ı O 1which according to Proposition 4.1 is equal toD x .3/ .S 2 .y .2/ /*g(S 1 .y .4/ //.ıS 2 .x .1/ ( O ı// ˝ Oı.y .1/ /.x .2/ y .3/ /( O ı 1D x .6/ .S 2 .x .2/ /S 2 .y .2/ /*g(S 1 .y .4/ /S 1 .x .4/ //.S 2 .x .5/ /ı/˝ Oı.x .1/ y .1/ /.x .3/ y .3/ /( ı O 1D .x .2/ ˝ x .1/ y/ gD T 1 .x ˝ y/ g:We conclude that T is a right A.H /-linear map as well.

Equivariant cyclic homology for quantum groups 165If M is an arbitrary AYD-module we define T W M ! M byT.F ˝ m/ D T.F/˝ mfor F ˝ m 2 A.H / y˝A.H / M . Due to Proposition 5.3 and Lemma 5.4 this definitionmakes sense.Proposition 5.5. The operator T defines a natural isomorphism T W id ! id of theidentity functor on the category of AYD-modules.Proof. It is clear from the construction that T W M ! M is an isomorphism for all M .If W M ! N is an AYD-map the equation T D T follows easily after identifying with the map id y˝ W A.H / y˝A.H / M ! A.H / y˝A.H / N . This yields the assertion.If the bornological quantum group H is unital one may construct the operator Ton an AYD-module M directly from the coaction M ! M y˝ H corresponding to theaction of yH . More precisely, one has the formulaT .m/ D S 1 .m .1/ / m .0/for every m 2 M .Using the terminology of [25] it follows from Proposition 5.5 that the category ofAYD-modules is a para-additive category in a natural way. This leads in particular tothe concept of a paracomplex of AYD-modules.Definition 5.6. A paracomplex C D C 0 ˚ C 1 consists of AYD-modules C 0 and C 1 andAYD-maps @ 0 W C 0 ! C 1 and @ 1 W C 1 ! C 0 such that@ 2 D idTwhere the differential @W C ! C 1 ˚ C 0 Š C is the composition of @ 0 ˚ @ 1 with thecanonical flip map.The morphism @ in a paracomplex is called a differential although it usually does notsatisfy the relation @ 2 D 0. As for ordinary complexes one defines chain maps betweenparacomplexes and homotopy equivalences. We always assume that such maps arecompatible with the AYD-module structures. Let us point out that it does not makesense to speak about the homology of a paracomplex in general.The paracomplexes we will work with arise from paramixed complexes in the followingsense.Definition 5.7. A paramixed complex M is a sequence of AYD-modules M n togetherwith AYD-maps b of degree 1 and B of degree C1 satisfying b 2 D 0, B 2 D 0 andŒb; B D bB C Bb D id T:If T is equal to the identity operator on M this reduces of course to the definitionof a mixed complex.

166 C. Voigt6 Equivariant differential formsIn this section we define equivariant differential forms and the equivariant X-complex.Moreover we discuss the properties of the periodic tensor algebra of an H -algebra.These are the main ingredients in the construction of equivariant cyclic homology.Let H be a bornological quantum group. If A is an H -algebra we obtain a leftaction of H on the space H y˝ n .A/ byt .x ˝ !/ D t .3/ xS.t .1/ / ˝ t .2/ !for t;x 2 H and ! 2 n .A/. Here n .A/ D A C y˝ A y˝n for n>0is the space ofnoncommutative n-forms over A with the diagonal H -action. For n D 0 one defines 0 .A/ D A. There is a left action of the dual quantum group yH on H y˝ n .A/ givenbyf .x ˝ !/ D .f * x/ ˝ ! D f.x .2/ /x .1/ ˝ !:By definition, the equivariant n-forms n H .A/ are the space H y˝ n .A/ together withthe H -action and the yH -action described above. We computet .f .x ˝ !// D t .f .x .2/ /x .1/ ˝ !/D f.x .2/ /t .3/ x .1/ S.t .1/ / ˝ t .2/ !D .S 2 .t .1/ /*f (S 1 .t .5/ // .t .4/ xS.t .2/ / ˝ t .3/ !/D .S 2 .t .1/ /*f (S 1 .t .3/ // .t .2/ .x ˝ !//and deduce that n H .A/ is an H -AYD-module. We let H .A/ be the direct sum of thespaces n H .A/.Now we define operators d and b H on H .A/ byd.x ˝ !/ D x ˝ d!andb H .x ˝ !da/ D . 1/ j!j .x ˝ !ax .2/ ˝ .S 1 .x .1/ / a/!/:The map b H should be thought of as a twisted version of the usual Hochschild operator.We computebH 2 .x ˝ !dadb/ D . 1/j!jC1 b H .x ˝ !dab x .2/ ˝ .S 1 .x .1/ / b/!da/D . 1/ j!jC1 b H .x ˝ !d.ab/ x ˝ !adb x .2/ ˝ .S 1 .x .1/ / b/!da/D .x ˝ !ab x .2/ ˝ S 1 .x .1/ / .ab/! x ˝ !ab C x .2/ ˝ .S 1 .x .1/ / b/!ax .2/ ˝ .S 1 .x .1/ / b/!a C x .2/ ˝ S 1 .x .1/ / .ab/!/ D 0which shows that bH 2 is a differential as in the nonequivariant situation. Let us discussthe compatibility of d and b H with the AYD-module structure. It is easy to check that

Equivariant cyclic homology for quantum groups 167d is an AYD-map and that the operator b H is yH -linear. Moreover we computeb H .t .x ˝ !da// D . 1/ j!j .t .4/ xS.t .1/ / ˝ .t .2/ !/.t .3/ a/t .6/ x .2/ S.t .1/ / ˝ .S 1 .t .5/ x .1/ S.t .2/ //t .4/ a/.t .3/ !//D . 1/ j!j .t .3/ xS.t .1/ / ˝ t .2/ .!a/ t .4/ x .2/ S.t .1/ / ˝ .t .2/ S 1 .x .1/ / a/.t .3/ !//D t b H .x ˝ !da/and deduce that b H is an AYD-map as well.Similar to the non-equivariant case we use d and b H to define an equivariant Karoubioperator H and an equivariant Connes operator B H byand H D 1 .b H d C db H /B H DnX j H don H n .A/. Let us record the following explicit formulas. For n>0we havej D0 H .x ˝ !da/ D . 1/ n1 x .2/ ˝ .S 1 .x .1/ / da/!on H n .A/ and in addition H .x ˝ a/ D x .2/ ˝ S 1 .x .1/ / a on H 0 .A/. For theConnes operator we computenXB H .x˝a 0 da 1 :::da n /D . 1/ ni x .2/˝S 1 .x .1/ /.da nC1 i :::da n /:da 0 :::da n iiD0Furthermore, the operator T is given byT.x˝ !/ D x .2/ ˝ S 1 .x .1/ / !on equivariant differential forms. Observe that all operators constructed so far areAYD-maps and thus commute with T according to Proposition 5.5.Lemma 6.1. On H n .A/ the following relations hold:a) nC1Hd D Td,b) H n D T C b H H n d,c) n H b H D b H T ,d) nC1H D .id db H /T ,e) . nC1HT /. n HT/D 0,f) B H b H C b H B H D id T .

168 C. VoigtProof. a) follows from the explicit formula for H . b) We compute n H .x ˝ a 0da 1 :::da n / D x .2/ ˝ S 1 .x .1/ / .da 1 :::da n /a 0D x .2/ ˝ S 1 .x .1/ / .a 0 da 1 :::da n /C . 1/ n b H .x .2/ ˝ S 1 .x .1/ / .da 1 :::da n /da 0 /D x .2/ ˝ S 1 .x .1/ / .a 0 da 1 :::da n / C b H H n d.x ˝ a 0da 1 :::da n /which yields the claim. c) follows by applying b H to both sides of b). d) Apply H tob) and use a). e) is a consequence of b) and d). f) We computeXn 1B H b H C b H B H D j H db H Cj D0where we use d) and b).nXn 1b H j H d D X j H .db H C b H d/C H n b H dj D0j D0D id H n .1 b H d/ D id H n . H C db H /D id T C db H T Tdb H D id TFrom the definition of B H and the fact that d 2 D 0 we obtain BH2summarize this discussion as follows.D 0. Let usProposition 6.2. Let A be an H -algebra. The space H .A/ of equivariant differentialforms is a paramixed complex in the category of AYD-modules.We remark that the definition of H .A/ for H D D.G/ differs slightly from thedefinition of G .A/ in [25] if the locally compact group G is not unimodular. However,this does not affect the definition of the equivariant homology groups.In the sequel we will drop the subscripts when working with the operators on H .A/introduced above. For instance, we shall simply write b instead of b H and B insteadof B H .The n-th level of the Hodge tower associated to H .A/ is defined byMn 1 n H .A/ D j H .A/ ˚ n H .A/=b.nC1 H.A//:j D0Using the grading into even and odd forms we see that n H .A/ together with theboundary operator B C b becomes a paracomplex. By definition, the Hodge tower H .A/ of A is the projective system . n H .A// n2N obtained in this way.From a conceptual point of view it is convenient to work with pro-categories in thesequel. The pro-category pro.C/ over a category C consists of projective systems inC. A pro-H -algebra is simply an algebra in the category pro.H -Mod/. For instance,every H -algebra becomes a pro-H -algebra by viewing it as a constant projective system.More information on the use of pro-categories in connection with cyclic homology canbe found in [10], [25].

Equivariant cyclic homology for quantum groups 169Definition 6.3. Let A be a pro-H -algebra. The equivariant X-complex X H .A/ of Ais the paracomplex 1 H .A/. Explicitly, we haveX H .A/W H 0 .A/ d 1 H.A/=b.H 2 .A//:bWe are interested in particular in the equivariant X-complex of the periodic tensoralgebra T A of an H -algebra A. The periodic tensor algebra T A is the even part of.A/ equipped with the Fedosov product given by! ı D ! . 1/ j!j d!dfor homogenous forms ! and . The natural projection .A/ ! A restricts to anequivariant homomorphism A W T A ! A and we obtain an extensionJA T A A Aof pro-H -algebras.The main properties of the pro-algebras T A and JA are explained in [20], [25]. Letus recall some terminology. We write n W N y˝n ! N for the iterated multiplicationin a pro-H -algebra N . Then N is called locally nilpotent if for every equivariant prolinearmap f W N ! C with constant range C there exists n 2 N such that f n D 0.It is straightforward to check that the pro-H -algebra JA is locally nilpotent.An equivariant pro-linear map l W A ! B between pro-H -algebras is called alonilcur if its curvature ! l W A y˝ A ! B defined by ! l .a; b/ D l.ab/ l.a/l.b/is locally nilpotent, that is, if for every equivariant pro-linear map f W B ! C withconstant range C there exists n 2 N such that fB n ! y˝n D 0. The term lonilcur is anlabbreviation for "equivariant pro-linear map with locally nilpotent curvature". SinceJA is locally nilpotent the natural map A W A ! T A is a lonilcur.Proposition 6.4. Let A be an H -algebra. The pro-H -algebra T A and the lonilcur A W A ! T A satisfy the following universal property. If l W A ! B is a lonilcur intoapro-H -algebra B there exists a unique equivariant homomorphism lW T A ! Bsuch that l A D l.An important ingredient in the Cuntz–Quillen approach to cyclic homology [8], [9],[10] is the concept of a quasifree pro-algebra. The same is true in the equivariant theory.Definition 6.5. A pro-H -algebra R is called H -equivariantly quasifree if there existsan equivariant splitting homomorphism R ! T R for the projection R .We state some equivalent descriptions of equivariantly quasifree pro-H -algebras.Theorem 6.6. Let H be a bornological quantum group and let R beapro-H -algebra.The following conditions are equivalent:a) R is H -equivariantly quasifree.

170 C. Voigtb) There exists an equivariant pro-linear map rW 1 .R/ ! 2 .R/ satisfyingfor all a 2 R and ! 2 1 .R/.r.a!/ D ar.!/; r.!a/ Dr.!/a !dac) There exists a projective resolution 0 ! P 1 ! P 0 ! R C of the R-bimodule R Cof length 1 in pro.H -Mod/.A map rW 1 .R/ ! 2 .R/ satisfying condition b) in Theorem 6.6 is also calledan equivariant graded connection on 1 .R/.We have the following basic examples of quasifree pro-H -algebras.Proposition 6.7. Let A be any H -algebra. The periodic tensor algebra T A is H -equivariantlyquasifree.An important result in theory of Cuntz and Quillen relates the X-complex of theperiodic tensor algebra T A to the standard complex of A constructed using noncommutativedifferential forms. The comparison between the equivariant X-complex andequivariant differential forms is carried out in the same way as in the group case [25].Proposition 6.8. There is a natural isomorphism X H .T A/ Š H .A/ such that thedifferentials of the equivariant X-complex correspond to@ 1 D b .id C/d on oddH .A/@ 0 DXn 1 2j b C Bj D0on 2nH .A/:Theorem 6.9. Let H be a bornological quantum group and let A be an H -algebra.Then the paracomplexes H .A/ and X H .T A/ are homotopy equivalent.For the proof of Theorem 6.9 it suffices to observe that the corresponding argumentsin [25] are based on the relations obtained in Proposition 6.1.7 Equivariant periodic cyclic homologyIn this section we define equivariant periodic cyclic homology for bornological quantumgroups.Definition 7.1. Let H be a bornological quantum group and let A and B be H -algebras.The equivariant periodic cyclic homology of A and B isHP H .A; B/ D H .Hom A.H / .X H .T .A Ì H Ì yH //; X H .T .B Ì H Ì yH ///:We write Hom A.H / for the space of AYD-maps and consider the usual differential fora Hom-complex in this definition. Using Proposition 5.5 it is straightforward to check

Equivariant cyclic homology for quantum groups 171that this yields indeed a complex. Remark that both entries in the above Hom-complexare only paracomplexes.Let us consider the special case that H D D.G/ is the smooth group algebraof a locally compact group G. In this situation the definition of HPH reduces tothe definition of HPG given in [25]. This is easily seen using the Takesaki–Takaiisomorphism obtained in Proposition 3.7 and the results from [27].As in the group case HPH is a bifunctor, contravariant in the first variable andcovariant in the second variable. We define HP H .A/ D HP H .C;A/to be the equivariantperiodic cyclic homology of A and HPH .A/ D HP H .A; C/ to be equivariantperiodic cyclic cohomology. There is a natural associative productHP Hi.A; B/ HP Hj.B; C / ! HPHiCj.A; C /;.x; y/ 7! x yinduced by the composition of maps. Every equivariant homomorphism f W A ! Bdefines an element in HP0 H .A; B/ denoted by Œf . The element Œid 2 HP 0 H .A; A/is denoted 1 or 1 A . An element x 2 HP H .A; B/ is called invertible if there exists anelement y 2 HP H .B; A/ such that x y D 1 A and y x D 1 B . An invertible elementof degree zero is called an HP H -equivalence. Such an element induces isomorphismsHP H .A; D/ Š HP H .B; D/ and HP H .D; A/ Š HP H .D; B/ for all H -algebras D.8 Homotopy invariance, stability and excisionIn this section we show that equivariant periodic cyclic homology is homotopy invariant,stable and satisfies excision in both variables. Since the arguments carry over from thegroup case with minor modifications most of the proofs will only be sketched. Moredetails can be found in [25].We begin with homotopy invariance. Let B be a pro-H -algebra and consider theFréchet algebra C 1 Œ0; 1 of smooth functions on the interval Œ0; 1. We denote byBŒ0; 1 the pro-H -algebra B y˝ C 1 Œ0; 1 where the action on C 1 Œ0; 1 is trivial. A(smooth) equivariant homotopy is an equivariant homomorphism ˆW A ! BŒ0; 1 ofH -algebras. Evaluation at the point t 2 Œ0; 1 yields an equivariant homomorphismˆt W A ! B. Two equivariant homomorphisms from A to B are called equivariantlyhomotopic if they can be connected by an equivariant homotopy.Theorem 8.1 (Homotopy invariance). Let A and B be H -algebras and let ˆW A !BŒ0; 1 be a smooth equivariant homotopy. Then the elements Œˆ0 and Œˆ1 inHP0 H .A; B/ are equal. Hence the functor HP H is homotopy invariant in both variableswith respect to smooth equivariant homotopies.Recall that 2 H .A/ is the paracomplex 0 H .A/ ˚ 1 H .A/ ˚ 2 H .A/=b.3 H .A//with the usual differential B C b and the grading into even and odd forms for anypro-H -algebra A. There is a natural chain map 2 W 2 H .A/ ! X H .A/.Proposition 8.2. Let A be an equivariantly quasifree pro-H -algebra. Then the map 2 W 2 H .A/ ! X H .A/ is a homotopy equivalence.

Equivariant cyclic homology for quantum groups 173andtr A .x ˝ .v 0 ˝ a 0 ˝ w 0 /d.v 1 ˝ a 1 ˝ w 1 //D b..S 1 .x .1/ / w 1 ;v 0 /b.w 0 ;v 1 /x .2/ ˝ a 0 da 1 :In these formulas we implicitly use the twisted trace tr x W l.b/ ! C for x 2 H definedby tr x .v ˝ w/ D b..S 1 .x/ w; v/. The twisted trace satisfies the relationtr x .T 0 T 1 / D tr x.2/ ..S 1 .x .1/ / T 1 /T 0 /for all T 0 ;T 1 2 l.b/. Using this relation one verifies that tr A defines a chain map. It isclear that tr A is yH -linear and it is straightforward to check that tr A is H -linear. Let usdefine t A D tr A ı X H . A / and show that Œt A is an inverse for Œ A . Using the fact that uis H -invariant one computes Œ A Œt A D 1.WehavetoproveŒt A Œ A D 1. Considerthe following equivariant homomorphisms l.bI A/ ! l.bI l.bI A// given byi 1 .v ˝ a ˝ w/ D u ˝ v ˝ a ˝ w ˝ u;i 2 .v ˝ a ˝ w/ D v ˝ u ˝ a ˝ u ˝ w:As above we see Œi 1 Œt l.bIA/ D 1 and we determine Œi 2 Œt l.bIA/ D Œt A Œ A . Let h tbe the linear map from l.bI A/ into l.bI l.bI A// given byh t .v ˝ a ˝ w/ D cos.t=2/ 2 u ˝ v ˝ a ˝ w ˝ u C sin.t=2/ 2 v ˝ u ˝ a ˝ u ˝ wi cos.t=2/ sin.t=2/u ˝ v ˝ a ˝ u ˝ wC i sin.t=2/ cos.t=2/v ˝ u ˝ a ˝ w ˝ u:The family h t depends smoothly on t and we have h 0 D i 1 and h 1 D i 2 . Since u isinvariant the map h t is in fact equivariant and one checks that h t is a homomorphism.Hence we have indeed defined a smooth homotopy between i 1 and i 2 . This yieldsŒi 1 D Œi 2 and hence Œt A Œ A D 1.We derive the following general stability theorem.Proposition 8.5 (Stability). Let H be a bornological quantum group and let A bean H -algebra. Moreover let V be an essential H -module and let b W V V ! Cbe a nonzero equivariant bilinear pairing. Then there exists an invertible element inHP0 G .A; l.bI A//. Hence there are natural isomorphismsHP H .l.bI A/; B/ Š HP Hfor all H -algebras A and B..A; B/ HPH .A; B/ Š HP H .A; l.bI B//Proof. Let us write ˇ W H H ! C for the canonical equivariant bilinear pairingintroduced in Section 3. Moreover we denote by b the pairing b W V V ! Cwhere V is the space V equipped with the trivial H -action. We have an equivariantisomorphism W l.b I l.ˇI A// Š l.ˇI l.bI A// given by.v ˝ .x ˝ a ˝ y/ ˝ w/ D x .1/ ˝ x .2/ v ˝ a ˝ y .1/ w ˝ y .2/

174 C. Voigtand using ˇ.x; y/ D .S.y/x/ as well as the fact that is right invariant one checksthat is an algebra homomorphism. Now we can apply Theorem 8.4 to obtain theassertion.We deduce a simpler description of equivariant periodic cyclic homology in certaincases. A bornological quantum group H is said to be of compact type if the dualalgebra yH is unital. Moreover let us call H of semisimple type if it is of compact typeand the value of the integral for yH on 1 2 yH is nonzero. For instance, the dual of acosemisimple Hopf algebra yH is of semisimple type.Proposition 8.6. Let H be a bornological quantum group of semisimple type. Thenwe haveHP H .A; B/ Š H .Hom A.H / .X H .T A/; X H .T B///for all H -algebras A and B.Proof. Under the above assumptions the canonical bilinear pairing ˇ W yH yH ! C isadmissible since the element 1 2 yH is invariant.Finally we discuss excision in equivariant periodic cyclic homology. Consider anextensionK E Qof H -algebras equipped with an equivariant linear splitting W Q ! E for the quotientmap W E ! Q.Let X H .T E W T Q/ be the kernel of the map X H .T / W X H .T E/ ! X G .T Q/ inducedby . The splitting yields a direct sum decomposition X H .T E/ D X H .T E WT Q/ ˚ X H .T Q/ of AYD-modules. The resulting extensionX H .T E W T Q/ X H .T E/ X H .T Q/of paracomplexes induces long exact sequences in homology in both variables. Thereis a natural covariant map W X H .T K/ ! X H .T E W T Q/ of paracomplexes and wehave the following generalized excision theorem.Theorem 8.7. The map W X H .T K/ ! X H .T E W T Q/ is a homotopy equivalence.This result implies excision in equivariant periodic cyclic homology.Theorem 8.8 (Excision). Let A be an H -algebra and let .; /W 0 ! K ! E !Q ! 0 be an extension of H -algebras with a bounded linear splitting. Then there aretwo natural exact sequencesHP0 H .A; K/ HP 0 H .A; E/ HPH0.A; Q/HP1 H .A; Q/ HP1 H .A; E/ HP1 H .A; K/

Equivariant cyclic homology for quantum groups 175andHP0 H .Q; A/ HP 0 H .E; A/ HPH0.K; A/HP1 H .K; A/ HP1 H .E; A/ HP1 H .Q; A/.The horizontal maps in these diagrams are induced by the maps in the extension.In Theorem 8.8 we only require a linear splitting for the quotient homomorphism W E ! Q. Taking double crossed products of the extension given in Theorem 8.8yields an extensionK y˝ K H E y˝ K H Q y˝ K Hof H -algebras with a linear splitting. Using Lemma 3.3 one obtains an equivariantlinear splitting for this extension. Now we can apply Theorem 8.7 to this extension andobtain the claim by considering long exact sequences in homology.For the proof of Theorem 8.7 one considers the left ideal L T E generated by K T E. The natural projection E W T E ! E induces an equivariant homomorphism W L ! K and one obtains an extensionN L Kof pro-H -algebras. The pro-H -algebra N is locally nilpotent and Theorem 8.7 followsfrom the following assertion.Theorem 8.9. With the notations as above we havea) The pro-H -algebra L is equivariantly quasifree.b) The inclusion map L ! T E induces a homotopy equivalence W X H .L/ !X H .T E W T Q/.In order to prove the first part of Theorem 8.9 one constructs explicitly a projectiveresolution of L of length one.9 Comparison to previous approachesIn this section we discuss the relation of the theory developed above to the previousapproaches due to Akbarpour and Khalkhali [1], [2] as well as Neshveyev and Tuset[22]. As a natural domain in which all these theories are defined we choose to workwith actions of cosemisimple Hopf algebras over the complex numbers. Note that acosemisimple Hopf algebra H is a bornological quantum group with the fine bornology.In the sequel all vector spaces are equipped with the fine bornology when viewed asbornological vector spaces.In their study of cyclic homology of crossed products [1] Akbarpour and Khalkhalidefine a cyclic module C H .A/ of equivariant chains associated to a unital H -module

176 C. Voigtalgebra A. The space Cn H .A/ in degree n of this cyclic module is the coinvariant spaceof H ˝ A˝nC1 with respect to a certain action of H . Recall that the coinvariant spaceV H associated to an H -module V is the quotient of V by the linear subspace generatedby all elements of the form t v .t/v. Using the natural identification n H .A/ D H ˝ A˝nC1 ˚ H ˝ A˝nthe action considered by Akbarpour and Khalkhali corresponds precisely to the actionof H on the first summand in this decomposition. Hence Cn H .A/ can be identifiedas a direct summand in the coinvariant space H n .A/ H . Moreover, the cyclic modulestructure of C H .A/ reproduces the boundary operators b and B of H .A/ H . We pointout that the relation T D id holds on the coinvariant space H .A/ H which means thatthe latter is always a mixed complex.It follows that there is a natural isomorphism of the cyclic type homologies associatedto the cyclic module C H .A/ and the mixed complex H .A/ H , respectively. Notealso that the complementary summand of C H .A/ in H .A/ H is obtained from thebar complex of A tensored with H . Since A is assumed to be a unital H -algebra, thiscomplementary summand is contractible with respect to the differential induced fromthe Hochschild boundary of H .A/ H .In the cohomological setting Akbarpour and Khalkhali introduce a cocyclic moduleCH .A/ for every unital H -module algebra. The definition of this cocyclic modulegiven in [2] is not literally dual to the one of the cyclic module C H .A/. In order toestablish the connection to our constructions let first A be an arbitrary H -algebra. Wedefine a modified action of H on H .A/ by the formulat ı .x ˝ !/ D t .2/ xS 1 .t .3/ / ˝ t .1/ !and write H .A/ for the space H .A/ equipped with this action. Let us comparethe modified action with the original actiont .x ˝ !/ D t .3/ xS.t .1/ / ˝ t .2/ !introduced in Section 6. In the space H .A/ H of coinvariants with respect to theoriginal action we havet ı .x ˝ !/ D t .2/ xS 1 .t .3/ / ˝ t .1/ ! D t .4/ xS 1 .t .5/ /t .1/ S.t .2/ / ˝ t .3/ !D t .2/ .xS 1 .t .3/ /t .1/ ˝ !/ D xS 1 .t .2/ /t .1/ ˝ ! D .t/ x ˝ !which implies that the canonical projection H .A/ ! H .A/ H factorizes over thecoinvariant space H .A/ Hwith respect to the modified action. Similarly, in the coinvariantspace H .A/ Hwe havet .x ˝ !/ D t .3/ xS.t .1/ / ˝ t .2/ ! D t .3/ xS.t .1/ /t .5/ S 1 .t .4/ / ˝ t .2/ !D t .2/ ı .xS.t .1/ /t .3/ ˝ !/ D xS.t .1/ /t .2/ ˝ ! D .t/ x ˝ !:As a consequence we see that the identity map on H .A/ induces an isomorphism H .A/ H Š H .A/ H

Equivariant cyclic homology for quantum groups 177between the coinvariant spaces.We may view a linear map H .A/ ! C as a linear map .A/ ! F.H/ whereF.H/ denotes the linear dual space of H . Under this identification an element inHom H . H .A/ ; C/ corresponds to a linear map f W .A/ ! F.H/ satisfying theequivariance conditionf.t !/.x/ D f .!/.S.t .2/ /xt .1/ /for all t 2 H and ! 2 .A/. In a completely analogous fashion to the case of homologydiscussed above, a direct inspection using the canonical isomorphismsHom H . H .A/ ; C/ Š Hom. H .A/ H ; C/ Š Hom. H .A/ H ; C/shows that the cyclic type cohomologies of the cocyclic module CH .A/ agree forevery unital H -algebra A with the ones associated to the mixed complex H .A/ H .Inparticular, the definition in [2] is indeed obtained by dualizing the construction givenin [1].The main difference between the cocyclic module used byAkbarbour and Khalkhaliand the definition in [22] is that Neshveyev and Tuset work with right actions instead ofleft actions. It is explained in [22] that the two approaches lead to isomorphic cocyclicmodules and hence to isomorphic cyclic type cohomologies.Now assume that H is a semisimple Hopf algebra. Then the coinvariant space H .A/ H is naturally isomorphic to the space H .A/ H of invariants. If A is a unital H -algebra then Theorem 6.9 and Proposition 8.6, together with the above considerations,yield a natural isomorphismHP .C H .A// Š HP H .C;A/which identifies the periodic cyclic homology of the cyclic module C H .A/ with theequivariant cyclic homology of A in the sense of Definition 7.1. Similarly, for asemisimple Hopf algebra H one obtains a natural isomorphismHP .C H .A// Š HP H .A; C/for every unital H -algebra A.Both of these isomorphisms fail to hold more generally, even in the classical settingof group actions. Let be a discrete group and consider the group ring H D C. Forthe H -algebra C with the trivial action one easily obtainsHP .C H .C// Š Hom H .H ad ; C/located in degree zero where H ad is the space H D C viewed as an H -module withthe adjoint action. On the other hand, a result in [25] shows that there is a naturalisomorphismHP H .C; C/ Š HP .H /which identifies the H -equivariant theory of the complex numbers with the periodiccyclic cohomology of H D C. This is the result one should expect from equivariant

178 C. VoigtKK-theory [16]. Hence, roughly speaking, the theory in [2], [22] only recovers thedegree zero part of the group cohomology. A similar remark applies to the homologygroups defined in [1].We mention that Akbarbour and Khalkhali have obtained an analogue of the Green–Julg isomorphismHP H .C;A/Š HP .A Ì H/if H is semisimple and A is a unital H -algebra [1]. This result holds in fact moregenerally in the case that H is the convolution algebra of a compact quantum groupand A is an arbitrary H -algebra. Similarly, there is a dual versionHP H .A; C/ Š HP .A Ì H/of the Green–Julg theorem for the convolution algebras of discrete quantum groups.The latter generalizes the identification of equivariant periodic cyclic cohomology fordiscrete groups mentioned above. These results will be discussed elsewhere.References[1] R. Akbarpour, M. Khalkhali, Hopf algebra equivariant cyclic homology and cyclic cohomologyof crossed product algebras, J. Reine Angew. Math. 559 (2003), 137–152.[2] R. Akbarpour, M. Khalkhali, Equivariant cyclic cohomology of H -algebras, K-theory 29(2003), 231–252.[3] S. Baaj, G. Skandalis, C -algèbres de Hopf et théorie de Kasparov équivariante, K-theory2 (1989), 683–721.[4] J. Block, E. Getzler, Equivariant cyclic homology and equivariant differential forms, Ann.Sci. École. Norm. Sup. 27 (1994), 493–527.[5] J.-L. Brylinski, Algebras associated with group actions and their homology, Brown Universitypreprint, 1986.[6] J.-L. Brylinski, Cyclic homology and equivariant theories, Ann. Inst. Fourier 37 (1987),15–28.[7] A. Connes, Noncommutative Geometry, Academic Press, 1994.[8] J. Cuntz, D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995),251–289.[9] J. Cuntz, D. Quillen, Cyclic homology and nonsingularity, J. Amer. Math. Soc. 8 (1995),373–442.[10] J. Cuntz, D. Quillen, Excision in bivariant periodic cyclic cohomology, Invent. Math. 127(1997), 67 - 98.[11] B. Drabant, A. van Daele, Y. Zhang, Actions of multiplier Hopf algebras, Comm. Algebra27 (1999), 4117–4172.[12] P. Hajac,M. Khalkhali, B. Rangipour, Y. Sommerhäuser, Stable anti-Yetter-Drinfeld modules,C. R. Acad. Sci. Paris 338 (2004), 587–590.[13] P. Hajac,M. Khalkhali, B. Rangipour, Y. Sommerhäuser, Hopf-cyclic homology and cohomologywith coefficients, C. R. Acad. Sci. Paris 338 (2004), 667–672.

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A Schwartz type algebra for the tangent groupoidPaulo Carrillo Rouse1 IntroductionThe concept of groupoid is central in non commutative geometry. Groupoids generalizethe concepts of spaces, groups and equivalence relations. It is clear nowadays thatgroupoids are natural substitutes of singular spaces. Many people have contributed torealizing this idea. We can find for instance a groupoid-like treatment in Dixmier’s workon transformation group, [11], or in Brown–Green–Rieffel’s work on orbit classificationof relations, [3]. In foliation theory, several models for the leaf space of a foliationwere realized using groupoids by several authors, for example by Haefliger ([12]) andWilkelnkemper ([24]), to mention some of them. There is also the case of orbifolds,these can be seen indeed as étale groupoids (see for example Moerdijk’s paper [17]).There are also some particular groupoid models for manifolds with corners and conicmanifolds, worked on by e.g. Monthubert [18], Debord–Lescure–Nistor ([10]) andAastrup–Melo–Monthubert–Schrohe ([1]).The way we treat ”singular spaces” in non commutative geometry is by associatingto them algebras. In the case when the ”singular space” is represented by a Lie groupoid,we can, for instance, consider the convolution algebra of differentiable functions withcompact support over the groupoid (see Connes’ or Paterson’s books [8] and [22]).This last algebra plays the role of the algebra of smooth functions over the ”singularspace” represented by the groupoid. From the convolution algebra it is also possible toconstruct a C -algebra, C .G /, that plays, in some sense, the role of the algebra ofcontinuous functions over the ”singular space”. The idea of associating algebras in thissense can be traced back in works of Dixmier ([11]) for transformation groups, Connes([6]) for foliations and Renault ([23]) for locally compact groupoids, to mention someof them.Using methods of non commutative geometry, we would like to get invariants ofthese algebras, and hence, of the spaces they represent. For that, Connes showedthat many groupoids and algebras associated to them appeared as ‘non commutativeanalogues‘ of spaces to which many tools of geometry (and topology) such asK-theory and characteristic classes could be applied ([7], [8]). One classical way toobtain invariants in classical geometry (topology) is through the index theory in thesense of Atiyah–Singer. In the Lie groupoid case there is a pseudodifferential calculus,developed by Connes ([6]), Monthubert–Pierrot ([19]) and Nistor–Weinstein–Xu ([21])in general. Some interesting particular cases were treated in the groupoid-spirit by Melrose([16]), Moroianu ([20]) and others (see [1]). Let G G .0/ be a Lie groupoid.Then there is an analytic index morphism (see [19]),ind a W K 0 .A G / ! K 0 .C .G //;

182 P. Carrillo Rousewhere AG is the Lie algebroid of G . The “C -index” ind a is a homotopy invariant ofthe G -pseudodifferential elliptic operators (G -PDO) and has proved to be very useful inmany different situations (see for example [2], [9], [10]). One way to define the aboveindex map is using Connes’ tangent groupoid associated to G as explained by Hilsumand Skandalis in [13] or by Monthubert and Pierrot in [19]. The tangent groupoid is aLie groupoidG T G .0/ Œ0; 1with G T WD AG f0g tG .0; 1, and the groupoid structure is given by the groupoidstructure of AG at t D 0 and by the groupoid structure of G for t ¤ 0. One of the mainfeatures of the tangent groupoid is that its C -algebra C .G T / is a continuous field ofC -algebras over the closed interval Œ0; 1, with associated fiber algebrasC 0 .A G / at t D 0; and C .G / for t ¤ 0.In fact, it gives a C -algebraic quantization of the Poisson manifold A G (in the senseof [14]), and this is the main point why it allows to define the index morphism as asort of ”deformation”. Thus, the tangent groupoid construction has been very useful inindex theory ([2], [10], [13]), but also for other purposes ([14], [21]).Now, to understand the purpose of the present work, let us first say that the indices (inthe sense of Atiyah–Singer–Connes) have not necessarily to be considered as elementsin K 0 .C .G //. Indeed, it is possible to consider indices in K 0 .Cc 1.G //. The C c 1-in-dices are more refined, but they have several inconveniences (see Alain Connes’ bookSection 9:ˇ for a discussion on this matter). Nevertheless this kind of indices have thegreat advantage that one can apply to them the existent tools (such as pairings withcyclic cocycles or Chern–Connes character) in order to obtain numerical invariants.In this work we begin a study of more refined indices. In particular we are lookingfor indices between the Cc1 and the C -levels, trying to keep the advantages of bothapproaches (see [5] for a more complete discussion). In the case of Lie groupoids,this refinement could mean to forget for a moment the powerful tools of the theory ofC -algebras, and instead working in a purely algebraic and geometric level. In thepresent article, we construct an algebra of C 1 functions over G T , denoted by S c .G T /.This algebra is also a field of algebras over the closed interval Œ0; 1, with associatedfiber algebrasS.AG / at t D 0, and Cc 1 .G / for t ¤ 0,where S.AG / is the Schwartz algebra of the Lie algebroid. Furthermore, we will haveC 1 c .G T / S c .G T / C .G T /; (1)as inclusions of algebras. Let us explain in some words why we define an algebra overthe tangent groupoid such that in zero it is Schwartz: The Schwartz algebras have ingeneral the good K-theory groups. For example we are interested in the symbols ofG -PDO, and, more precisely, in their homotopy classes in K-theory, that is, we areinterested in the group K 0 .A G / D K 0 .C 0 .A G //. Here it would not be enough totake the K-theory of Cc1 .AG / (see the example [8], p. 142), however it is enough to

A Schwartz type algebra for the tangent groupoid 183consider the Schwartz algebra S.A G /. Indeed, the Fourier transform shows that thislast algebra is stable under holomorphic calculus on C 0 .A G / and so it has the “good”K-theory, meaning that K 0 .A G / D K 0 .S.A G //. None of the inclusions in (1) isstable under holomorphic calculus, but that is precisely what we wanted because ouralgebra S c .G T / has the remarkable property that its evaluation at zero is stable underholomorphic calculus, while its evaluation at one (for example) is not.The algebra S c .G T / is, as a vector space, a particular case of a more generalconstruction that we do for “deformation to the normal cone manifolds” from which thetangent groupoid is a special case (see [4] and [13]). A deformation to the normal conemanifold (DNC for short) is a manifold associated to an injective immersion X,! Mthat is considered as a sort of blow up in differential geometry. The constructionof a DNC manifold has very nice functorial properties (Section 3) which we exploitto achieve our construction. We think that our construction could be used also forother purposes, for example, it seems that it could help to give more understanding inquantization theory (see again [4]).The article is organized as follows. In the second section we recall the basic factsabout Lie groupoids. We explain very briefly how to define the convolution algebraCc1 .G /. In the third section we explain the “deformation to the normal cone” constructionassociated to an injective immersion. Even if this could be considered as classicalmaterial, we do it in some detail since we will use in the sequel very explicit descriptionsthat we could not find elsewhere. We also review some functorial properties associatedto these deformations. A particular case of this construction is the tangent groupoid associatedto a Lie groupoid. In the fourth section we start by constructing a vector spaceS c .DX M / for any deformation to the normal cone manifold DM X; this space alreadyexhibits the characteristic of being a field of vector spaces over the closed interval Œ0; 1whose fiber at zero is a Schwartz space while for values different from zero it is equalto Cc1.M /. We then define the algebra S c.G T /; the main result is that its product iswell-defined and associative. The last section is devoted to motivate the constructionof our algebra by explaining shortly some further developments that will immediatelyfollow from this work. All the results of the present work are part of the author’s PhDthesis.I want to thank my PhD advisor, Georges Skandalis, for all the ideas that heshared with me, and for all the comments and remarks he made on the present work.I would also like to thank the referee for the useful comments he made for improvingthis paper.2 Lie groupoidsLet us recall what a groupoid is.Definition 2.1. A groupoid consists of the following data: two sets G and G .0/ , andmaps• s; r W G ! G .0/ , called the source and target map respectively,

184 P. Carrillo Rouse• mW G .2/ ! G , called the product map (where G .2/ Df.; / 2 G G W s./ Dr./g),• uW G .0/ ! G the unit map, and• i W G ! G the inverse mapsuch that, if we write m.; / D , u.x/ D x and i./ D 1 ,wehave1. . ı/ D . / ı for all ; ; ı 2 G when this is possible,2. x D and x D for all ; 2 G with s./ D x and r./ D x,3. 1 D u.r.// and 1 D u.s.// for all 2 G ,4. r. / D r./ and s. / D s./.Generally, we denote a groupoid by G G .0/ where the parallel arrows are the sourceand target maps and the other maps are given.Now, a Lie groupoid is a groupoid in which every set and map appearing inthe last definition is C 1 (possibly with borders), and the source and target mapsare submersions. For A; B subsets of G .0/ we use the notation GA B for the subsetf 2 G W s./ 2 A; r./ 2 Bg.Throughout this paper, G G .0/ is a Lie groupoid. We recall how to define analgebra structure in Cc1 .G / using smooth Haar systems.Definition 2.2. A smooth Haar system over a Lie groupoid consists of a family ofmeasures x in G x for each x 2 G .0/ such that• for 2 Gx y we have the following compatibility condition:ZZf./d x ./ D f. ı / d y ./;G x G y• for each f 2 Cc 1.G / the map Zx 7! f./d x ./G xbelongs to C 1 c .G .0/ /.A Lie groupoid always possesses a smooth Haar system. In fact, if we fix a smooth(positive) section of the 1-density bundle associated to the Lie algebroid we obtain asmooth Haar system in a canonical way. The advantage of using 1-densities is that themeasures are locally equivalent to the Lebesgue measure. We suppose for the rest ofthe paper an underlying smooth Haar system given by 1-densities (for complete details

A Schwartz type algebra for the tangent groupoid 185see [22]). We can now define a convolution product on Cc 1.G /. Forf; g 2 C c 1.G /we setZ.f g/./ D f. 1 /g./ d s./ ./:G s./This gives a well-defined associative product.Remark 2.3. There is a way to avoid the Haar system when one works with Liegroupoids, using half densities (see Connes’ book [8]).3 Deformation to the normal coneLet M be a C 1 manifold and X M be a C 1 submanifold. We denote by NXMnormal bundle to X in M , i:e:, NXM WD T XM=TX. We define the following settheD M X WD N M X 0 t M .0; 1: (2)The purpose of this section is to recall how to define a C 1 -structure with boundary inDX M . This is more or less classical, for example it was extensively used in [13]. Herewe are only going to do a sketch.Let us first consider the case where M D R n and X D R p f0g (where we identifycanonically X D R p ). We denote by q D n p and by Dp n for DRnRas above. In thispcase we clearly have that Dp n D Rp R q Œ0; 1 (as a set). Consider the bijectiongiven by‰ W R p R q Œ0; 1 ! D n p (3)‰.x; ; t/ D´.x;;0/ if t D 0;.x; t; t/ if t>0;whose the inverse is given explicitly by´‰ 1 .x;;0/ if t D 0;.x;;t/D.x; 1 t ;t/ if t>0:We can consider the C 1 -structure with border on Dp n induced by this bijection.In the general case, let .U;/be a local chart in M and suppose it is an X-slice,so that it satisfies1) W U Š ! U R p R q ;2) if V D U \ X, V D U \ .R p 0/, then we have V D 1 .V /.With this notation we have that D U V Dn pis an open subset. We may define a functionQ W D U V! DU V

186 P. Carrillo Rousein the following way. For x 2 V we have .x/ 2 R p f0g. If we write .x/ D. 1 .x/; 0/, then 1 W V ! V R pis a diffeomorphism, where V D U \.R p f0g/. Set Q.v;;0/ D . 1 .v/; d N v ./; 0/and Q.u; t/ D ..u/; t/ for t ¤ 0. Here d N v W N v ! R q is the normal componentof the derivative d v for v 2 V. It is clear that Q is also a bijection (in particular itinduces a C 1 structure with border over D U V ).Let us define, with the same notations as above, the following set: U V Df.x;;t/2 Rp R q Œ0; 1 W .x; t / 2 U g:This is an open subset of R p R q Œ0; 1 and thus a C 1 manifold (with border). It isimmediate that DV U is diffeomorphic to U Vthrough the restriction of ‰ given in (3).Now we consider an atlas f.U˛;˛/g˛2 of M consisting of X-slices. It is clear thatD M X D [˛2 D U˛V˛(4)and if we take D U˛V˛'˛! U˛ defined as the compositionV˛D U˛V˛then we obtain the following result.˛! D U˛V˛‰ 1˛! U˛V˛Proposition 3.1. f.D U˛ ;'˛/g˛2 is a C 1 atlas with border over D M V˛ X .In fact the proposition can be proved directly from the following elementary lemma.Lemma 3.2. Let F W U ! U 0 be a C 1 diffeomorphism where U R p R q andU 0 R p R q are open subsets. We write F D .F 1 ;F 2 / and we suppose thatF 2 .x; 0/ D 0. Then the function FQW U V ! U 0Q F.x;;t/DVdefined by0´.F1 .x; 0/; @F 2.x; 0/ ;0/@if t D 0;.F 1 .x; t/; 1 t F 2.x;t/;t/ if t>0:is a C 1 map.Proof. Since the result will hold if and only if it is true in each coordinate, it is enoughto prove that if we have a C 1 map F W U ! R with F.x;0/ D 0, then the mapQF W U V ! R given by Q F.x;;t/Dis a C 1 map. For that we write´@F@1t.x; 0/ if t D 0;F.x;t/ if t>0F.x;/ D @F .x; 0/ C h.x; / @

A Schwartz type algebra for the tangent groupoid 187with hW U ! R q a C 1 map such that h.x; 0/ D 0. Then1@FF.x;t/ D .x; 0/ C h.x; t/ t @from which we immediately get the result.Definition 3.3 (DNC). Let X M be as above. The set DX M 1provided with the Cstructure with border induced by the atlas described in the last proposition is called thedeformation to normal cone associated to X M . We will often write DNC insteadof deformation to the normal cone.Remark 3.4. Following the same steps, it is possible to define a deformation to thenormal cone associated to an injective immersion X,! M .Let us mention some basic examples of DCN manifolds D M X .Examples 3.5. 1. Consider the case when X D;. We have that D;M D M .0; 1with the usual C 1 structure on M .0; 1. We used this fact implicitly to cover DXMas in (4).2. Consider the case when X M is an open subset. Then we do not have anydeformation at zero and we immediately see from the definition that DXM is just theopen subset of M Œ0; 1 consisting of the union of X Œ0; 1 and M .0; 1.The most important feature of the DNC construction is that it is in some sensefunctorial. More explicitly, let .M; X/ and .M 0 ;X 0 / be C 1 -couples as above and letF W .M; X/ ! .M 0 ;X 0 / be a couple morphism, i.e., a C 1 map F W M ! M 0 withF.X/ X 0 . We define D.F /W DX M ! DM 0Xby the following formulas:0D.F /.x; ; 0/ D .F .x/; d N F x ./; 0/ and D.F /.m; t/ D .F .m/; t/ for t ¤ 0;where d N F x is by definition the map.N M X / xd N F x! .N M 0X 0/ F.x/induced by T x M dF x! T F.x/ M 0 .We have the following proposition, which is also an immediate consequence of thelemma above.Proposition 3.6. The map D.F /W DX M ! DM 0Xis C 1 .0Remark 3.7. If we consider the category C21 of C 1 pairs given by a C 1 manifold anda C 1 submanifold and pair morphisms as above, we can reformulate the propositionand say that we have a functorD W C 1 2 ! C 1where C 1 denotes the category of C 1 manifolds with border.

188 P. Carrillo Rouse3.1 The tangent groupoidDefinition 3.8 (Tangent groupoid). Let G G .0/ be a Lie groupoid. The tangentgroupoid associated to G is the groupoid that has D G as the set of arrows andG .0/G .0/ Œ0; 1 as the units with• s T .x;;0/D .x; 0/ and r T .x;;0/D .x; 0/ at t D 0,• s T .; t/ D .s./; t/ and r T .; t/ D .r./; t/ at t ¤ 0,• the product is given by m T ..x; ; 0/; .x; ; 0// D .x; C ;0/ andm T ..; t/; .ˇ; t// D .m.; ˇ/; t/ if t ¤ 0 and if r.ˇ/ D s./,• the unit map u T W G .0/ ! G T is given by u T .x; 0/ D .x; 0/ and u T .x; t/ D.u.x/; t/ for t ¤ 0.We denote G T WD D G G .0/ .As we have seen above G T can be considered as a C 1 manifold with border. As aconsequence of the functoriality of the DNC construction we can show that the tangentgroupoid is in fact a Lie groupoid. Indeed, it is easy to check that if we identify in acanonical way D G .2/with .G T / .2/ , thenG .0/m T D D.m/; s T D D.s/; r T D D.r/; u T D D.u/where we are considering the following pair morphisms:mW ..G / .2/ ; G .0/ / ! .G ; G .0/ /;s; r W .G ; G .0/ / ! .G .0/ ; G .0/ /;uW .G .0/ ; G .0/ / ! .G ; G .0/ /:Finally, if f x g is a smooth Haar system on G , then, setting• .x;0/ WD x at .G T / .x;0/ D T x G x , and• .x;t/ WD t q x at .G T / .x;t/ D G x for t ¤ 0, where q D dim G x ,one obtains a smooth Haar system for the tangent groupoid (details may be found in[22]).We finish this section with some interesting examples of groupoids and their tangentgroupoids.Examples 3.9. 1. The tangent groupoid of a group. Let G be a Lie group consideredas a Lie groupoid, G WD G feg. In this case the normal bundle to the inclusionfeg ,! G is of course identified with the Lie algebra of the group. Hence, the tangentgroupoid is a deformation of the group in its Lie algebra:G T D g f0g tG .0; 1:

A Schwartz type algebra for the tangent groupoid 1892. The tangent groupoid of a smooth vector bundle. Let E p ! X be a smoothvector bundle over a (connected) C 1 manifold X. We can consider the Lie groupoidE X induced by the vector structure of the fibers, i:e:, s./ D p./ D r./ and thecomposition is given by the vector sum ı D C . In this case the normal vectorbundle associated to the zero section can be identified to E itself. Hence, as a set thetangent groupoid is E Œ0; 1, but the C 1 -structure at zero is given locally as in (3).3. The tangent groupoid of a C 1 -manifold. Let M a C 1 -manifold. We canconsider the product groupoid G M WD M M M . The tangent groupoid in thiscase takes the following form:G T MD TM f0g tM M .0; 1:This is called the tangent groupoid to M and it was introduced by Connes for giving avery conceptual proof of the Atiyah–Singer index theorem (see [8] and [10]).4 An algebra for the tangent groupoidIn this section we will show how to construct an algebra for the tangent groupoid whichconsists of C 1 functions that satisfy a rapid decay condition at zero while out of zerothey satisfy a compact support condition. This algebra is the main construction in thiswork.4.1 Schwartz type spaces for deformation to the normal cone manifolds. Ouralgebra for the tangent groupoid will be a particular case of a construction associated toany deformation to the normal cone. We start by defining a space for DNCs associatedto open subsets of R p R q .Definition 4.1. Let p; q 2 N and U R p R q an open subset, and let VU \ .R p f0g/.D(1) Let K U Œ0; 1 be a compact subset. We say that K is a conic compact subsetof U Œ0; 1 relative to V ifK 0 D K \ .U f0g/ V:(2) Let g 2 C 1 . U V/. We say that f has compact conic support K, if there exists aconic compact K of U Œ0; 1 relative to V such that if t ¤ 0 and .x; t; t/ … Kthen g.x; ; t/ D 0.(3) We denote by S c . U V / the set of functions g 2 C 1 . U V/ that have compact conicsupport and that satisfy the following condition:.s 1 ) for all k; m 2 N, l 2 N p and ˛ 2 N q there exists C .k;m;l;˛/ >0such that.1 Ckk 2 / k k@ l x @˛ @m t g.x; ; t/k C .k;m;l;˛/:

190 P. Carrillo RouseNow, the spaces S c . U V/ are invariant under diffeomorphisms. More precisely ifF W U ! U 0 is a C 1 diffeomorphism as in Lemma 3.2 then we can prove the nextresult.Proposition 4.2. Let g 2 S c . U 0V/, then Qg WD g ı FQ2 S 0 c . U V /.Proof. The first observation is that Qg 2 C 1 . U V/, thanks to Lemma 3.2. Let us checkthat it has compact conic support. For that, let K 0 U 0 Œ0; 1 be the conic compactsupport of g. We letK D .F 1 Id Œ0;1 / U Œ0; 1;which is a conic compact subset of U Œ0; 1 relative to V , and it is immediate bydefinition that Qg.x; ; t/ D 0 if t ¤ 0 and .x; t ;t/ … K, that is, Qg has compact conicsupport K.We now check the rapid decay property .s 1 /: To simplify the proof we first introducesome useful notation. Writing F D .F 1 ;F 2 / as in Lemma 3.2, we denoteF 1 .x; / D .A 1 .x;/;:::;A p .x; // and F 2 .x; / D .B 1 .x;/;:::;B q .x; //.We also write w D w.x; ; t/ D .A 1 .x;t/;:::;A p .x; t// and D .x; ; t/ D. BQ1 .x;;t/;:::; BQq .x;;t//where BQj is as above, i:e:,Q B j .x;;t/D8ˆ

A Schwartz type algebra for the tangent groupoid 191are bounded in K . For the expressions k@ ı z j .z/k, we proceed first by developing asin Lemma 3.2, that is, @Bj j .x;;t/D@ .x; 0/ C hj .x; t/ :Now, since we are only considering points in K , it is immediate that we can findconstants C j >0such thatk@ ı z j .z/k C j kk m ı:In the same way (remember that F is a diffeomorphism) we can have constants C i >0such thatk i .!;;t/kC i kk:Putting all together, and using the property .s 1 / for g, we get bounds C>0such thatand this concludes the proof.kk k k@˛zQg.z/k C;Remark 4.3. We can resume the last invariance result as follows: If .U; V/ is aC 1 pair diffeomorphic to .U; V / with U E, an open subset of a vector spaceE, and V D U \ E, then S c .D U V/ is well defined and does not depend on the pairdiffeomorphism.With the last compatibility result in hand we are ready to give the main definitionin this work.Definition 4.4. Let g 2 C 1 .D M X /.(a) We say that g has compact conic support K, if there exists a compact subsetK M Œ0; 1 with K 0 WD K \ .M f0g/ X (conic compact relative to X)such that if t ¤ 0 and .m; t/ … K then g.m; t/ D 0.(b) We say that g is rapidly decaying at zero if for every X-slice chart .U;/ andfor every 2 Cc1.U Œ0; 1/, the map g 2 C 1 . U V/ given byg .x;;t/D .g ı ' 1 /.x;;t/ . ı p ı ' 1 /.x;;t/is in S c . U V /, where p is the projection p W DM X.x;;0/7! .x; 0/, and .m; t/ 7! .m; t/ for t ¤ 0.! M Œ0; 1 given byFinally, we denote by S c .DX M / the set of functions g 2 C 1 .DX M / that are rapidlydecaying at zero with compact conic support.Remark 4.5. (a) It follows from the definition of S c .DX M / that the latter containsCc1.DM X/ as a vector subspace.(b) It is clear that Cc1.M .0; 1/ can be considered as a subspace of S c.DX M / byextending by zero the functions at NX M .

192 P. Carrillo RouseFollowing the lines of the last remark we are going to precise a possible decompositionof our space S c .DX M / that will be very useful in the sequel. Let f.U˛;˛/g˛2a family of X-slices covering X. Consider the open cover of M Œ0; 1 consisting off.U˛ Œ0; 1; ˛/g˛2 together with M .0; 1. We can take a partition of the unitysubordinated to the last cover,f˛;g˛2 :That is, we have the following properties:• 0 ˛; 1,• supp ˛ U˛ Œ0; 1 and supp M .0; 1,• P˛ ˛ C P D 1.Let f 2 S c .D M X/. If we denotef˛ WD f j DU˛V˛ .˛ ı p/ 2 C 1 .D U˛V˛/andf WD f j M .0;1 . ı p/ 2 C 1 .M .0; 1/;then we obtain the following decomposition:f D X˛f˛ C f :Now, since f is conic compactly supported we can suppose, without lost of generality,that• f 2 C 1 c.M .0; 1/, and• that ˛ is compactly supported in U˛ Œ0; 1.What we conclude of all this, is that we can decompose our space S c .D M X/ as follows:S c .D M X / D X˛2ƒ S c .D U˛V˛/ C C 1 c.M .0; 1/: (5)As we mentioned in the introduction, we want to see the space S c .DX M / as a fieldof vector spaces over the interval Œ0; 1, where at zero we talk about Schwartz spaces.In our case we are interested in Schwartz functions on the vector bundle NXM . Let usfirst recall the notion of the Schwartz space associated to a vector bundle.Definition 4.6. Let .E;p;X/be a smooth vector bundle over a C 1 manifold X. Wedefine the Schwartz space S.E/ as the set of C 1 functions g 2 C 1 .E/ such that g is aSchwartz function at each fiber (uniformly) and g has compact support in the directionof X, i:e:, if there exists a compact subset K X such that g.E x / D 0 for x … K.

A Schwartz type algebra for the tangent groupoid 193The vector space S.E/ is an associative algebra with the product given as follows:for f; g 2 S.E/,weputZ.f g/./ D f. /g./ d p./ ./; (6)E p./where is a smooth Haar system of the Lie groupoid E X. A classical Fourierargument can be applied to show that the algebra S.E/ is isomorphic to .S.E /; /(punctual product). In particular this implies that K 0 .S.E// Š K 0 .E /.In the case we are interested in, we have a couple .M; X/ and a vector bundleassociated to it, that is, the normal bundle over X, NXM . The reason why we gave thelast definition is because we get evaluation linear mapse 0 W S c .D M X / ! S.N M X/; (7)ande t W S c .D M X / ! C 1 c.M / (8)for t ¤ 0. Consequently, we have that the vector space S c .DX M / is a field of vectorspaces over the closed interval Œ0; 1 whose fiber spaces are S.NXM / at t D 0 andCc 1 .M / for t ¤ 0.Let us finish this subsection by giving the examples of spaces S c .DX M / correspondingto the DCN manifolds in Examples 3.5.Examples 4.7. 1. For X D;, we have that S c .D; M / Š C c 1 .M .0; 1/.2. For X M an open subset we have that S c .DX M / Š C c 1 .W / where W M Œ0; 1 is the open subset consisting of the union of X Œ0; 1 and M .0; 1.4.2 Schwartz type algebra for the tangent groupoid. In this section we define analgebra structure on S c .G T /. We start by defining a function m c W S c .D G .2// !G .0/S c .D G / by the following formulas:G .0/For F 2 S c .D G .2//,weletG .0/ Zm c .F /.x; ; 0/ D F.x; ; ; 0/ d x ./T x G xandZm c .F /.; t/ D F. ı ı 1 ;ı;t/tq d s./ .ı/:G s./If we canonically identify D G .2/with .G T / .2/ , the map above is nothing else than theG .0/integration along the fibers of m T W .G T / .2/ ! G T . We have the following proposition:Proposition 4.8. m c W S c ..G T / .2/ / ! S c .G T / is a well-defined linear map.

194 P. Carrillo RouseThe interesting part of the proposition is that the map is well defined since it isevidently be linear. Let us suppose for the moment that the proposition is true. Underthis assumption, we will define the product in S c .G T /.Definition 4.9. Let f; g 2 S c .G T /, we define a function f g in G T byZ.f g/.x;;0/D f.x; ; 0/g.x; ; 0/ d x ./T x G xandZ.f g/.; t/ D f. ı ı 1 ; t/g.ı; t/tq d s./ .ı/G s./for t ¤ 0.We can enounce our main result.Theorem 4.10. defines an associative product on S c .G T /.Proof. Remember that we are assuming for the moment the validity of Proposition 4.8.Let f; g 2 S c .G T /. We let F WD .f; g/ be the function in .G T / .2/ defined by.f;g/.x;;;0/D f.x;;0/ g.x; ; 0/and.f; g/..; t/; .ı; t// D f.;t/ g.ı; t/for t ¤ 0. Now, from the Leibnitz formula for the derivative of a product it is immediatethat .f; g/ 2 S c ..G T / .2/ /. Finally, by definition we have thatm c ..f; g// D f g;hence, by Proposition 4.8, f g is a well-defined element in S c .G T /.For the associativity of the product, let us remark that when one restricts the productto C 1 c .G T /, this coincides with the product classically considered on C 1 c .G T / (whichis associative, see for example [22]). The associativity for S c .G T / is proved exactly inthe same way that for C 1 c .G T /.It remains to prove Proposition 4.8. We are going to start locally. Let U 2 R p R q R q be an open set and V D U \R p f0gf0g. Let P W R p R q R q ! R p R qthe canonical projection .x;;/ 7! .x; /. We set U 0 D P.U/ 2 R p R q , then U 0is also an open subset, V Š U 0 \ R p f0g and P j V D Id V . We denote also by P therestriction P W U ! U 0 . As in Lemma 3.2 we have a C 1 map PQW U V ! U 0Vwhichin this case is explicitly written asP.x;;;t/D Q.x;;t/:We define PQc W S c . U V / ! S c. U 0/ as follows:Let us prove a lemma.Q P c .F /.x; ; t/ DVZf2R q W.x;;;t/2 U V g F .x; ; ; t/ d:

A Schwartz type algebra for the tangent groupoid 195Lemma 4.11. PQc W S c . U V / ! S c. U 0V/ is well defined.Proof. The first observation is that the integral in the definition of PQc is always welldefined. Indeed, this is a consequence of the following two facts:• for t D 0, 7! F .x; ; ; 0/ 2 S.R q /,• for t ¤ 0, 7! F.x;;;t/ 2 Cc 1.Rq /.Taking derivatives under the integral sign, we obtain that PQc .F / 2 C 1 . U 0V/. Then,we just have to show that PQc .F / verifies the two conditions of Definition 4.1. For thefirst, if K U Œ0; 1 is the compact conic support of F , then it is enough to putK 0 D .P Id Œ0;1 /.K/in order to obtain a conic compact subset of U 0 Œ0; 1 relative to V and to check thatK 0 is the compact conic support of PQc .F / . Let us now verify the condition .s 1 /. Letk; m 2 N, l 2 N p and ˇ 2 N q . We want to find C .k;m;l;ˇ/ >0such that.1 Ckk 2 / k k@ l x @ˇ @m Qt P c .F /.x; ; t/k C .k;m;l;˛/ :For k 0 k C q 2 and ˛ D .0; ˇ/ 2 Rq R q we have by hypothesis that there exists>0such thatC 0.k 0 ;m;l;˛/Then we also have thatwithk@ l x @ˇ @m tk@ l x @ˇ @m t F.x;;;t/kC 0 1.1 Ck.; /k 2 / k0 :ZPQc .F /.x; ; t/k C 0 1df2R q W.x;;;t/2 U V g .1 Ck.; /k 2 / k0 C 0 111d C.1 Ckk 2 / q 2Zf2R k0 q g .1 Ckk 2 / k0 .1 Ckk 2 / kZC D C 0 1d:f2R q g .1 Ckk 2 / k0We can now give the proof of Proposition 4.8.Proof of Proposition 4.8. Let us first fix some notation. We suppose dim G D p C qand dim G .0/ D p, in particular this implies that dim G .2/ D p C q C q. Let .U;/and .U 0 ; 0 / be G .0/ -slices in G .2/ and G , respectively, such that the following diagramcommutes:U m U 0 0 U PU 0 ,

196 P. Carrillo Rousewhere P W R p R q R q ! R p R q is the canonical projection (as above) andP.U/ D U 0 . This is possible since m is a surjective submersion. Now, we apply theDNC construction to the diagram above to obtain, thanks to the functoriality of theconstruction, the following commutative diagram:D U VD.m/ D U0V 0 Q 0QDV U DU 0D.P / V0‰ ‰ 0 QP 0,where ‰ and Q are as in Section 3. Let g 2 S c .DV U / and define8 Rˆ< Rg.x; ; ; 0/ d q if t D 0;P c .g/.x; ; t/ Dˆ: Rf2R q W.x;;;t/2 U V g g.x; ; ; t/t q d if t ¤ 0:Then, from the last commutative diagram, we get thatP c .g/ D PQc .g ı ‰ / ı .‰ 0/ 1 ;hence, thanks to Lemma 4.11, we can conclude that we have a well-defined linear mapP c W S c .DV U / ! S c.D U 0V 0 /:We now use the Proposition 4.2 to writeS c .D U V / Dfh 2 C 1 .D U V / W h ı Q 1 2 S c .D U V /g;and so for h 2 S c .D U V/ we see thatP c .h ı Q 1 / ı Q 0 2 S c .D U0V/: 0We use again the second commutative diagram to see thatWe then have a well-defined linear mapm c .h/ D P c .h ı Q 1 / ı Q 0 :m c W S c .D U V / ! S c.D U0V 0 /:To pass to the global case we only have to use the decomposition of S c .D G .2/G .0/ / andof S c .D G G .0/ / as in (5), and of course the invariance under diffeomorphisms (Proposition4.2).

A Schwartz type algebra for the tangent groupoid 197Recall that we have well-defined evaluation morphisms as in (7) and (8). In thecase of the tangent groupoid they are by definition morphisms of algebras. Hence, thealgebra S c .G T / is a field of algebras over the closed interval Œ0; 1, with associatedfiber algebras,S.AG / at t D 0, and Cc 1 .G / for t ¤ 0.It is very interesting to see what this means in the examples given in 3.9.5 Further developmentsLet G G .0/ be a Lie groupoid. In index theory for Lie groupoids the tangent groupoidhas been used to define the analytic index associated to the group, as a morphismK 0 .A G / ! K 0 .Cr .G // (see [19]) or as a KK-element in KK.C 0.A G /; Cr .G //(see [13]), this can be done because one has the following short exact sequence ofC -algebras0 ! Cr .G .0; 1/ ! C r .G T / e 0! C 0 .A G / ! 0; (9)and because of the fact that the K-groups of the algebra Cr .G .0; 1/ vanish (homotopyinvariance). The index defined at the C -level has proven to be very useful (seefor example [9]) but extracting numerical invariants from it, with the existent tools, isvery difficult. In non commutative geometry, and also in classical geometry, the toolsfor obtaining more explicit invariants are more developed for the ’smooth objects’; inour case this means the convolution algebra Cc1 .G /, where we can for example applyChern–Weil–Connes theory. Hence, in some way, the indices defined in K 0 .Cc 1 .G //are more refined objects and for some cases it would be preferable to work with them.Unfortunately these indices are not good enough, since for example they are not homotopyinvariant; in [8] Alain Connes discusses this and also other reasons why it is notenough to keep with the Cc 1-indices. The main reason to construct the algebra S c.G T /is that it gives an intermediate way between the Cc1-level and the C -level and willallow us in [5] to define another analytic index morphism associated to the groupoid,with the advantage that this index will take values in a group that allows to do pairingswith cyclic cocycles and in general to apply Chern–Connes theory to it. The way weare going to define our index is by obtaining first a short exact sequence analogous to(9), that is, a sequence of the following kind0 ! J ! S c .G T / e 0! S.A G // ! 0: (10)The problem here will be that we do not dispose of the advantages of the K-theory forC -algebras, since the algebras we are considering are not of this type (we do not havefor example homotopy invariance).References[1] J. Aastrup, S. T. Melo, , B. Monthubert, E. Schrohe, Boutet de Monvel’s Calculus andGroupoids I, Preprint 2006, arxiv:math.KT/0611336.

198 P. Carrillo Rouse[2] J. Aastrup, R. Nest, and E. Schrohe, A continuous field of C -algebras and the tangentgroupoid for manifolds with boundary, J. Funct. Anal. 237 (2006), 482–506.[3] L. Brown, P. Green, and M. Rieffel, Stable isomorphism and stable Morita equivalence forC -algebras, Pacific J. Math. 71 (1977), 349–363.[4] J. Carinena, J. Clemente-Gallardo, E. Follana, J. M. Gracia-Bondia, A. Rivero, and J. C.Varilly, Connes’ tangent groupoid and strict quantization, J. Geom. Phys. 32 (1999), 79–96.[5] P. Carrillo, An analytic index for Lie groupoids, Preprint 2006, arxiv:math.KT/0612455.[6] A. Connes, Sur la théorie non commutative de l’intégration, in Algèbres d’opérateurs,Lecture Notes in Math. 725, Springer-Verlag, Berlin 1979, 19–143.[7] A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62(1985), 257–360.[8] A. Connes, Non commutative geometry, Academic Press, San Diego, CA, 1994.[9] A. Connes, and G. Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst.Math 6 (1984), 1139–1183.[10] C. Debord, J. M. Lescure, and V. Nistor, Groupoids and an index theorem for conicalpseudomanifolds, Preprint 2008, arxiv:math.OA/0609438.[11] J. Dixmier, Les C -algèbres et leurs représentations, Cahiers Scientifiques XXIX,Gauthier-Villars, Paris 1964.[12] A. Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1982), 70–97.[13] M. Hilsum and G. Skandalis, Morphismes K-orientés d’espaces de feuilles et fonctorialitéen théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup.(4) 20 (1987), 325–390.[14] N. P. Landsman, Quantization and the tangent groupoid, in Operator algebras and mathematicalphysics, Theta, Bucharest 2003, 251–265.[15] R. Lauter, B. Monthubert, and V. Nistor, Pseudodifferential analysis on continuous familygroupoids, Doc. Math 5 (2000), 625–655.[16] R. Melrose, The Atiyah-Patodi-Singer index theorem, Res. Notes in Math. 4, AK Peters,Wellesley, MA, 1993.[17] I. Moerdijk, Orbifolds as groupoids: an introduction, in Orbifolds in mathematics andphysics (Madison, WI, 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, RI,205–222.[18] B. Monthubert, Groupoids and Pseudodifferential calculus on manifolds with corners, J.Funct. Anal. 199 (2003), 243–286.[19] B. Monthubert and F. Pierrot, Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris325 (1997), 193–198.[20] S. Moroinau, Adiabatic limits of eta and zeta functions of elliptic operators, Math. Z. 246(2004), 441–471.[21] V. Nistor, A. Weinstein, P. and Xu, Pseudodifferential operators on differential groupoids,Pacific J. Math 189 (1999), 117–152.[22] A. Paterson, Groupoids and inverse semigroups, and their operator algebras, Progr. Math.170, Birkhäuser, Boston, MA, 1999.

A Schwartz type algebra for the tangent groupoid 199[23] J. Renault, A groupoid approach to C -algebras, Lecture Notes in Math. 793, Springer-Verlag, Berlin 1980.[24] H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1 (1983), 51–75.

C -algebras associated with the ax C b-semigroupover NJoachim Cuntz 1 IntroductionIn this note we present a C -algebra (denoted by Q N ) which is associated to the axCbsemigroupover N. It is in fact a natural quotient of the C -algebra associated withthis semigroup by some additional relations (which make it simple and purely infinite).These relations are satisfied in representations related to number theory. The studyof this algebra is motivated by the construction of Bost–Connes in [1]. Our C -algebracontains the algebra considered by Bost–Connes, but in addition a generatorcorresponding to translation by the additive group Z.As a C -algebra, Q N has an interesting structure. It is a crossed product of theBunce–Deddens algebra associated to Q by the action of the multiplicative semigroupN . It has a unique canonical KMS-state. We also determine its K-theory, whosegenerators turn out to be determined by prime numbers.On the other hand, Q N can also be obtained as a crossed product of the commutativealgebra of continuous functions on the completion OZ by the natural action of the axCbsemigroupover N. More interestingly, its stabilization is isomorphic to the crossedproduct of the algebra C 0 .A f / of continuous functions on the space of finite adeles bythe natural action of the ax C b-group P C Qover Q. Somewhat surprisingly one obtainsexactly the same C -algebra Q N (up to stabilization) working with the completion Rof Q at the infinite place and taking the crossed product by the natural action of theax C b-group P C Q on C 0.R/.In the last section we consider the analogous construction of a C -algebra replacingthe multiplicative semigroup N by Z , i.e., omitting the condition of positivity on themultiplicative part of the ax C b-(semi)group. We obtain a purely infinite C -algebraQ Z which can be written as a crossed product of Q N by Z=2. The fixed point algebrafor its canonical one-parameter group . t / is a dihedral group analogue of the Bunce–Deddens algebra. Its K-theory involves a shift of parity from K 0 to K 1 and vice versa.Its stabilization is isomorphic to the natural crossed product C 0 .A f / Ì P Q .2 A canonical representation of the ax C b-semigroupWe denote by N the set of natural numbers including 0. N will normally be regardedas a semigroup with addition. Research supported by the Deutsche Forschungsgemeinschaft.

202 J. CuntzWe denote by N the set of natural numbers excluding 0. N will normally beregarded as a semigroup with multiplication.The natural analogue, for a semigroup S, of an unitary representation of a group isa representation of S by isometries, i.e. by operators s g , g 2 S on a Hilbert space thatsatisfy sg s g D 1.On the Hilbert space `2.N/ consider the isometries s n , n 2 N and v k , k 2 Ndefined byv k . m / D mCk ; s n . m / D mnwhere m ;m2 N denotes the standard orthonormal basis.We have s n s m D s nm and v n v m D v nCm , i.e. s and v define representations of N and N respectively, by isometries. Moreover we have the following relations n v k D v nk s nwhich expresses the compatibility between multiplication and addition.In other words, the s n and v k define a representation of the ax C b-semigroup 1 kP N D j n 2 N ;k2 N0 nover N, where the matrix0n 1k is represented by .v k s n / .Note that, for each n, the operators s n ;vs n ;:::;v n 1 s n generate a C -algebraisomorphic to O n , [4].3 A purely infinite simple C -algebra associated with theax C b-semigroupThe C -algebra A generated by the elements s n and v considered in Section 2 containsthe algebra K of compact operators on `2.N/. Denote by u, resp. u k the image of v,resp. v k in the quotient A=K. We also still denote by s n the image of s n in the quotient.Then the u k are unitary and are defined also for k 2 Z and they furthermore satisfythe characteristic relation P n 1kD0 uk e n u k D 1 where e n D s n sn denotes the rangeprojection of s n . This relation expresses the fact that N is the union of the sets ofnumbers which are congruent to k mod n for k D 0;:::;n 1.We now consider the universal C -algebra generated by elements satisfying theserelations.Definition 3.1. We define the C -algebra Q N as the universal C -algebra generated byisometries s n , n 2 N with range projections e n D s n sn , and by a unitary u satisfyingthe relationsXn 1s n s m D s nm ; s n u D u n s n ; u k e n u k D 1for n; m 2 N .kD0

C -algebras associated with the ax C b-semigroup over N 203Lemma 3.2. In Q N we have(a) e n D P m 1iD0 uin e nm u in for all n; m 2 N ;(b) e p s q D e pq s p D s q e p and e p e q D e pq D e q e p when p and q are relativelyprime;(c) sn s m D s m sn for n, m relatively prime.Proof. (a) This follows by conjugating the identity 1 D P u i e m u i by s n t sn andusing the fact that s n e m sn D e nm.(b) Since the u i e pq u i are pairwise orthogonal for 0 i < pq, we see thatu lp e pq u lp ? u kq e pq u kq if 0

204 J. CuntzConsider the action of T n on B n given by˛.t1 ;:::;t n /.s pi / D t i s piand denote by F n the fixed-point algebra for ˛ (i.e. F n D F \ B n ). There is a naturalfaithful conditional expectation E W B n ! F n defined by E.x/ D R T n ˛t.x/dt.Now D n is the fixed point algebra, in F n , for the action ˇ of T on F n given byˇt.e k / D e k ;ˇt.u/ D e it uand there is an associated expectation F W F n ! D n defined by F.x/ D R T ˇt.x/dt.The composition G D F ı E gives a faithful conditional expectation A n ! D n .These conditional expectations extend to the inductive limit and thus give conditionalexpectations E W Q N ! F , F W F ! D and G W Q N ! D.Theorem 3.4. The C -algebra Q N is simple and purely infinite.Proof. Since inductive limits of purely infinite simple C -algebras are purely infinitesimple [8], 4.1.8 (ii), it suffices to show that each B n is purely infinite simple.For each N , denote, as above, by D n.N / the subalgebra of D n generated byfu k e l u k ;k2 Zg where l D p1 N pN 2 :::pN n . The natural map Spec D n ! Spec D n.N /is surjective and, by the proof of Lemma 3.3, D .N /n Š C l .Choose 1 ; 2 ;:::; l in Spec D n such that f 1 ; 2 ;:::; l g!Spec D n.N / is bijectiveand such that i .s k t s k / ¤ i.t/for all k of the form k D p m 11:::p m nn with m i N . We can choose pairwise orthogonalprojections f 1 ;f 2 ;:::;f l in D n with sufficiently small support around the i such thatf i s k f i D 0 for all 1 i l and k as above and such that f i xf i D i .x/f i for allx 2 Dn N /. Then the map ' W D.N n ! C .f 1 ;:::;f l / Š C l defined by x 7! P f i xf iis an isomorphism.We denote by P .N / the set of linear combinations of all products of the formu k s m 1p 1:::s m np nwith m i N .Forx 2 P .N / we have that'.G.x// D X f i xf i :Let now 0 x 2 B n be different from 0. Since G is faithful, G.x/ ¤ 0 and wenormalize x such that kG.x/k D1. Let y 2 P .N / , for sufficiently large N , be suchthat kx yk

C -algebras associated with the ax C b-semigroup over N 205As a consequence of the simplicity of Q N we see that the canonical representationon `2.Z/ bys n . k / D nk ; u n . k / D kCnis faithful. Similarly, if we divide the C -algebra generated by the analogous isometrieson `2.N/, discussed in Section 2, by the canonical ideal K, we get an algebraisomorphic to Q N .The subalgebra F is generated by u and the projections e n , thus by a weightedshift. Thus we recognize F as the Bunce–Deddens algebra of the type where everyprime appears with infinite multiplicity. This well known algebra has been introducedin [3] in exactly that form. It is simple and has a unique tracial state. It can also berepresented as inductive limit of the inductive system .M n C.S 1 // with mapsM n .C.S 1 // ! M nk .C.S 1 //mapping the unitary u generating C.S 1 / to the k k-matrix010 0 ::: u1 0 ::: 0B0 1 ::: 0C@ ::: A :0 ::: 1 0From this description it immediately follows that K 0 .F / D Q and K 1 .F / D Z.Remark 3.5. (a) Just as O n is a crossed product of a UHF -algebra by N, we see thatQ N is a crossed product F Ì N by the multiplicative semigroup N . The algebraQ N also contains the commutative subalgebra D. Since D is the inductive limit of theD n , we see from Lemma 3.3 that Spec D D OZ WD Q O pZ p . The Bost–Connes algebraC Q [1] can be described as a crossed product C. OZ/ Ì N and is a natural subalgebraof Q N (using the natural inclusion C. OZ/ ! F ).We can also obtain Q N by adding to the generators n ;n 2 N (our s n ) and e , 2 Q=Z for C Q , described in [1], Proposition 18, one additional unitary generator usatisfyingue D e u and u n n D n u(here we identify an element of Q=Z with the corresponding complex number ofmodulus 1).(b) The algebra generated by s n (for a single n) and u has also been considered in[5]. It has been shown there that this algebra is simple and contains a Bunce–Deddensalgebra.4 The canonical action of R on Q NSo far, in our discussion, the isometries associated with each prime number appear asgenerators on the same footing and it is a priori not clear how to determine, for two

206 J. Cuntzprime numbers p and q, the size of p and q (or even the bigger number among p and q)from the corresponding generators s p and s q . In fact, the C -algebra generated by thes n , n 2 N is the infinite tensor product of one Toeplitz algebra for each prime numberand in this C -algebra there is no way to distinguish the s p for different p.However, the fact that we have added u to the generators allows to retrieve the nfrom s n using the KMS-condition.Definition 4.1. Let .A; t / be a C -algebra equipped with a one-parameter automorphismgroup . t /, a state on A and ˇ 2 .0; 1. We say that satisfies the ˇ-KMSconditionwith respect to . t /, if for each pair x;y of elements in A, there is a holomorphicfunction F x;y , continuous on the boundary, on the strip fz 2 C j Im z 2 Œ0; ˇgsuch thatF x;y .t/ D .x t .y//; F x;y .t C iˇ/ D . t .y/x/for t 2 R.Proposition 4.2. Let 0 be the unique tracial state on F and define on Q N by D 0 ı E. Let . t / denote the one-parameter automorphism group on Q N definedby t .s n / D n it s n and t .u/ D u. Then is a 1-KMS-state for . t /.Proof. According to [2] 5.3.1, it suffices to check that.x i .y// D .yx/for a dense *-subalgebra of analytic vectors for . t /. Here . z / denotes the extensionto complex variables z 2 C of . t / on the set of analytic vectors.However it is immediately clear that this identity holds for x;y linear combinationsof elements of the form as n or sm b, a; b 2 F . Such linear combinations are analyticand dense.Theorem 4.3. There is a unique state on Q N with the following property.There exists a one-parameter automorphism group . t / t2R for which is a1-KMS-state and such that t .u/ D u, t .e n / D e n for all n and t. Moreover wehave• is given by D 0 ı E where 0 is the canonical trace on F ,• the one-parameter group for which is 1-KMS, is unique and is the standardautomorphism group considered above, determined by t .s n / D n it s n and t .u/ D u:Proof. Since t acts as the identity on e n and on u, it is the identity on F D C .u; fe n g/.If is a KMS-state for . t /, it therefore has to be a trace on F . It is well-known (andclear) that there is a unique trace 0 on F . For instance, the relationXu k e n u k D 10k

C -algebras associated with the ax C b-semigroup over N 207shows that 0 .e n / D 1=n and 0 .u/ D 0.The relation.s n s n/ D n.s n s n /and the 1-KMS-condition show that t .s n / D n it s n .5 The K-groups of Q NThe K-groups of Q N can be computed using the fact that Q N is a crossed product of theBunce–Deddens algebra F by the semigroup N . The K-groups of a Bunce–Deddensalgebra are well known and easy to determine using its representation as an inductivelimit of algebras of type M n .C.S 1 //. Specifically, for F ,wehaveK 0 .F / D Q; K 1 .F / D Z:We consider Q N as an inductive limit of the subalgebras B n D C .F ;s p1 ;:::;s pn /where 2 D p 1

208 J. Cuntz6 Representations as crossed productsFor each n, define the endomorphism ' n of Q N by ' n .x/ D s n xs n . Since ' n' m D ' nmthis defines an inductive system.Definition 6.1. We define xQ N as the inductive limit of the inductive system .Q N ;' n /.By construction we have a family n of natural inclusions of Q N into the inductivelimit xQ N satisfying the relations nm ' n D m . We denote by 1 k the element k .1/ ofxQ N . We have that 1 k D kl .e l / 1 kl . The union of the subalgebras 1 kxQ N 1 k is densein xQ N and 1 k 1 kl for all k; l. In order to define a multiplier a of xQ N it thereforesuffices to define a1 k and 1 k a for all k.We can extend the isometries s n naturally to unitaries Ns n in the multiplier algebraM. xQ N / of xQ N by requiringNs n 1 k D k .s n /; 1 k Ns n D kn .s n /:Note that this is well defined because, using the identities s n e l D e nl s n and ' l .s n / Ds n e l ,wehave.1 k Ns n /1 l D kn .s n / l .1/ D knl .s n e l e kn / D knl .e nl s n e kn / D k .1/ l .s n / D 1 k .Ns n 1 l /:Proposition 6.2. The elements Ns n , n 2 N, define unitaries in M. xQ N / such that Ns n Ns m DNs nm and such that Ns n Ns m DNs m Ns n.Defining Ns a DNs n Ns m for a D n=m 2 Q C we define unitaries in M. xQ N / such thatNs a Ns b DNs ab for a; b 2 Q C .Proof. In order to show that Ns n is unitary it suffices to show that 1 k Ns n Ns n1 k Ns n Ns n D 1 k for all k, n.D 1 k andWe can also extend the generating unitary u in Q N to a unitary in the multiplieralgebra M. xQ N /. We define the unitary Nu in M. xQ N / by the identityNu 1 k D k .u k /:We can also define fractional powers of Nu by settingNu 1=n 1 kn D kn .u k /:Proposition 6.3. For all a 2 Q Cand b 2 Q we have the identityProof. Check that Ns 1=n Nu DNu 1=n Ns 1=nNs a Nu b DNu ab Ns a :Following Bost–Connes we denote by P C Qthe ax C b-group P C 1 bQ D j a 2 Q 0 aC ;b2 Q :

C -algebras associated with the ax C b-semigroup over N 209It follows from the previous proposition that we have a representation of P C Qin theunitary group of M. xQ N /. Denote by A f the locally compact space of finite adeles overQ, i.e.,A f Df.x p / p2P j x p 2 OQ p and x p 2 OZ p for almost all pgwhere P is the set of primes in N.The canonical commutative subalgebra D of Q N is by the Gelfand transform isomorphicto C.X/ where X is the compact space OZ D Q O p2PZ p . It is invariant underthe endomorphisms ' n and we obtain an inductive system of commutative algebras.D; ' n /. The inductive limit xD of this system is a canonical commutative subalgebraof xQ N . It is isomorphic to C 0 .A f /. In fact the spectrum of xD is the projective limit ofthe system (Spec D; O' n / and ' n corresponds to multiplication by n on OZ.Theorem 6.4. The algebra xQ N is isomorphic to the crossed product of C 0 .A f / by thenatural action of the ax C b-group P C Q .Proof. Denote by B the crossed product and consider the projection e 2 C 0 .A f / Bdefined by the characteristic function of the maximal compact subgroup Q O p2PZ p A f . Consider also the multipliers Ns n and Nu of B defined as the images of the elements100nand1101of PCQ , respectively. Then the elements s n D Ns n e and u D Nuesatisfy the relations defining Q N . Moreover, the C -algebra generated by the s n and ucontains the subalgebra eC 0 .A f / of B. This shows that this subalgebra equals eBeand, by simplicity of Q N , it follows that eBe Š Q N .It is a somewhat surprising fact that we get exactly the same C -algebra, eventogether with its canonical action of R if we replace the completion A f of Q at thefinite places by the completion R at the infinite place.Theorem 6.5. The algebra xQ N is isomorphic to the crossed product of C 0 .R/ by thenatural action of the ax+b-group P C Q . There is an isomorphism xQ N ! C 0 .R/ Ì P C Qwhich carries the natural one parameter group t to a one parameter group 0 t suchthat 0 t .f u g/ D a it fu g where u g denotes the unitary multiplier of C 0 .R/ Ì P C Qassociated with an element g 2 P C Qof the form 1 bg D0 a(the KMS-state is carried to the state which extends Lebesgue measure on C 0 .R/).In order to prove this we need a little bit of preparation concerning the representationof the Bunce–Deddens algebra F as an inductive limit. As noted above it is an inductivelimit of the inductive system .M n .C.S 1 /// with maps sending the unitary generator zof C.S 1 / to a unitary v in M k .C/ ˝ C.S 1 / satisfying v k D 1 ˝ z. We observe nowthat such a v is unique up to unitary equivalence.Lemma 6.6. Let v 1 ;v 2 be two unitaries in M k .C/˝C.S 1 / such that v k 1 D vk 2 D 1˝z.Then there is a unitary w in M k .C/ ˝ C.S 1 / such that v 2 D wv 1 w .

210 J. CuntzProof. The spectral projections p i .t/ for the different k th roots of e 2it 1, given byv 1 .t/ and v 2 .t/, have to be continuous functions of t. This implies that all the p i .t/ areone-dimensional and that, after possibly relabelling, we must have the situation wherep i .t C 1/ D p iC1 .t/:This means that the p i combine to define a line bundle on the k-fold covering of S 1by S 1 . However any two such line bundles are unitarily equivalent.We now determine the crossed product C 0 .R/ Ì Q where Q acts by translation.Lemma 6.7. The crossed product C 0 .R/ Ì Q is isomorphic to the stabilized Bunce–Deddens algebra K ˝ F .Proof. The algebra C 0 .R/ Ì Q is an inductive limit of algebras of the form C 0 .R/ Ì Zwith respect to the mapsˇk W C 0 .R/ Ì Z! C 0 .R/ Ì Zobtained from the embeddings Z Š Z 1 k,! Q. It is well known that C 0.R/ Ì Z isisomorphic to K ˝ C.S 1 /. An explicit isomorphism is obtained from the mapXf n u n 7 !X k .f n /e k;kCnn2Zn2Z;k2Zwhere k denotes translation by k, u k denotes the unitary in the crossed product implementingthis automorphism and e ij denote the matrix units in K Š K.`2Z/.This map sends C 0 .R/ Ì Z to the mapping torus algebraff 2 C.R; K/ j f.t C 1/ D Uf .t/U gŠK ˝ C.S 1 /where U is the multiplier of K D K.`2.Z// given by the bilateral shift on `2.Z/.A projection p corresponding, under this isomorphism, to e ˝ 1 in K ˝ C.S 1 /with e a projection of rank 1 can be represented in the form p D ug C f C gu with appropriate positive functions f and g with compact support on R. Underthe map C 0 .R/ Ì Z Š C 0 .R/ Ì kZ ! C 0 .R/ Ì Z, the projection p is mapped top 0 D u k g k C f k C g k u k where g k .t/ WD g.t=k/, f k .t/ WD f.t=k/. Now, p 0corresponds to a projection of rank k in K ˝ C.S 1 /.Let z be the unitary generator of C.S 1 /. Then the element e ˝ z corresponds thefunction e 2it .ug C f C gu / which is mapped to v D e 2it=k .u k g k C f k C g k u k /.Thus v k D p 0 and p 0 corresponds to a projection of rank k in K ˝ C.S 1 /.On the other hand F D lim A n where A n D M n .C.S 1 // and the inductive limit!is taken relative to the maps ˛k W M n .C.S 1 // ! M kn .C.S 1 // which map the unitarygenerator z of C.S 1 / to an element v such that v k D 1 ˝ z. Compare this now to theinductive system A 0 n D C 0.R/ Ì Z 1 n with respect to the maps ˇk considered above.

C -algebras associated with the ax C b-semigroup over N 211From our analysis of ˇk and from Lemma 6.6 we conclude that there are unitaries W kin M.A 0 / such that the following diagram commuteskn˛k A knA nA 0 nAd W kˇkA 0 knwhere the vertical arrows denote the natural inclusions M n .C.S 1 // ! K ˝ C.S 1 /.We conclude that these natural inclusions induce an isomorphism from K ˝ F Dlim!K ˝ A n to C 0 .R/ Ì Q D lim!A 0 n .Proof of Theorem 6.5. Consider the commutative diagram and the injection F ,!C 0 .R/ Ì Q constructed at the end of the proof of Lemma 6.7. Note also that theˇk are -unital and therefore extend to the multiplier algebra. This shows that theinjection transforms ˛p into ˇp times an approximately inner automorphism, for eachprime p. Therefore the injection transforms t , which is the dual action on the crossedproduct for the character n 7! n it of N , to the restriction of the corresponding dualaction on the crossed product .C 0 .R/ Ì Q/ Ì N .Finally, note that the endomorphisms ˇn of C 0 .R/ Ì Q are in fact automorphismsand that therefore ˇ extends from a semigroup action to an action of the group Q C .This shows that .C 0 .R/ Ì Q/ Ì N D .C 0 .R/ Ì Q/ Ì Q C D C 0.R/ Ì P C Q .Remark 6.8. If we consider the full space of adeles Q A D A f R rather than the finiteadeles A f or the completion at the infinite place R, we obtain the following situation.By [9], IV, §2, Lemma 2, Q is discrete in Q A and the quotient is the compact space. OZ R/=Z. Thus, the crossed product C 0 .Q A / Ì Q which would be analogous tothe Bunce–Deddens algebra is Morita equivalent to C.. OZ R/=Z/. Now,. OZ R/=Zhas a measure which is invariant under the action of the multiplicative semigroup N .Therefore the crossed product C 0 .Q A / Ì P C Q has a trace and is not isomorphic to Q N .7 The case of the multiplicative semigroup Z We consider now the analogous C -algebras where we replace the multiplicative semigroupN by the semigroup Z . This case is also important in view of generalizationsfrom Q to more general number fields (in fact the construction of Q Z and much of theanalysis of its structure carries over to the ring of integers in an arbitrary algebraic numberfield, at least if the class number is 1). Thus, on `2.Z/ we consider the isometriess n ;n2 Z and the unitaries u m ;m2 Z defined by s n . k / D nk and u m . k / D kCm .As above, these operators satisfy the relationss n s m D s nm ; s n u m D u nm s n ;Xn 1u i e n u i D 1: (1)iD0

212 J. CuntzThe operator s 1 plays a somewhat special role and we therefore denote it by f . Thenthe s n ;n2 Z and u generate the same C -algebra as the s n ;n2 N together with u andf . The element f is a selfadjoint unitary so that f 2 D 1 and we have the relationsfs n D s n f; f uf D u 1 :We consider now again the universal C -algebra generated by isometries s n ;n 2 Zand a unitary u subject to the relations (1). We denote this C -algebra by Q Z . We seefrom the discussion above that we get a crossed product Q Z Š Q N Ì Z=2 where Z=2acts by the automorphism ˛ of Q N that fixes the s n ;n2 N and ˛.u/ D u 1 .Theorem 7.1. The algebra Q Z is simple and purely infinite.Proof. Composing the conditional expectation G W Q N ! D used in the proof ofTheorem 3.4 with the natural expectation Q Z D Q N Ì Z=2 ! Q N we obtain again afaithful expectation G 0 W Q Z ! D. The rest of the proof follows exactly the proof ofTheorem 3.4, using in addition the fact that the f i in that proof can be chosen such thatf i ff i D 0.Denote by P Q the full ax C b-group over Q, i.e. 1 bP Q D j a 2 Q ;b2 Q :0 aTheorem 7.2. Q Z is isomorphic to the crossed product of C 0 .A f / or of C 0 .R/ by thenatural action of P Q .Proof. This follows from Theorems 6.4 and 6.5 since P Q D P C Q Ì Z=2.On Q Z we can define the one-parameter group . t / by t .s n / D n it s n ;n 2 N .The fixed point algebra is the crossed product F Ì Z=2 of the Bunce–Deddens algebraby Z=2.In order to compute the K-groups of Q Z we first determine the K-theory for F 0 DF Ì Z=2. This algebra is the inductive limit of the subalgebras A 0 n D C .u; f; e n /.Lemma 7.3. (a) The C -algebra C .u; f / is isomorphic to C .D/, where D is thedihedral group D D Z Ì Z=2 (Z=2 acts on Z by a 7! a). For each n D 1;2;:::,the algebra A 0 n is isomorphic to M n.C .D//.(b) We have K 0 .A 0 n / D K 0.C .D// D Z 3 and K 1 .A 0 n / D K 1.C .D// D 0 forall n. The generators of K 0 .C .D// are given by Œ1 and by the classes of the spectralprojections .uf / C and f C of uf and f , for the eigenvalue 1.(c) Let p be prime. If p D 2, then the map K 0 .A 0 n / ! K 0.Apn 0 / is described, withrespect to the basis Œ1, Œ.uf / C and Œf C of K 0 .C .D// Š Z 3 , by the matrix02 110@ 0 0 1A0 0 1

C -algebras associated with the ax C b-semigroup over N 213If p is odd, then the map K 0 .A 0 n / ! K 0.Apn 0 / is described by the matrix0@ p p 1 1p 12 20 0 1 A :0 1 0Proof. (a) It is clear that the universal algebra generated by two unitaries u; f satisfyingf 2 D 1 and fuf D u is isomorphic to C .D/.In the decomposition of C .u; f; e 2 / with respect to the orthogonal projections e 2and ue 2 u 1 , the elements u and f correspond to matrices 0w1 0and f0 00 f 1where w is unitary and f 0 ;f 1 are symmetries (selfadjoint unitaries).The relations between u and f imply that wf 1 D f 0 and that wf 1 is a selfadjointunitary, whence f 1 wf 1 D w . Thus A 0 2 is isomorphic to M 2.C .w; f 1 // and w; f 1satisfy the same relations as u; f .If p is an odd prime, then in the decomposition of C .u; f; e p / with respect to thepairwise orthogonal projections e p ;ue p u 1 ;:::;u p 1 e p u .p 1/ , u and f correspondto the following p p-matrices0101 f0 0 ::: w0 0 ::: 01 0 ::: 00 0 ::: f 1B0 1 ::: 0C@ ::: A and 0 0 ::: f2 0B :::C@ 0 0 f0 ::: 1 02 ::: A0 f 1 ::: 0 0where w and the f 1 ;:::;fp 12are unitary and f 0 is a symmetry.The relation fuf D u implies that f 0 D wf 1 , f 1 D f 2 DDfp 12and f 2i1 for all i. From .wf 1 / 2 D 1 we derive that f 1 wf 1 D w . Thus Ap 0 Š M p.C .w; f 1 //and w; f 1 satisfy the same relations as u; f .The case of general n is obtained by iteration using the fact that C .u; f; e nm / ŠM m .C .u; f; e n //.(b) This follows for instance from the well known fact that C .D/ Š .C ? C/ .(c) Let p D 2. Using the description of K 0 .C .u; f // under (b) and the descriptionof the map C .u; f / ! C .u; f; e 2 / from the proof of (a), we see that the generatorsŒ1, Œ.uf / C and Œf C are respectively mapped to 2Œ1, Œ1 and Œf C C Œ.uf / C .Let p be an odd prime. The matrix corresponding to f in the proof of (a) is conjugateto the matrix where all the f i , i D 1;:::;.p 1/=2 are replaced by 1. Thus the classof f C is mapped to Œ.wf 1 / C C p 12 Œ1.The matrix corresponding to uf is conjugate to a selfadjoint matrix of the sameform as the matrix representing the image of f , but with the upper left entry f 0 replacedby f 1 . Thus the class of .uf / C is mapped to Œf C1 C p 12 Œ1.D

214 J. CuntzProposition 7.4. The K-groups of the algebra F 0 D F Ì Z=2 are given by K 0 .F 0 / DQ ˚ Z and K 1 .F 0 / D 0.Proof. This follows from the fact that F 0 is the inductive limit of the algebras A 0 n andthe description of the maps K 0 .A 0 n / ! K 0.Apn 0 / given in Lemma 7.3 (c). The elementŒf C C Œuf C Œ1 is invariant under the map K 0 .A 0 n / ! K 0.Apn 0 / and maps to thegenerator of Z in the inductive limit.Remark 7.5. Since F 0 is a simple C -algebra with unique trace in a classifiableclass (in the sense of the classification program), this computation of its K-theory incombination with the classification result in [6] shows that F 0 is an AF -algebra. Thuswe obtain an example of a simple AF -algebra F 0 admitting an action by Z=2 suchthat the fixed point algebra is the Bunce–Deddens algebra F and thus not AF . Thisexample had been considered before in [7].In analogy to the case over N we denote by B 0 n the C -subalgebra of Q Z generatedby u; f; s p1 ;:::;s pn . We now immediately deduceTheorem 7.6. We haveandK 0 .B 0 n / D Z2n 1 ; K 1 .B 0 n / D Z2n 1K 0 .Q Z / D Z 1 ; K 1 .Q Z / D Z 1 :Note added in proof. The arguments in the computation of the K-theory are notcomplete. This problem will be addressed in a subsequent paper.References[1] J.-B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneoussymmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), 411–457.[2] Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics,Vol. 2, Equilibrium states. Models in quantum statistical mechanics, sec. ed., Texts Monogr.Phys., Springer-Verlag, Berlin 1997.[3] John W. Bunce and James A. Deddens, A family of simple C -algebras related to weightedshift operators, J. Functional Analysis 19(1975), 13–24.[4] Joachim Cuntz, Simple C -algebras generated by isometries, Comm. Math. Phys. 57 (1977),173–185.[5] Ilan Hirshberg, On C -algebras associated to certain endomorphisms of discrete groups,New York J. Math. 8 (2002), 99–109.[6] Xinhui Jiang and Hongbing Su, A classification of simple limits of splitting interval algebras,J. Funct. Anal. 151 (1997), 50–76.[7] A. Kumjian, An involutive automorphism of the Bunce-Deddens algebra, C. R. Math. Rep.Acad. Sci. Canada 10 (1988), 217–218.

C -algebras associated with the ax C b-semigroup over N 215[8] M. Rørdam, Classification of nuclear, simple C -algebras, in Classification of nuclearC -algebras. Entropy in operator algebras, Encyclopaedia Math. Sci. 126, Springer-Verlag,Berlin 2002, 1–145.[9] André Weil, Basic number theory, Grundlehren Math. Wiss. 144, Springer-Verlag, NewYork1967.

On a class of Hilbert C*-manifoldsWend Werner1 IntroductionThis note had initially two objectives, an explicit calculation, for all vector fields, ofthe invariant connection on a certain type of infinite dimensional symmetric space and,using results from [1], to characterize those invariant cone fields on a similar kindof spaces that can be thought of as the result of some kind of ‘quantization’. Bothquestions are related since the invariance of the cone fields is intimately connected tothe behavior of parallel transport along geodesics.Invariant connections for finite dimensional symmetric spaces have long been knownto exist and to be unique. One has to be a little bit more careful in the infinite dimensional,Banach manifold setting since there the existence of a sufficient amount ofsmooth functions no longer can be proven. The type of calculations we are interestedin here have been carried out under similar circumstances in [2].Important for our approach is to use an invariant Hilbert C*-structure on the fibersof the tangent bundle. We show that the symmetric space we are dealing with canbe defined in terms of the automorphism group of this structure. For the underlyinginvariant (operator) Finsler structure, the analogous result holds.The theory of invariant cone fields on finite dimensional spaces is very well developed.A comprehensive account is [5]. We will see in the last section that the infinitedimensional theory might behave slightly differently.2 Hilbert C*-manifolds2.1. Recall that a (left) Hilbert C*-module over a C*-algebra A is a complex vectorspace E which is a left A-module with a sesquilinear pairing E E ! A satisfying,for r; s 2 E and a 2 A, the following requirements:(i) har;siDahr; si;(ii) hr; si Dhs; ri ;(iii) hs; si >0for s ¤ 0;(iv) equipped with the normE is a Banach space.ksk D p khs; sik;Right Hilbert C*-modules are defined similarly. Whenever we want to refer to thealgebra A explicitly, we speak of a Hilbert A-module.

218 W. Werner2.2. The objects defined above coincide with the so called ternary rings of operators(TRO), which are intrinsically characterized in [9], [11]. On such a space E, a tripleproduct f; ; g is given in such a way that E, up to (the obvious definition of) TROisomorphisms,is a subspace of a space of bounded Hilbert space operators L.H /,invariant under the triple productfx;y;zg Dxy z:The relation to (left) Hilbert C*-modules is based on the equationfx;y;zg Dhx;yiz;connecting triple product to module action as well as scalar product of a HilbertA-module. Here, A is the algebra of linear mappingsA D EE D lin fx 7! fe 1 ;e 2 ;xgje 1;2 2 Eg ;which is independent of the chosen embedding. Note that the norm of an element e 2 Emust coincide with kek Dkfe; e; egk 1=3 .2.3. TRO-morphisms will be those mappings that preserve the product f; ; g. Thesemappings differ in general from what is considered to be the natural choice for HilbertA-morphisms, the so called adjointable maps. The latter are in particular A-modulemorphisms, a property that would be too restrictive for our purposes. We will hencestick in the following to TRO-morphisms.Definition 2.4. Let M be a Banach manifold and A a C*-algebra. M is said to bea(right-, left-) Hilbert A-manifold if on each tangent space T p .M / there is given thestructure of a Hilbert A-module depending smoothly on base points.Definition 2.5. Let M be a Hilbert C*-manifold. The group of automorphisms,Aut M consists of all diffeomorphisms ˆW M ! M so that dˆ is (pointwise) aTRO-morphism.2.6. It is not clear under which circumstances a Banach manifold can be given thestructure of a Hilbert C*-manifold. This is so because, first, no characterization seemsto be known of Banach spaces that are (topologically linear) isomorphic to HilbertC*-modules, and, second, because in general, there is no smooth partition of the unitthat would permit the step from local to global.We will use group actions instead. This will bring up homogeneous Hilbert C*-manifolds in the following sense.Definition. A Banach manifold M is called homogeneous Hilbert C*-manifold iff it isa Hilbert C*-manifold for which Aut M acts transitively.2.7. Suppose M is a homogeneous space with respect to a smooth Banach Lie groupaction. Fix a base point o 2 M , and denote the isotropy subgroup at o by H . Supposethat T o .M / carries the structure of a Hilbert A-module with form h; i o , module

On a class of Hilbert C*-manifolds 219map x 7! a o x, and TRO-structure fx;y;zg o Dhx;yi o o z. If for all h 2 H ,x;y;z 2 T o .M / we haved o hfx;y;zg o Dfd o h.x/; d o h.y/; d o h.z/g othen a Hilbert A-module on T p .M / for any p D g.o/ 2 M , is well-defined throughfx;y;zg p D d o gfd p g 1 .x/; d p g 1 .y/; d p g 1 .z/g o :With this structure, M becomes a homogeneous Hilbert C*-manifold2.8. Our example here is the following. Fix a TRO E and denote by U its open unitball. If we follow the path laid out above, we find the following invariant HilbertC*-structure on U . Define a triple product for T a M at a 2 U byfx;y;zg a D x.1 a a/ 1 y .1 aa / 1 z;so thatas well ashx;yi a D .1 aa / 1=2 x.1 a a/ 1 y .1 aa / 1=2 a z D .1 aa / 1=2 .1 aa / 1=2 ; 2 EE :We will refer to this structure as the canonical Hilbert C*-structure on U .2.9. This definition is motivated in the following way. As shown in [4], Hol U , thegroup of all biholomorphic automorphisms U , consists of mappings of the form T ıM a ,where for any a 2 U ,M a .x/ WD .1 aa / 1=2 .x C a/.1 C a x/ 1 .1 a a/ 1=2 ;and T is a (linear) isometry of E, restricted to U . Then Hol U acts transitively on U ,and the group of (linear) isometries is the isotropy subgroup at the point 0. For lateruse, we include here the fact thatas well asd x M a .h/ D .1 aa / 1=2 .1 C xa / 1 h.1 C a x/ 1 .1 a a/ 1=2dx 2 M a.h 1 ;h 2 /D .1 aa / 1=2 .1 C xa / 1 h 2 a .1 C xa /h 1 .1 C a x/ 1 .1 a a/ 1=2.1 aa / 1=2 .1 C xa / 1 h 1 .1 C a x/ 1 a h 2 .1 C a x/ 1 .1 a a/ 1=2 :The Hilbert C*-structure from 2.8 is in fact constructed according to the constructionin 2.6 for a group G, somewhat smaller than Hol U .

220 W. WernerDefinition 2.10. Suppose X is a (closed) subspace of L.H /. EquipM n .X/ D ˚.x ij / j x ij 2 X for i;j D 1;:::;n with the norm it carries as a subspace of M n .L.H // D L.H ˚˚H/, and, for abounded operator T W X ! X, denote by T .n/ D id Mn .C/ ˝ X W M n .X/ ! M n .X/the operator .x ij / 7 ! .T x ij /. Then T is said to be completely bounded, iffkT k cb WD sup kT .n/ k < 1:n2NSimilarly, T is called a complete isometry iff each of the maps T .n/ is an isometry.We have the followingTheorem 2.11 ([3], [9]). Let E be a TRO. Then M n .E/ carries a distinguished TROstructure,and the group of TRO-automorphisms of E coincides with the group ofcomplete isometries.Theorem 2.12. Let U be the open unit ball of a TRO E, equipped with the canonicalHilbert C*-structure, and suppose M n .E/ carries the standard TRO-structure foreach n 2 N. Then a diffeomorphism ˆW U ! U is a Hilbert C*-automorphism iffˆ D T ıM a , where a 2 U , and T is the restriction of a linear and completely isometricmapping of E to U .Proof. That each map of the form T ı M a is in Aut U follows from the constructionof the Hilbert C*-structure on U as well as from Theorem 2.11. Since the derivativeof any element ˆ 2 Aut U is complex linear by definition, Aut U Hol U , and henceˆ D T ı M a for some a 2 U and an isometry T . Because the linear map T fixes theorigin, dT must be a TRO-automorphism, and the result follows, by another applicationof Theorem 2.11.2.13. Let M again be a Hilbert C*-manifold. Then the tangent space at each pointm of M carries an essentially unique norm, naturally connected to the TRO-structureby kxk m Dkfx;x;xg m k 1=3 . Since TROs embed completely isometrically into somespace of bounded Hilbert space operators, this norm naturally extends to the spacesM n .T m M/. We will call a Banach manifold M an operator Finsler manifold in caseeach tangent space carries the structure of an operator space depending continuously onbase points. In this sense, any Hilbert C*-manifold carries an operator Finsler structurein a natural way. If, as before, U is the open unit ball of a fixed TRO .E; f; ; g; kk/,furnished with its invariant Hilbert C*-structure then, using that complete isometriesand automorphisms of a TRO are the same mappings, we haveTheorem. If the open unit ball of a TRO is equipped with the natural invariant operatorFinsler structure its automorphism groups coincides with the automorphism group ofthe underlying homogeneous Hilbert C*-manifold.

On a class of Hilbert C*-manifolds 2213 The invariant connection3.1. The definition of a connection for Banach manifolds cannot be, due to the scarcityof smooth functions, the usual one. We follow [6], 1.5.1 Definition.Definition 3.2. Let M be a manifold, modeled over the Banach space E, and denotethe space of bounded bilinear mappings E E ! E by L 2 .E; E/. Then M is saidto possess a connection iff there is an atlas U for M so that for each U 2 U there is asmooth mapping W U ! L 2 .E; E/, called the Christoffel symbol of the connectionon U , which under a change of coordinates ˆ transforms according to.ˆ0X;ˆ0Y/D ˆ00 .X; Y / C ˆ0.X;Y/:The covariant derivative of a vector field Y in the direction of the vector field X is,locally, defined to be the principal part ofr X Y D dX.Y /.X;Y/;where, in a chart, the principal part of .u; X/ 2 U T.U/is X.The reader should note that this definition is equivalent to specifying a smoothvector subbundle H of TTM with the property that H p is for each p 2 TM closedand complementary to the tangent space ker.d p / of the fiber E .p/ through p. Itisnot, however, equivalent to the requirement that r be C 1 .M /-linear in its first variableand a derivation w.r.t. the action of C 1 .M / on vector fields, although connections asdefined above do have this property.3.3. We can keep the notion of invariance of a connection under the smooth action ofa (Banach) Lie group, however. In fact, if such a group G acts on M then, for eachg 2 G a connection g r is defined by lettingg r X Y Dr g Xg Y;g X.gm/ D d m gX.m/:Christoffel symbols then transform as in the definition above, g.m/ .g 0 .m/X.m/; g 0 .m/Y.m// D g 00 .m/.X.m/; Y.m// C g 0 .m/ m .X.m/; Y.m//;and we call r invariant under the action of G whenever g rDrfor all g 2 G.3.4. If M is the Hilbert C*-manifold U that was defined in the last section, therecan be at most one connection which is invariant under the action of the group ofbiholomorphic self-maps of U . This is because the difference of two of them is thedifference of their Christoffel symbols which have to vanish at each point according tothe way they transform under the reflections a D M a o M a , o .x/ D x. To findone, we let o .x; y/ D 0:Then, for any g leaving the origin fixed, g 00 D 0 and so 0 remains zero when transformedunder g. Using the transformation rule for Christoffel symbols we find

222 W. WernerTheorem 3.5. On the Hilbert C*-manifold U there exists exactly one invariant connectionwhose Christoffel symbol at a is given by a .x; y/ D y.1 a a/ 1=2 a .1 aa / 1=2 xC x.1 a a/ 1=2 a .1 aa / 1=2 y D 2fx;Ma 0 .0/a; ygs a ;where fx;y;zg sproduct on E.D 1 .fz;y;xg Cfx;y;zg/ denotes the (symmetric) Jordan triple23.6. The reader should observe that the form f; ; g defined above is parallel for theinvariant connection, i.e. for all vector fields X; Y; Z and W we haver W fX; Y; Zg Dfr W XYZgCfXr W YZgCfXYr W Zg:This follows either from direct calculation (using the Jacobi identity for f; ; g), orfrom the fact that the covariant derivative of f; ; g is invariant under reflections. Thiscondition shows that r behaves like the Levi-Civitá connection with respect to theHilbert C*-structure on M . Under what conditions such a connection exists undermore general circumstances, is investigated in [10].4 Causality4.1. It is customary in a number of physical theories to pass from a Lorentzian to aRiemannian manifold (‘Wick Rotation’). This is due to the difficulties one is facedwith in a truly Lorentzian situation and which disappear in the Riemannian set-up. Onepeculiarity of Lorentzian manifolds can still be modeled in the Riemannian situation:Light cones in the fibers of the tangent bundle which specify those pairs of points on Mthat may interact with each other and are thus intimately connected to the notion ofcausality.Definition 4.2. A cone field on a manifold M is, for each m 2 M , an assignment of acone C m T m .M /.Here a cone is always supposed to be pointed but not necessarily to be generating.4.3. Our interest here is with cone fields that are invariant under the action of a group G,i.e. that C gm D d m g.C m / for all m 2 M and g 2 G. In our case, G will be a certainsubgroup of Aut U . Though it might be tempting to see invariance under Hol U (ora large subgroup) as a substitute for diffeomorphism invariance in the complex case,the main motivation here comes from the property that the connection defined in theprevious section has the property that for an invariant cone field .C m / parallel transportalong a curve from .a/to .b/ends in C .b/ if it began in C .a/ . This is well knownin the finite dimensional situation (see [7], [8]) and, in this regard, not much is changingin passing to infinite dimensions.

On a class of Hilbert C*-manifolds 2234.4. In the following we will suppose that ‘space’ is represented by a certain set D ofself-adjoint operators, and that on this set there is defined an invariant cone field. Thequestion we would like to answer is this: How can such fields be characterized, whichcome from interpreting the points of D as bounded Hilbert space operators?In the following, denote by E a fixed (abstract) ternary ring of operators, and by Uits unit ball.4.5. On top of the causal structure of U we need ‘selfadjointness’, which for us will bethe existence of a ‘real form’ for U , compatible with the (almost) complex structure.We do this by requiring that E carries an involutory real automorphism ‘*’ so that.ix/ D ix for all x 2 E. Since we will be studying TRO-embeddings into L.H /that preserve the real form, it will be necessary to impose the additional condition that,for all x;y;z 2 E, fx;y;zg Dfz ;y ;x g. (Note that then each x 2 E has a uniquedecomposition into real and imaginary parts, with some norm estimates.) A ternaryring of operators E that meets all these conditions, will be called a *-ternary ring ofoperators. We will suppose in the following that E is a space of this kind.4.6. The ‘space manifold’here will be the open unit ball U sa of the selfadjoint part of E.U sa is itself a symmetric space. If G sa consists of all elements in Aut U which leaveU sa invariant, then U sa D G sa =H sa , where H sa comprises the TRO-automorphisms thatare *-selfadjoint. In fact, it follows from fx;y;zg Dfz ;y ;x g for all x;y;z 2 E(and an expansion into power series) that M a .x/ D M a .x / for all x 2 U and soM a 2 G sa iff a D a. A TRO-automorphism T is in H sa iff T.x / D T.x/ for allx 2 U , and soG sa D fT ı M a j T 2 H sa ;a 2 U sa g ;as well as U sa D G sa =H sa .4.7. In order to comply with the requirement that causality be invariant under paralleltransport we have to impose the condition that the field of cones we fix in TU sa mustbe invariant under the action of G sa . We consider smooth embeddings ˆW U ! L.H /which preserve• the Hilbert C*-structure,• the complex structure as well as the (canonical) real forms,• the action of the automorphism groups.And we want to know: What characterizes the G sa invariant cone fields that are pulledback to U via ˆ? Whenever a cone field meets these properties we will call it natural.4.8. Since a natural cone field is supposed to be invariant under the action of G sa ,we may restrict our attention to cones in T o U D E. Furthermore, any cone in Ethat gives rise to a G sa invariant field of cones has to be invariant under the action ofH sa . It can also be shown that under the assumptions made above, dˆ has to preservethe ternary structure of each tangent space T p U . The question we were asking thusbecomes: What properties must an H sa -invariant cone in E sa possess so that it is of the

224 W. Wernerform ‰ 1 .L.H / C / for a *-ternary monomorphism ‰ W E ! L.H /? For the sake ofsimplicity, we restrict our attention to the case where E is a dual Banach space. ThenTheorem 4.9 ([1]). Let E be a *-ternary ring of operators, which is a dual Banachspace. Define the center of E byZ.E/ D fe 2 E j exy D xye for all x;y 2 Egand call u 2 E tripotent whenever fu; u; ug Du. Then a cone C E sa is natural iffthere is a central, selfadjoint tripotent element u 2 E so thatC D C u WD feue j e 2 Eg :Definition 4.10. If E is a TRO with real form *, and u 2 E sa , then u is called rigid iffˆ.u/ D u for all *-selfadjoint TRO-automorphisms ˆ of E.The following result is now easy to prove.Theorem 4.11. The only cones C u in E that give rise to a G sa -invariant causal structureon U sa , coming from an embedding of E into some space L.H /, are those for whichthe central, selfadjoint element u is rigid.Note that the existence of a rigid selfadjoint central tripotent u is impossible in finitedimensions. In fact, Z.E/ is a weak*-closed commutative von Neumann algebra (seeLemma 4.10 in [1]) in which the tripotents are projections, which is invariant under anyTRO-automorphism ˆ of E, and for which ˆj Z.E/is a C*-automorphism whose propertiesessential here are best understood by means of the underlying homeomorphismof the spectrum of Z.E/.References[1] D. Blecher and W. Werner, Ordered C -modules, Proc. London Math. Soc. 92 (2006),682–712.[2] G. Corach, H. Porta, and L. Recht, The geometry of spaces of projections in C*-algebras,Adv. Math. 101 (1993), 59–77.[3] M. Hamana, Triple envelopes and Shilov boundaries of operator spaces, Math. J. ToyamaUniv. 22 (1999), 77–93.[4] L.A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, inInfinite Dimensional Holomorphy (T. L. Hayden and T. J. Su Suffridge, eds.), Lecture Notesin Math. 364, Springer-Verlag, Berlin 1974, 13–40.[5] J. Hilgert and G. Olafsson, Causal symmetric spaces, Perspect. Math. 18, Academic Press,San Diego, CA, 1997.[6] W. Klingenberg, Riemannian Geometry, second edition, de Gruyter Stud. Math. 1, Walterde Gruyter, Berlin 1995.[7] O. Loos, Symmetric Spaces I, Benjamin, New York, Amsterdam 1969.

On a class of Hilbert C*-manifolds 225[8] D. Mittenhuber and K.-H. Neeb, On the exponential function of an ordered manifold withaffine connection, Math. Z. 218 (1995), 1–23.[9] M. Neal and B. Russo, Operator space characterizations of C*-algebras and ternary rings,Pacific J. Math. 209 (2003), 339–364.[10] W. Werner, Hilbert C*-manifolds with Levi-Civitá connection, in preparation.[11] H. Zettl, A characterization of ternary rings of operators, Adv. in Math. 48 (1983), 117–143.

Duality for topological abelian group stacksand T -dualityUlrich Bunke, Thomas Schick, Markus Spitzweck, and Andreas ThomContents1 Introduction ....................................... 2281.1 A sheaf theoretic version of Pontrjagin duality .................. 2281.2 Picard stacks ................................... 2291.3 Duality of Picard stacks .............................. 2301.4 Admissible groups ................................ 2321.5 T -duality ..................................... 2342 Sheaves of Picard categories .............................. 2362.1 Picard categories ................................. 2362.2 Examples of Picard categories .......................... 2382.3 Picard stacks and examples ............................ 2392.4 Additive functors ................................. 2402.5 Representation of Picard stacks by complexes of sheaves of groups ....... 2423 Sheaf theory on big sites of topological spaces ..................... 2483.1 Topological spaces and sites ........................... 2483.2 Sheaves of topological groups .......................... 2483.3 Restriction ..................................... 2513.4 Application to sites of topological spaces ..................... 2613.5 Z mult -modules ................................... 2674 Admissibility of sheaves represented by topological abelian groups .......... 2694.1 Admissible sheaves and groups .......................... 2694.2 A double complex ................................. 2724.3 Discrete groups .................................. 2794.4 Admissibility of the groups R n and T n ..................... 2864.5 Profinite groups .................................. 2894.6 Compact connected abelian groups ........................ 3075 Duality of locally compact group stacks ........................ 3155.1 Pontrjagin Duality ................................. 3155.2 Duality and classification ............................. 3196 T -duality of twisted torus bundles ........................... 3236.1 Pairs and topological T -duality .......................... 3236.2 Torus bundles, torsors and gerbes ......................... 3276.3 Pairs and group stacks .............................. 3326.4 T -duality triples and group stacks ........................ 337References .......................................... 346

228 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1 Introduction1.1 A sheaf theoretic version of Pontrjagin duality1.1.1 A character of an abelian topological group G is a continuous homomorphismW G ! Tfrom G to the circle group T. The set of all characters of G will be denoted by yG. Itis again a group under point-wise multiplication.Assume that G is locally compact. The compact-open topology on the space ofcontinuous maps Map.G; T/ induces a compactly generated topology on this space ofmaps, and hence a topology on its subset yG Map.G; T/. The group yG equippedwith this topology is called the dual group of G.1.1.2 An element g 2 G gives rise to a character ev.g/ 2 yG defined by ev.g/./ WD.g/ for 2 yG. In this way we get a continuous homomorphismThe main assertion of Pontrjagin duality isev W G ! yG:Theorem 1.1 (Pontrjagin duality). If G is a locally compact abelian group, thenev W G ! yG is an isomorphism of topological groups.Proofs of this theorem can be found e.g. in [Fol95] or [HM98].1.1.3 In the present paper we use the language of sheaves in order to encode thetopology of spaces and groups. Let X; B be topological spaces. The space X givesrise to a sheaf of sets X on B which associates to an open subset U B the setX.U / D C.U;X/ of continuous maps from U to X. If G is a topological abeliangroup, then G is a sheaf of abelian groups on B.The space X or the group G is not completely determined by the sheaf it generatesover B. As an extreme example take B D ¹º. Then X and the underlying discretespace X ı induce isomorphic sheaves on B.For another example, assume that B is totally disconnected. Then the sheavesgenerated by the spaces Œ0; 1 and ¹º are isomorphic.1.1.4 But one can do better and consider sheaves which are defined on all topologicalspaces or at least on a sufficiently big subcategory S TOP. We turn S into aGrothendieck site determined by the pre-topology of open coverings (for details aboutthe choice of S see Section 3). We will see that the topology in S is sub-canonical sothat every object X 2 S represents a sheaf X 2 ShS. By the Yoneda Lemma the spaceX can be recovered from the sheaf X 2 ShS represented by X. The evaluation of thesheaf X on A 2 S is defined by X.A/ WD Hom S .A; X/. Our site S will in particularcontain all locally compact spaces.

Duality for topological abelian group stacks and T -duality 2291.1.5 We can reformulate Pontrjagin duality in sheaf theoretic terms. The circle groupT belongs to S and gives rise to a sheaf T. Given a sheaf of abelian groups F 2 Sh Ab Swe define its dual byD.F / WD Hom ShAb S .F; T/and observe thatD.G/ Š yGfor an abelian group G 2 S.The image of the natural pairingunder the isomorphismF ˝Z D.F / ! THom ShAb S.F ˝Z D.F /; T/ Š Hom ShAb S.F; Hom ShAb S .D.F /; T//D Hom ShAb S.F; D.D.F ///gives the evaluation mapev F W F ! D.D.F //:Definition 1.2. We call a sheaf of abelian groups F 2 Sh Ab S dualizable, if the evaluationmap ev F W F ! D.D.F // is an isomorphism of sheaves.The sheaf theoretic reformulation of Pontrjagin duality is now:Theorem 1.3 (Sheaf theoretic version of Pontrjagin duality). If G is a locally compactabelian group, then G is dualizable.1.2 Picard stacks1.2.1 A group gives rise to a category BG with one object so that the group appears asthe group of automorphisms of this object. A sheaf-theoretic analog is the notion of agerbe.1.2.2 A set can be identified with a small category which has only identity morphisms.In a similar way a sheaf of sets can be considered as a strict presheaf of categories. Inthe present paper a presheaf of categories on S is a lax contravariant functor S ! Cat.Thus a presheaf F of categories associates to each object A 2 S a category F .A/, andto each morphism f W A ! B a functor f W F.B/ ! F .A/. The adjective lax means,that in addition for each pair of composable morphisms f; g 2 S we have specifiedan isomorphism of functors g ı f f;gŠ .f ı g/ which satisfies higher associativityrelations. The sheaf of categories is called strict if these isomorphisms are identities.1.2.3 A category is called a groupoid if all its morphisms are isomorphisms. A prestackon S is a presheaf of categories on S which takes values in groupoids. A prestack is astack if it satisfies in addition descent conditions on the level of objects and morphisms.For details about stacks we refer to [Vis05].

230 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1.2.4 A sheaf of groups F 2 Sh Ab S gives rise to a prestack p BF which associates toU 2 S the groupoid p BF.U/ WD BF .U /. This prestack is not a stack in general. Butit can be stackified (this is similar to sheafification) to the stack BF .1.2.5 For an abelian group G the category BG is actually a group object in Cat. Thegroup operation is implemented by the functor BG BG ! BG which is obvious onthe level of objects and given by the group structure of G on the level of morphisms.In this case associativity and commutativity is strictly satisfied. In a similar manner,for F 2 Sh Ab S the prestack p BF on S becomes a group object in prestacks on S.1.2.6 A group G can of course also be viewed as a category G with only identitymorphisms. This category is again a strict group object in Cat. The functor G G ! Gis given by the group operation on the level of objects, and in the obvious way on thelevel of morphisms.1.2.7 In general, in order to define group objects in two-categories like Cat or the categoryof stacks on S, one would relax the strictness of associativity and commutativity.For our purpose the appropriate relaxed notion is that of a Picard category which wewill explain in detail in 2. The corresponding sheafified notion is that of a Picard stack.We let PIC S denote the two-category of Picard stacks on S.1.2.8 A sheaf of groups F 2 S Ab S gives rise to a Picard stack in two ways.First of all it determines the Picard stack F which associates to U 2 S the Picardcategory F.U/ in the sense of 1.2.6. The other possibility is the Picard stack BFobtained as the stackification of the Picard prestack p BF .1.2.9 Let A 2 PIC S be a Picard stack. The presheaf of its isomorphism classes ofobjects generates a sheaf of abelian groups which will be denoted by H 0 .A/ 2 Sh Ab S.The Picard stack A gives furthermore rise to the sheaf of abelian groups H 1 .A/ 2Sh Ab S of automorphisms of the unit object.A Picard stack A 2 PIC S fits into an extension (see 2.5.12)BH 1 .A/ ! A ! H 0 .A/: (1)1.3 Duality of Picard stacks1.3.1 It is an essential observation by Deligne that the two-category of Picard stacksPIC S admits an interior HOM PIC S . Thus for Picard stacks A; B 2 PIC S we have aPicard stackHOM PIC S .A; B/ 2 PICSof additive morphisms from A to B (see 2.4).

Duality for topological abelian group stacks and T -duality 2311.3.2 Since we can consider sheaves of groups as Picard stacks in two different waysone can now ask how Pontrjagin duality is properly reflected in the language of Picardstacks. It turns out that the correct dualizing object is the stack BT 2 PIC S, and notT 2 PIC S as one might guess first. For a Picard stack A we define its dual byD.A/ WD HOM PIC S .A; BT/:We hope that using the same symbol for the dual sheaf and dual Picard stack does notintroduce to much confusion (see the two footnotes attached to the formulas (2) and(3) below). One can ask what the duals of the Picard stacks F and BF look like. Ingeneral we have (see 5.5)D.BF/Š D.F /: 1 (2)One could expect thatD.F / Š BD.F /; 2 (3)but this only holds under the condition that Ext 1 Sh Ab S .F; T/ D 0 (see 5.6). This conditionis not always satisfied, e.g.Ext 1 Sh Ab S .˚n2NZ=2Z; T/ 6D 0(see 4.29). But the main effort of the present paper is made to show thatExt 1 Sh Ab S .G; T/ D 0for a large class of locally compact abelian groups (this condition is part of admissibility,see Definition 1.5).1.3.3 Let A 2 PICS be a Picard stack. There is a natural evaluationev A W A ! D.D.A//:In analogy to 1.2 we make the following definition.Definition 1.4. We call a Picard stack dualizable if ev A W A ! D.D.A// is an equivalenceof Picard stacks.1.3.4 If G is a locally compact abelian group andthen the evaluation mapsExt 1 Sh Ab S .G; T/ Š Ext1 Sh Ab S . yG; T/ Š 0; (4)ev G W G ! D.D.G//;ev BG W BG ! D.D.BG//are isomorphisms by applying (2) and (3) twice, and using the sheaf-theoretic versionof Pontrjagin duality 1.3. In other words, under the condition (4) above G and BG aredualizable Picard stacks.1 On the right-hand side D.F / is the dual sheaf as in 1.1.5, considered as a Picard stack as in 1.2.8.2 The symbol D.F / on the left-hand side of this equation denotes the dual of the Picard stack given bythe sheaf F , while D.F / on the right-hand side is the dual sheaf.

232 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1.3.5 Given the structureBH 1 .A/ ! A ! H 0 .A/of A 2 PIC S, we ask for a similar description of the dual D.A/ 2 PIC S.We will see under the crucial conditionthat D.A/ fits intoExt 1 Sh Ab S .H 0 .A/; T/ Š Ext 2 Sh Ab S .H 1 .A/; T/ Š 0;BD.H 0 .A// ! D.A/ ! D.H 1 .A//:Without the condition the description of H 1 .D.A// is more complicated, and we referto (45) for more details.1.3.6 This discussion now leads to one of the main results of the present paper.Definition 1.5. We call the sheaf F 2 Sh Ab S admissible, iffExt 1 Sh Ab S .F; T/ Š Ext2 Sh Ab S .F; T/ Š 0:Theorem 1.6 (Pontrjagin duality for Picard stacks). If A 2 PIC S is a Picard stacksuch that H i .A/ and D.H i .A// are dualizable and admissible for i D 1; 0, then Ais dualizable.1.4 Admissible groups1.4.1 Pontrjagin duality for locally compact abelian groups implies that the sheaf Gassociated to a locally compact abelian group G is dualizable. In order to apply Theorem1.6 to Picard stacks A 2 PIC S whose sheaves H i .A/, i D 1; 0 are represented bylocally compact abelian groups we must know for a locally compact group G whetherthe sheaf G is admissible.Definition 1.7. We call a locally compact group G admissible, if the sheaf G is admissible.1.4.2 Admissibility of a locally compact group is a complicated property. Not everylocally compact group is admissible, e.g. a discrete group of the form ˚n2N Z=pZ forsome integer p is not admissible (see 4.29).1.4.3 Admissibility is a vanishing conditionExt 1 Sh Ab S .G; T/ Š Ext2 Sh Ab S .G; T/ Š 0:This condition depends on the site S. In the present paper we shall also consider the subsitesS lc-acyc S lc S of locally compact locally acyclic spaces and locally compactspaces.

Duality for topological abelian group stacks and T -duality 233For these sites the Ext-functor commutes with restriction (we verify this propertyin 3.4). Admissibility thus becomes a weaker condition on a smaller site. We will refineour notion of admissibility by saying that a group G is admissible on the site S lc (orsimilarly for S lc-acyc ), if the corresponding restrictions of the extension sheaves vanish,e.g.Ext 1 Sh Ab S .G; T/ jS lcŠ Ext 2 Sh Ab S .G; T/ jS lcŠ 0in the case S lc .1.4.4 Some locally compact abelian groups are admissible on the site S. This appliese.g. to finitely generated groups like Z; Z=nZ, but also to T n and R n .In the case of profinite groups G we need the technical assumption that it does nothave too much two-torsion and three-torsion.Definition 1.8 (4.6). We say that the topological abelian group G satisfies the two-threecondition, if1. it does not admit Q n2NZ=2Z as a sub-quotient,2. the multiplication by 3 on the component G 0 of the identity has finite cokernel.We can show that a profinite abelian group which satisfies the two-three conditionis also admissible. We conjecture that it is possible to remove the condition using othertechniques.A compact connected abelian group is divisible. Hence, if p 2 N is a prime, thenthe multiplication p W G ! G is set-theoretically surjective.Definition 1.9. We say that G is locally topologically p-divisible, if the map p has acontinuous local section. The group G is locally topologically divisible, if it is locallytopologically p-divisable for all primes p.For a general connected compact group which satisfies the two-three condition wecan only show that it is admissible on S lc-acyc . If it is locally topologically divisible,then it is admissible on the larger site S lc .We think that the two-tree condition and the restriction to a locally compact site isof technical nature. The condition of local compactness enters the proof since at oneplace we want to calculate the cohomology of the sheaf Z on the space A G using aKünneth formula. For this reason we want that A is locally compact.As our counterexample above shows, a general (infinitely generated) discrete groupis not admissible unless we restrict to the site S lc-acyc . We do not know if the restrictionto the site S lc-acyc is really necessary for a general compact connected groups.Using that the class of admissible groups is closed under finite products and extensionswe get the following general theorem.Theorem 1.10 (4.8). 1. If G is a locally compact abelian group which satisfies thetwo-three condition, then it is admissible over S lc-acyc .2. Assume that

234 U. Bunke, T. Schick, M. Spitzweck, and A. Thom(a) G satisfies the two-three condition,(b) G admits an open subgroup of the form C R n with C compact such that G=C R nis finitely generated(c) the connected component of the identity of G is locally topologically divisible.Then G is admissible over S lc .The whole Section 4 is devoted to the proof of this theorem. This section is verylong and technical. A reader who is interested in the extension of Pontrjagin duality totopological abelian group stacks and applications to T -duality is advised to skip thissection in a first reading and to take Theorem 4.8 for granted.1.5 T -duality1.5.1 The aim of topological T -duality is to model the underlying topology of mirrorsymmetry in algebraic geometry and T -duality in string theory. For a detailed motivationwe refer to [BRS]. The objects of topological T -duality over a space B are pairs.E; G/ of a T n -principal bundle E ! B and a gerbe G ! E with band T. A gerbewith band T over a space E is a map of stacks G ! E which is locally isomorphicto BT jE ! E. Topological T -duality associates to .E; G/ dual pairs . yE; yG/. Forprecise definitions we refer to [BRS] and 6.1.1.5.2 The case n D 1 (n is the dimension of the fibre of E ! B) is quite easyto understand (see [BS05]). In this case every pair admits a unique T -dual (up toisomorphism, of course). The higher dimensional case is more complicated since onthe one hand not every pair admits a T -dual, and on the other hand, in general a T -dualis not unique, compare [BS05], [BRS].1.5.3 The general idea is that the construction of a T -dual pair of .E; G/ needs thechoice of an additional structure. This structure might not exist, and in this case thereis no T -dual. On the other hand, if the additional structure exists, then it might not beunique, so that the T -dual is not unique, too.1.5.4 The additional structure in [BRS] was the extension of the pair to a T -dualitytriple. We review this notion in 6.1.One can also interpret the approach to T -duality via non-commutative topology[MR05], [MR06] in this way. The gerbe G ! E determines an isomorphism class ofa bundle of compact operators (by equality of the Dixmier–Douady classes). The additionalstructure which determines a T -dual is an R n -action on this bundle of compactoperators which lifts the T n -action on E. Let us call a bundle of algebras of compactoperators on E together with such a R n -action a dynamical triple.The precise relationship between T -duality triples and dynamical triples is studiedin the thesis of Ansgar Schneider [Sch07].

Duality for topological abelian group stacks and T -duality 2351.5.5 The initial motivation of the present paper was a third choice for the additionalstructure of the pair which came to live in a discussion with T. Pantev in spring 2006.It was motivated by the analogy with some constructions in algebraic geometry, seee.g. [DP].The starting point is the observation that a T -dual exists if and only if the restrictionof the gerbe G ! E to Ej B .1/, the restriction of E to a one-skeleton of B, is trivial.In particular, the restriction of G to the fibres of E has to be trivial. Of course, theT n -bundle E ! B is locally trivial on B, too. Therefore, locally on the base B, thesequence of maps G ! E ! B is equivalent to.BT T n / jB ! T n jB ! B jB ;where we identify spaces over B with the corresponding sheaves. The stack .BT T n / jB is a Picard stack.Our proposal for the additional structure on G ! E ! B is that of a torsor overthe Picard stack .BT T n / jB .1.5.6 In order to avoid the definition of a torsor over a group object in the two-categorialworld of stacks we use the following trick (see 6.2 for details). Note that a torsor X overan abelian group G can equivalently be described as an extension of abelian groups0 ! G ! U ! p Z ! 0so that X Š p 1 .1/. We use the same trick to describe a torsor over the Picard stack.BT T n / jB as an extensionof Picard stacks..BT T n / jB ! U ! Z jB1.5.7 The sheaf of sections of E ! B is a torsor over T n . Let0 ! T n jB ! E ! Z jB ! 0be the corresponding extension of sheaves of abelian groups. The filtration (1) of thePicard stack U has the formBT jB ! U ! E;and locally on BU Š .BT T n Z/ jB : (5)1.5.8 The proposal 1.5.5 was motivated by the hope that the Pontrjagin dual D.U / ofU determines the T -dual pair. In fact, this can not be true directly since the structureof D.U / (this uses admissibility of T n and Z n ) is given locally on B byD.U / Š .BT BZ n Z/ jB :Here the factor Z in (5) gives rise to BT, the factor BT yields Z, and T n yields BZ naccording to the rules (2) and (3). The problematic factor is BZ n in a place wherewe expect a factor T n . The way out is to interpret the gerbe BZ n as the gerbe ofR n -reductions of the trivial principal bundle T n B ! B.

236 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1.5.9 We have canonical isomorphisms H 0 .D.U // Š D.H 1 .U // Š D.T jB / ŠZ jB and let D.U / 1 D.U / be the pre-image of ¹1º jB Z jB under the natural mapD.U / ! H 0 .D.U //. The quotient D.U / 1 =.BT/ jB is a Z n -gerbe over B which weinterpret as the gerbe of R n -reductions of a T n -principal bundle yE ! B which iswell-defined up to unique isomorphism. The full structure of D.U / supplies the dualgerbe yG ! yE. The details of this construction are explained in 6.4.1.5.10 In this way we use Pontrjagin duality of Picard stacks in order to construct aT -dual pair to .E; G/. Schematically the picture is.E; G/ 1 Ý U2 Ý D.U / 3 Ý . yE; yG/;where the steps are as follows:1. choice of the structure on G of a torsor over .BT T n / jB2. Pontrjagin duality D.U / WD Hom PIC S .U; BT/3. Extraction of the dual pair from D.U / as explained in 1.5.9.1.5.11 Consider a pair .E; G/. In Subsection 6.4 we provide two constructions:1. The construction ˆ starts with the choice of a T -duality triple extending .E; G/and constructs the structure of a torsor over .BT T n / jB on G.2. The construction ‰ starts with the structure on G of a torsor over .BT T n / jBand constructs an extension of .E; G/ to a T -duality triple.Our main result Theorem 6.23 asserts that the constructions ‰ and ˆ are inverses toeach other. In other words, the theories of topological T -duality via Pontrjagin dualityof Picard stacks and via T -duality triples are equivalent.2 Sheaves of Picard categories2.1 Picard categories2.1.1 In a cartesian closed category one can define the notion of a group object in thestandard way. It is given by an object, a multiplication, an identity, and an inversionmorphism. The group axioms can be written as a collection of commutative diagramsinvolving these morphisms.Stacks on a site S form a two-cartesian closed two-category. A group object in atwo-cartesian closed two-category is again given by an object and the multiplication,identity and inversion morphisms. In addition each of the commutative diagrams fromthe category case is now filled by a two-morphisms. These two-morphisms must satisfyhigher associativity relations.Instead of writing out all these relations we will follow the exposition of DeligneSGA4, exposé XVIII [Del], which gives a rather effective way of working with group

Duality for topological abelian group stacks and T -duality 237objects in two-categories. We will not be interested in the most general case. For ourpurpose it suffices to consider a notion which includes sheaves of abelian groups andgerbes with band given by a sheaf of abelian groups. We choose to work with sheavesof strictly commutative Picard categories.2.1.2 Let C be a category,F W C C ! Cbe a bi-functor, andbe a natural isomorphism of tri-functors. W F.F.X;Y/;Z/ ! F.X;F.Y;Z//Definition 2.1. The pair .F; / is called an associative functor if the following holds.For every family .X i / i2I of objects of C let e W I ! M.I/ denote the canonicalmap into the free monoid (without identity) on I . We require the existence of amap F W M.I/ ! Ob.C / and isomorphisms a i W F .e.i// ! X i , and isomorphismsa g;h W F .gh/ ! F.F.g/; F .h// such that the following diagram commutes:F .f .gh// a f;ghF.F.f /; F .gh//a g;hF.F.f /; F .F .g/; F .h//F ..fg/h/ afg;hF.F.fg/; F .h// F.F.F.f /; F .g//; F .h//.a f;g(6)2.1.3 Let .F; / be as above. Let in addition be given a natural transformation ofbi-functors W F.X;Y/ ! F.Y;X/:Definition 2.2. .F;;/is called a commutative and associative functor if the followingholds. For every family .X i / i2I of objects in C let e W I ! N.I/ denote the canonicalmap into the free abelian monoid (without identity) on I . We require that there existF W N.I/ ! C , isomorphisms a i W F .e.i// ! X i and isomorphisms a g;h W F .gh/ !F.F.g/; F .h// such that (6) andF .gh/ a g;hF .hg/ ah;gF.F.g/; F .h//F.F.h/; F .g//(7)commute.

238 U. Bunke, T. Schick, M. Spitzweck, and A. Thom2.1.4 We can now define the notion of a strict Picard category. Instead of F will usethe symbol “C”.Definition 2.3. A Picard category is a groupoid P together with a bi-functor CW P P ! P and natural isomorphisms and as above such that .C;;/is a commutativeand associative functor, and such that for all X 2 P the functor Y 7! X C Y is anequivalence of categories.2.2 Examples of Picard categories2.2.1 Let G be an abelian group. We consider G as a category with only identitymorphisms. The addition CW G G ! G is a bi-functor. For and we choose theidentity transformations. Then .G; C;;/ is a strict Picard category which we willdenote again by G.2.2.2 Let us now consider the abelian group G as a category BG with one object and Mor BG .; / WD G. We let CW BG BG ! BG be the bi-functor which acts asaddition Mor BG ./Mor BG ./ ! Mor BG ./. For and we again choose the identitytransformations. We will denote this strict Picard category .BG; C;;/shortly by BG.2.2.3 Let G; H be abelian groups. We define the category EXT.G; H / as the categoryof short exact sequencesE W 0 ! H ! E ! G ! 0:Morphisms in EXT.G; H / are isomorphisms of complexes which reduce to the identityon G and H . We define the bi-functor CW EXT.G; H / EXT.G; H / ! EXT.G; H /as the Baer additionE 1 C E 2 W 0 ! H ! F ! G ! 0;where F WD zF=H, zF E 1 E 2 is the pre-image of the diagonal in G G, andthe action of H on zF is induced by the anti-diagonal action on E 1 E 2 . The associativitymorphism W .E 1 C E 2 / C E 3 ! E 1 C .E 2 C E 3 / is induced by the canonicalisomorphism .E 1 E 2 / E 3 ! E 1 .E 2 E 3 /. Finally, the transformation W E 1 CE 2 ! E 2 CE 1 is induced by the flip E 1 E 2 ! E 2 E 1 . The triple .C;;/defines on EXT.G; H / the structure of a Picard category.2.2.4 Let G be a topological abelian group, e.g. G WD T. On a space B we considerthe category of G-principal bundles BG.B/. Given two G-principal bundles Q ! B,P ! B we can define a new G-principal bundle Q ˝G P WD Q B P=G. Thequotient is taken by the anti-diagonal action, i.e. we identify .q; p/ .qg; pg 1 /. TheG-principal structure on Q ˝G P is induced by the action Œq; pg WD Œq; pg.We define the associativity morphism W .Q ˝G P/˝G R ! Q ˝G .P ˝G R/

Duality for topological abelian group stacks and T -duality 239as the map ŒŒq; p; r ! Œq; Œp; r. Finally we let W Q ˝G P ! P ˝G Q be givenby Œq; p ! Œp; q.We claim that .˝G;;/is a strict Picard category. Let I be a collection of objectsof BG.B/. We choose an ordering of I . Then we can write each element of j 2 N.I/in a unique way as ordered product f D P 1 P 2 :::P r with P 1 P 2 P r . Wedefine F .f / WD P 1˝G .P 2˝G .˝G P r / /. We let a P W F .P / ! P , P 2 I ,bethe identity. Furthermore, for f; g 2 N.I/ we define a f;g W F .fg/ ! F .f / ˝B F .g/as the map induced the permutation which puts the factors of f and g in order (and sothat the order to repeated factors is not changed). Commutativity of (6) is now clear.In order to check the commutativity of (7) observe that W Q ˝G Q ! Q ˝G Q is theidentity, as Œqg; q 7! Œq; qg D Œqg; q.2.3 Picard stacks and examples2.3.1 We consider a site S. A prestack on S is a lax associative functor P W S op !groupoids, where from now on by a groupoid we understand an essentially small category3 in which all morphisms are isomorphisms. Lax associative means that as a part ofthe data for each composable pair of maps f; g in S there is an isomorphism of functorsl f;g W P.f/ı P.g/ ! P.g ı f/, and these isomorphisms satisfy higher associativityrelation. A prestack is a stack if it satisfies the usual descent conditions one the levelof objects and morphisms. For a reference of the language of stacks see e.g. [Vis05].Definition 2.4. A Picard stack P on S is a stack P together with an operationCW P P ! P and transformations , which induce for each U 2 S the structureof a Picard category on P.U/.2.3.2 Let Sh Ab S denote the category of sheaves of abelian groups on S. Extending theexample 2.2.1 to sheaves we can view each object F 2 Sh Ab S as a Picard stack, whichwe will again denote by F .2.3.3 We can also extend Example 2.2.2 to sheaves. In this way every sheaf F 2 Sh Ab Sgives rise to a Picard stack BF .2.3.4 We now extend the example 2.2.3 to sheaves. First of all note that one can definea Picard category EXT.G ; H/ of extensions of sheaves 0 ! H ! E ! G ! 0 as adirect generalization of 2.2.3. Then we define a prestack EXT.G ; H/ which associatesto U 2 S the Picard category EXT.G ; H/.U / WD EXT.G jU ; H jU /. One checks thatEXT.G ; H/ is a stack.2.3.5 Let G 2 Sh Ab S be a sheaf of abelian groups. We consider G as a group object inthe category ShS of sheaves of sets on S. AG -torsor is an object T 2 ShS with an actionof G such that T is locally isomorphic to G . We consider the category of G -torsorsTors.G / and their isomorphisms. We define the functor CW Tors.G / Tors.G / !3 Later, e.g. in 2.5.3, we need that the isomorphism classes in a groupoid form a set.

240 U. Bunke, T. Schick, M. Spitzweck, and A. ThomTors.G / by T 1 C T 2 WD T 1 T 2 =G , where we take the quotient by the anti-diagonalaction. The structure of a G -torsor is induced by the action of G on the second factor T 2 .We let and be induced by the associativity transformation of the cartesian productand the flip, respectively. With these structures Tors.G / becomes a Picard category.We define a Picard stack T ors.G / by localization, i.e. we set T ors.G /.U / WDTors.G jU /.We have a canonical map of Picard stacksU W EXT.Z; G / ! T ors.G /which maps the extension E W 0 ! G ! E ! Z ! 0 to the G-torsor U.E/ D 1 .1/.Here Z denotes the constant sheaf on S with value Z. One can check that U is anequivalence of Picard stacks (see 2.4.3 for precise definitions).2.3.6 The example 2.2.4 has a sheaf theoretic interpretation. We assume that S is somesmall subcategory of the category of topological spaces which is closed under takingopen subspaces, and such that the topology is induced by the coverings of spaces byfamilies of open subsets.Let G be a topological abelian group. We have a Picard prestack p BG on Swhich associates to each space B 2 S the Picard category p BG.B/ of G-principalbundles on B. As in 2.2.4 the monoidal structure on p BG.B/ is given by the tensorproduct of G-principal bundles (this uses that G is abelian). We now define BG as thestackification of p BG.The topological group G gives rise to a sheaf G 2 Sh Ab S which associates to U 2 Sthe abelian group G.U / WD C.U;G/. We have a canonical transformation W BG ! T ors.G/:It is induced by a transformation p W p BG ! T ors.G/. We describe the functorp B W p BG.B/ ! T ors.G/.B/ for all B 2 S. Let E ! B be a G-principal bundle.Then we define p B .E/ 2 Tors.G jB / to be the sheaf which associates to each. W U ! B/ 2 S=B the set of sections of E ! U .One can check that is an equivalence of Picard stacks (see 2.4.3 for precisedefinitions).2.4 Additive functors2.4.1 We shall now discuss the notion of an additive functor between Picard stacksF W P 1 ! P 2 .Definition 2.5. An additive functor between Picard categories is a functor F W P 1 ! P 2and a natural transformation F.xCy/ ! F.x/CF.y/such that the following diagramscommute:

Duality for topological abelian group stacks and T -duality 2411.F.x C y/F./F.y C x/F.x/C F.y/ F.y/C F.x/;2.F ..x C y/ C z/F.x C y/ C F.z/ .F .x/ C F.y//C F.z/F./F.x C .y C z// F.x/C F.y C z// F.x/C .F .y/ C F.z//.Definition 2.6. An isomorphism between additive functors uW F ! G is an isomorphismof functors such thatF.x C y/u xCy G.x C y/F.x/C F.y/ u xCu y G.x/ C G.y/commutes.Definition 2.7. We let Hom.P 1 ;P 2 / denote the groupoid of additive functors from P 1to P 2 .ByPIC we denote the two-category of Picard categories.2.4.2 The groupoid Hom.P 1 ;P 2 / has a natural structure of a Picard category. We setand define the transformationsuch that.F 1 C F 2 /.x/ WD F 1 .x/ C F 2 .x/.F 1 C F 2 /.x C y/ ! .F 1 C F 2 /.x/ C .F 1 C F 2 /.y/.F 1 C F 2 /.x C y/ .F1 C F 2 /.x/ C .F 1 C F 2 /.y/F 1 .x C y/ C F 2 .x C y/ .F1 .x/ C F 1 .y// C .F 2 .x/ C F 2 .y// ˛ıı˛ F1 .x/ C F 2 .x/ C F 1 .y/ C F 2 .y/commutes. The associativity and commutativity constraints are induced by those of P 2 .

242 U. Bunke, T. Schick, M. Spitzweck, and A. Thom2.4.3 Let now P 1 ;P 2 be two Picard stacks on S.Definition 2.8. An additive functor F W P 1 ! P 2 is a morphism of stacks togetherwith a two-morphismP 1 P C 1 P 1FF F P 2 P C 2 P 2which induces for each U 2 S an additive functor F U W P 1 .U / ! P 2 .U / of Picard categories.An isomorphism between additive functors uW F 1 ! F 2 is a two-isomorphismof morphisms of stacks which induces for each U 2 S an isomorphism of additivefunctors u U W F 1;U ! F 2;U .As in the case of Picard categories, the additive functors between Picard stacks formagain a Picard category.Definition 2.9. We let Hom.P 1 ;P 2 / denote the groupoid of additive functors from P 1 ,P 2 .We get a two-category PIC.S/ of Picard stacks on S.Definition 2.10. We let HOM.P 1 ;P 2 / denote the Picard category of additive functorsbetween Picard stacks P 1 and P 2 .2.4.4 Let P;Q be Picard stacks on the site S. By localization we define a Picard stackHOM.P; Q/.Definition 2.11. The Picard stack HOM.P; Q/ is the sheaf of Picard categories givenbyHOM PIC.S/ .P; Q/.U / WD HOM.P jU ;Q jU /:A priori this describes a prestack, but one easily checks the stack conditions.2.5 Representation of Picard stacks by complexes of sheaves of groups2.5.1 There is an obvious notion of a Picard prestack. Furthermore, there is an associatedPicard stack construction a such that for a Picard prestack P and a Picard stackQ we have a natural equivalenceHom.aP; Q/ Š Hom.P; Q/ (8)2.5.2 Let S be a site. We follow Deligne, SGA 4.3, Expose XVIII, [Del]. Let C.S/ bethe two-category of complexes of sheaves of abelian groupsK W 0 ! K 1 d ! K 0 ! 0which live in degrees 1; 0. Morphisms in C.S/ are morphisms of complexes, andtwo-isomorphisms are homotopies between morphisms.To such a complex we associate a Picard stack ch.K/ on S as follows. We firstdefine a Picard prestack p ch.K/.

Duality for topological abelian group stacks and T -duality 2431. For U 2 S we define the set of objects of p ch.K/.U / as K 0 .U /.2. For x;y 2 K 0 .U / we define the set of morphisms byHomp ch.K/.U /.x; y/ WD ¹f 2 K 1 .U / j df D x3. The composition of morphisms is addition in K 1 .U /.4. The functor CW p ch.K/.U / p ch.K/.U / ! p ch.K/.U / is given by addition ofobjects and morphisms. The associativity and the commutativity constraints arethe identities.Definition 2.12. We define ch.K/ as the Picard stack associated to the prestack p ch.K/,i.e. ch.K/ WD a p ch.K/.2.5.3 If P is a Picard stack, then we can define a presheaf p H 0 .P / which associatesto U 2 S the group of isomorphism classes of P.U/. We let H 0 .P / be the associatedsheaf. Furthermore we have a sheaf H 1 .P / which associates to U 2 S the group ofautomorphisms Aut.e U /, where e U 2 P.U/is a choice of a neutral object with respectto C. Since the neutral object e U is determined up to unique isomorphism, the groupAut.e U / is also determined up to unique isomorphism. Note that Aut.e U / is abelian, asshows the following diagramyº:e C eidCf e C egCid e C eŠ f g e e eand the fact that .id C f/ı .g C id/ D g C f D .g C id/ ı .id C f/.If V ! U is a morphism in S, then we have a unique isomorphism f W .e U / jV ! eVwhich induces a group homomorphism f ııf 1 W Aut.e U / jV ! Aut.e V /. Thishomomorphism is the structure map of the sheaf H 1 .P / which is determined up tounique isomorphism in this way.We now observe thatH 1 .ch.K// Š H 1 .K/; H 0 .ch.K// Š H 0 .K/: (9)2.5.4 A morphism of complexes f W K ! L in C.S/, i.e. a diagramŠŠ0 K1d K K0 0f 1d0 L1 L L0 0,induces an additive functor of Picard prestacks p ch.f /W p ch.K/ ! p ch.L/ and, by thefunctoriality of the associated stack construction, an additive functorch.f /W ch.K/ ! ch.L/:In view of (9), it is an equivalence if and only if f is a quasi isomorphism.f 0

244 U. Bunke, T. Schick, M. Spitzweck, and A. Thom2.5.5 We consider two morphisms f; g W K ! L of complexes in C.S/. A homotopyH W f ! g is a morphism of sheaves H W K 0 ! L 1 such that f 0 g 0 D d L ı Hand f 1 g 1 D H ı d K . It is easy to see that H induces an isomorphism of additivefunctors p ch.H /W p ch.f / ! p ch.g/ and thereforech.H /W ch.f / ! ch.g/by the associated stack construction and (8).One can show that the isomorphisms p ch.f / ! p ch.g/ correspond precisely tohomotopies H W f ! g. This implies that the morphisms ch.f / ! ch.g/ alsocorrespond bijectively to these homotopies.2.5.6 We have the following result.Lemma 2.13 ([Del, 1.4.13]). 1. For every Picard stack P there exists a complexK 2 C.S/ such that P Š ch.K/.2. For every additive functor F W ch.K/ ! ch.L/ there exists a quasi isomorphismk W K 0 ! K and a morphism l W K 0 ! L in C.S/ such that F Š ch.l/ ı ch.k/ 1 .2.5.7 Let PIC [ .S/ be the category of Picard stacks obtained from the two-categoryPIC S by identifying isomorphic additive functors.Proposition 2.14 ([Del, 1.4.15]). The construction ch gives an equivalence of categories1;0 .Sh Ab S/ ! PIC [ .S/;D Œwhere D Œ 1;0 .Sh Ab S/ is the full subcategory of D C .Sh Ab S/ of objects whose cohomologyis trivial in degrees 62 ¹ 1; 0º.Lemma 2.15 ([Del, 1.4.16]). Let K; L 2 C.S/ and assume that L 1 is injective.1. p ch.L/ is already a stack.2. For every morphism F 2 Hom PIC.S/ .ch.K/; ch.L// there exists a morphismf 2 Hom C.ShAb S/.K; L/ such that ch.f / Š F .Let C 0 .S/ C.S/ denote the full sub-two-category of complexes 0 ! K 1 !K 0 ! 0 with K 1 injective.Lemma 2.16 ([Del, 1.4.17]). The construction ch induces an equivalence of the twocategoriesPIC.S/ and C 0 .S/.2.5.8 Finally we give a characterization of the Picard stack Hom.ch.K/; ch.L//.Lemma 2.17 ([Del, 1.4.18.1]). Assume that L 1 is injective. Then we have an equivalencech. 0 Hom ShAb S .K; L// ! HOM PIC.S/ .ch.K/; ch.L//:

Duality for topological abelian group stacks and T -duality 245Lemma 2.18. Let K; L 2 C.S/. Then we have an isomorphismfor i D 1; 0.H i .HOM PIC.S/ .ch.K/; ch.L/// Š R i Hom ShAb S .K; L/Proof. First observe that by the discussion above the left hand side, and by the definitionof R Hom the right side both only depend on the quasi-isomorphism type of the complexL of length 2. Without loss of generality we can therefore assume that L 1 is injective.We now choose an injective resolution I W 0 ! L 1 ! I 0 ! I 1 ::: of L startingwith the choice of an embedding L 0 ! I 0 . Then we have Hom.K; I / Š RHom.K; L/.We now observe that H i Hom.K; I / Š H i Hom.K; L/ for i D 0; 1. While the casei D 1 is obvious, for i D 0 observe that a 0-cycle in Hom.K; I / is a morphism ofcomplexes and necessarily factors over L ! I . We thus have for i 2¹ 1; 0ºR i Hom.K; L/ Š H i Hom.K; L/ Š H i .Hom.ch.K/; ch.L///:2.5.9 For A; B 2 Sh Ab S we have canonical isomorphismsExt 2 Sh Ab S .B; A/ Š R0 Hom ShAb S.B; AŒ2/ Š Hom D.ShAb S/.B; AŒ2/:In the following we recall two eventually equivalent ways how an exact complexrepresents an elementK W 0 ! A ! X ! Y ! B ! 0Y.K/ 2 Hom D.ShAb S/.B; AŒ2/ Š Ext 2 Sh Ab S .B; A/(the letter Y stands for Yoneda who investigated this construction first). Let K A be thecomplexK A W 0 ! X ! Y ! B ! 0;where B sits in degree 0. The obvious inclusion ˛ W AŒ2 ! K A induced by A ! Xis a quasi-isomorphism. Furthermore, we have a canonical map ˇ W B ! K A . Theelement Y.K/ 2 Hom D.ShAb S/.B; AŒ2/ is by definition the compositionWe can also consider the complex K B given byY.K/W B ˇ! K A˛ 1! AŒ2: (10)0 ! A ! X ! Y ! 0where A is in degree 2. The projection Y ! B induces a quasi-isomorphism W K B ! B. We furthermore have a canonical map ı W K B ! AŒ2. We consider thecomposition Y 0 .K/ 2 Hom D.ShAb S/.B; AŒ2/Y 0 .K/W B 1 ! K Bı! AŒ2: (11)

246 U. Bunke, T. Schick, M. Spitzweck, and A. ThomLemma 2.19. In Hom D.ShAb S/.B; AŒ2/ we have the equalityY.K/ D Y 0 .K/:Proof. We consider the morphism of complexes W K B ! K A given by obvious mapsin the diagramX Y B It fits intoBA X Y .ˇ K A ˛AŒ2B ı AŒ2.K BIt suffices to show that the two squares commute. In fact we will show that ishomotopic to ˇ ı and ˛ ı ı. Note that ˇ ı is given byX Y B Aid X00Y .The dotted arrows in this diagram gives the zero homotopy of this difference.Similarly ˛ ı ı is given byX Y B 0A0 XidY ,and we have again indicated the required zero homotopy.2.5.10 The equivalence between the two-categories PIC.S/ and C 0 .S/ (see 2.16) allowsus to classify equivalence classes of Picard stacks with fixed H i .P / Š A i , i D 1; 0,A i 2 Sh Ab S. An equivalence of such Picard stacks is an equivalence which inducesthe identity on the cohomology.Lemma 2.20. The set Ext PIC.S/ .A 0 ;A 1 / of equivalence classes of Picard stacks Pwith given isomorphisms H i .P / Š A i , i D 0; 1, A i 2 Sh Ab S is in bijection withExt 2 Sh Ab S .A 0;A 1 /.Proof. We define a map W Ext PIC.S/ .A 0 ;A 1 / ! Ext 2 Sh Ab S .A 0;A 1 / (12)

Duality for topological abelian group stacks and T -duality 247as follows.Consider an exact complexK W 0 ! A 1 ! K 1 ! K 0 ! A 0 ! 0such that K 1 is injective, and let K 2 C.S/ be the complex0 ! K 1 ! K 0 ! 0Then we define .ch.K// WD Y.K/.We must show that is well-defined. Indeed, if ch.K/ Š ch.L/, then (see 2.16)K Š L by an equivalence which induces the identity on the level of cohomology. Itfollows that Y.K/ D Y.L/.Since chW C 0 .S/ ! PIC [ .S/ is an equivalence, hence in particular surjectiveon the level of equivalence classes, the map is well-defined. Since every elementof Ext 2 Sh Ab S .A 1;A 0 / can be written as Y.K/ for a suitable complex K asabove we conclude that is surjective. If .ch.K// Š .ch.L//, then there existsM 2 C.S/ with given isomorphisms H i .M / Š A i together with quasi-isomorphismsK M ! L inducing the identity on cohomology. But this diagram induces anequivalence ch.K/ Š ch.M / Š ch.L/.2.5.11 We continue with a further description of BF for F 2 Sh Ab S, compare 1.2.8.Note that the sheaf of groups of automorphisms of BF is F , whereas the group ofobjects is trivial. Therefore we can represent BF as ch.F Œ1/, where F Œ1 is thecomplex 0 ! F ! 0 ! 0 concentrated in degree 1.2.5.12 Let P 2 PIC.S/ be a Picard stack and G H 1 .P /. Let us assume thatP D ch.K/ for a suitable complex K W 0 ! K 1 ! K 0 ! 0. We have a naturalinjection G,! ker.K 1 ! K 0 /. We consider the quotient xK 1 defined by the exactsequence 0 ! G ! K 1 ! xK 1 ! 0 and obtain a new Picard stack xP Š ch. xK/,where xK W 0 ! xK 1 ! K 0 ! 0. The following diagram represents a sequence ofmorphisms of Picard stacksBG ! P ! xP;0 G 00 K1 K 0 0 0 0 xK 1 K0 0.The Picard stack xP can be considered as a quotient of P by BG. We will employ thisconstruction in 6.4.5.

248 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3 Sheaf theory on big sites of topological spaces3.1 Topological spaces and sites3.1.1 In this paper a topological space will always be compactly generated and Hausdorff.We will define categorical limits and colimits in the category of compactlygenerated Hausdorff spaces. Furthermore, we will equip mapping spaces Map.X; Y /with the compactly generated topology obtained from the compact-open topology. Inthis category we have the exponential lawMap.X Y; Z/ Š Map.X; Map.Y; Z//:By Map.X; Y / ı we will denote the underlying set.category of topological spaces we refer to [Ste67].For details on this convenient3.1.2 The sheaf theory of the present paper refers to the Grothendieck site S. Theunderlying category of S is the category of compactly generated topological Hausdorffspaces. The covering families of a space X 2 S are coverings by families of opensubsets.We will also need the sites S lc and S lc-acyc given by the full subcategories of locallycompact and locally compact locally acyclic spaces (see 3.4.6).3.1.3 We let Pr S and ShS denote the category of presheaves and sheaves of sets on S.Then we have an adjoint pair of functorsi ] W Pr S , ShS W iwhere i is the inclusion of sheaves into presheaves, and i ] is the sheafification functor.If F 2 Pr S, then sometimes we will write F ] WD i ] F .As before, by Pr Ab S and Sh Ab S we denote the categories of presheaves and sheavesof abelian groups.3.2 Sheaves of topological groups3.2.1 In this subsection we collect some general facts about sheaves generated byspaces and topological groups. We formulate the results for the site S. But they remaintrue if one replaces S by S lc or S lc-acyc .3.2.2 Every object X 2 S represents a presheaf X 2 Pr S of sets defined byon objects, and byon morphisms.S 3 U 7! X.U / WD Hom S .U; X/ 2 SetsHom S .U; V / 3 f 7! f W X.V / ! X.U /

Duality for topological abelian group stacks and T -duality 249Lemma 3.1. X is a sheaf.Proof. Straightforward.Note that a topology is called sub-canonical if all representable presheaves aresheaves. Hence S carries a sub-canonical topology.3.2.3 We will also need the relative version. For Y 2 S we consider the relative siteS=Y of spaces over Y . Its objects are morphisms A ! Y , and its morphisms arecommutative diagramsA B Y .The covering families of A ! Y are induced from the coverings of A by open subsets.An object X ! Y 2 S=Y represents the presheaf X ! Y 2 Pr S=Y (by a similardefinition as in the absolute case 3.2.2). The induced topology on S=Y is again subcanonical.In fact, we have the following lemma.Lemma 3.2. For all .X ! Y/2 S the presheaf X ! Y is a sheaf.Proof. Straightforward.3.2.4 Let I be a small category and X 2 S I . Then we havelim i2I X.i/ Š lim i2I X.i/:One can not expect a similar property for arbitrary colimits. But we have the followingresult.Lemma 3.3. Let I be a directed partially ordered set and X 2 S I be a direct systemof discrete sets such that X.i/ ! X.j / is injective for all i j . Then the canonicalmapcolim i2I X.i/ ! colim i2I X.i/is an isomorphism.Proof. Let p colim i2I X.i/ denote the colimit in the sense of presheaves. Then we havean inclusionp colim i2I X.i/ ,! colim i2I X.i/(this uses injectivity of the structure maps of the system X). Since the target is a sheafand sheafification preserves injections it induces an inclusioncolim i2I X.i/ ,! colim i2I X.i/: (13)

250 U. Bunke, T. Schick, M. Spitzweck, and A. ThomLet now A 2 S and f 2 colim i2I X.i/.A/DHom S .A; colim i2I X.i//. Ascolim i2I X.i/is discrete and f is continuous the family of subsets ¹f1 .x/ A j x 2 colim i2I X.i/ºis an open covering of A by disjoint open subsets. The familyYf jf 1 .x/ 2 Y p colim i2I X.i/.f1 .x//xxrepresents a section over A of the sheafification colim i2I X.i/ of p colim i2I X.i/ whichof course maps to f under the inclusion (13). Therefore (13) is also surjective.3.2.5 If G 2 S is a topological abelian group, then Hom S .U; G/ WD G.U / has thestructure of a group by point-wise multiplications. In this case G is a sheaf of abeliangroups G 2 Sh Ab S.3.2.6 We can pass back and forth between (pre)sheaves of abelian groups and(pre)sheaves of sets using the following adjoint pairs of functorsp Z.:::/W Pr S , Pr Ab S W F ; Z.:::/W ShS , Sh Ab S W F :The functor F forgets the abelian group structure. The functors p Z.:::/and Z.:::/are called the linearization functors. The presheaf linearization associates to a presheafof sets H 2 Pr S the presheaf p Z.H / 2 Pr Ab S which sends U 2 S to the free abeliangroup Z.H /.U / generated by the set H.U/. The linearization functor for sheaves is thecomposition Z.:::/WD i ] ı p Z.:::/of the presheaf linearization and the sheafification.Lemma 3.4. Consider an exact sequence of topological groups in S1 ! G ! H ! L ! 1:Then 1 ! G ! H ! L is an exact sequence of sheaves of groups.The map of topological spaces H ! L has local sections if and only ifis an exact sequence of sheaves of groups.1 ! G ! H ! L ! 1Proof. Exactness at G and H is clear.Assume the existence of local sections of H ! L. This implies exactness at L.On the other hand, evaluating the sequence at the object L 2 S we see that theexactness of the sequence of sheaves implies the existence of local sections to H ! L.3.2.7 For sheaves G; H 2 ShS we define the sheafHom Sh S .G; H / 2 ShS; Hom Sh S .G; H /.U / WD Hom Sh S .G jU ;H jU /:Again, this a priori defines a presheaf, but one checks the sheaf conditions in a straightforwardmanner.

Duality for topological abelian group stacks and T -duality 2513.2.8 If X; H 2 S and H is in addition a topological abelian group, then Map.X; H /is again a topological abelian group. For a topological abelian group G 2 S we letHom top-Ab .G; H / Map.G; H / be the closed subgroup of homomorphisms. Recallthe construction of the internal Hom ShAb S .:::;:::/, compare 3.2.7.Lemma 3.5. Assume that G; H 2 S are topological abelian groups. Then we have acanonical isomorphism of sheaves of abelian groupse W Hom top-Ab .G; H / ! Hom ShAb S .G;H/:Proof. We first describe the morphism e. Let 2 Hom.G; H /.U /. We define theelement e./ 2 Hom ShAb S .G;H/.U /, i.e. a morphism of sheaves e./W G jU ! H jU ,such that it sends f 2 G. W V ! U/to ¹V 3 v 7! ..v//f .v/º2 H. W V ! U/.Let us now describe the inverse e 1 . Let 2 Hom ShAb S .G;H/.U / be given. Since.pr U W U G ! U/2 S=U it gives rise to a mapG W G.pr U W U G ! U/! H .pr U W U G ! U/:We now consider .pr G W U G ! G/ 2 G.pr U W U G ! U/ and set G WDG.pr G / 2 H .pr U W U G ! U/ D Hom S .U G; H/. We now invoke the exponentiallaw isomorphism expW Hom S .U G; H/ ! Hom S .U; Map.G; H //. Sincewas a homomorphism and using the sheaf property, we actually get an elemente 1 . / WD exp. G / 2 Hom S .U; Hom top-Ab .G; H // D Hom top-Ab .G; H /.U /. (Detailsof the argument for this fact are left to the reader.)If S is replaced by one of the sub-sites S lc or S lc-acyc , then it may happen thatHom top-Ab .G; H / does not belong to the sub-site. In this case Lemma 3.5 remains trueif one interprets Hom top-Ab .G; H / as the restriction of the sheaf on S represented byHom top-Ab .G; H / to the corresponding sub-site.3.3 Restriction3.3.1 In this subsection we prove a general result in sheaf theory (Proposition 3.12)which is probably well-known, but which we could not locate in the literature. Let S bea site. For Y 2 S we can consider the relative site S=Y . Its objects are maps .U ! Y/in S, and its morphisms are diagramsU V Y .The covering families for S=Y are induced by the covering families of S, i.e. for.U ! Y/ 2 S=Y a covering family WD .U i ! U/ i2I induces in S=Y a covering

252 U. Bunke, T. Schick, M. Spitzweck, and A. Thomfamily0 jY WD B@U i Y1 UC Ai2I:3.3.2 There is a canonical functor f W S=Y ! S given on objects by f.U ! Y/WD U .It induces adjoint pairs of functorsf W ShS=Y , ShS W f ;p f W Pr S=Y , Pr S W p f :We will often write f .F / DW F jY for the restriction functor. For F 2 Pr S it is givenin explicit terms byF jY .U ! Y/WD F.U/:The functor f is in fact the restriction of p f to the subcategory of sheaves ShS Pr S. IfF 2 ShS, then one must check that p f F is a sheaf. To this end note that thedescent conditions for p f F with respect to the induced coverings described in 3.3.1immediately follow from the descent conditions for F .3.3.3 Let us now study how the restriction acts on representable sheaves. We assumethat S has finite products. Let X; Y 2 S.Lemma 3.6. If S has finite products, then we have a natural isomorphismX jY Š X Y ! Y :Proof. By the universal property of the product for W U ! Y we haveX Y ! Y .U ! Y/D¹h 2 Hom S .U; X Y/j pr Y ı h D ºŠ Hom S .U; X/D X.U /D X jY .U ! Y/:3.3.4 Let i ] W Pr S ! ShS and i ] YW Pr S=Y ! ShS=Y be the sheafification functors.Lemma 3.7. We have a canonical isomorphism f ı i ] Š i ] Y ı p f .Proof. The argument is similar to that in the proof of [BSS, Lemma 2.31]. The mainpoint is that the sheafifications i ] F.U/ and i ] Y F jY .U ! Y/ can be expressed interms of the category of covering families cov S .U / and cov S=Y .U ! Y/. Comparedwith [BSS, Lemma 2.31] the argument is simplified by the fact that the induction ofcovering families described in 3.3.1 induces an isomorphism of categories cov S .U / !cov S=Y .U ! Y/.

Duality for topological abelian group stacks and T -duality 2533.3.5 Recall that a functor is called exact if it commutes with limits and colimits.Lemma 3.8. The restriction functor f W ShS ! ShS=Y is exact.Proof. The functor f is a right-adjoint and therefore commutes with limits. Sincecolimits of presheaves are defined object-wise, i.e. for a diagramC ! Pr S;c 7! F cof presheaves we have. p colim c2C F c /.U / D colim c2C F c .U /;it follows from the explicit description of p f , that it commutes with colimits ofpresheaves. Indeed, for a diagram of sheaves F W C ! ShS we havecolim c2C F c D i ]p colim c2C F c :By Lemma 3.7 we get f colim c2C F c Š f i ]p colim c2C F c Š i ] Y p f p colim c2C F c Ši ]p colim c2C p f F c Š colim c2C f F c :3.3.6 Let H 2 ShS and X 2 S.Lemma 3.9. If S has finite products, then for U 2 S we have a natural bijectionHom Sh S .X; H /.U / Š H.U X/:Proof. We have the following chain of isomorphismsHom Sh S .X; H /.U / Š Hom Sh S=U .X jU ;H jU /Š Hom Sh S=U .U X ! U ;H jU / (by Lemma 3.6)Š H.U X/:We will need the explicit description of this bijection. We define a map‰ W Hom Sh S .X; H /.U / ! H.U X/as follows. An element h 2 Hom Sh S=U .X; H /.U / D Hom Sh S .X jU ;H jU / induces amap Q hW X jU .U X ! U/ ! H jU .U X ! U/. Now we have X jU .U X !U/ D X.U X/ D Map.U X; X/ and H jU .U X ! U/ D H.U X/. Letpr X W U X ! X be the projection. We define ‰.h/ WD Q h.pr X /.We now define ˆW H.UX/ ! Hom Sh S .X; H /.U /. Let g 2 H.UX/. For each.e W V ! U/2 S=U we must define a map Og .V !U/ W X jU .V ! U/! H jU .V ! U/.Note that X jU .V ! U/D X.V / D Map.V; X/ ı . Let x 2 Map.V; X/. Then we havean induced map .e; x/W V ! U X. We define ‰.g/ WD H.e; x/.g/. One checksthat this construction is functorial in e and therefore defines a morphism of sheaves.A straight forward calculation shows that ‰ and ˆ are inverses to each other andinduce the bijection above.

254 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.3.7 For a map U ! V we have an isomorphism of sites S=U Š .S=V /=.U ! V/.Several formulas below implicitly contain this identification. For example, we have acanonical isomorphismF jU Š .F jV / jU !Vfor F 2 ShS. ForG; H 2 ShS this induces isomorphismsHom Sh S .G; H / jV Š Hom Sh S=V .G jV ;H jV /: (14)3.3.8 We now consider a sheaf of abelian groups H 2 Sh Ab S. If 2 cov S .U / is acovering family of U and H 2 Sh Ab S, then we can define the Čech complex LC .I H/(see [BSS, 2.3.5]).Definition 3.10. The sheaf H is called flabby if H i . LC .I H// Š 0 for all i 1,U 2 S and 2 cov S .U /.For A 2 S we have the section functor.AI :::/W Sh Ab S ! Ab;.AI H/ WD H.A/:This functor is left exact and admits a right-derived functor R.AI :::/which can becalculated using injective resolutions. Note that .AI :::/is acyclic on flabby sheaves,i.e. R i .AI H/ Š 0 for i 1 if H is flabby. Hence, the derived functor R.AI :::/can also be calculated using flabby resolutions. Note that an injective sheaf is flabby.Lemma 3.11. The restriction functor Sh Ab S ! Sh Ab S=Y preserves flabby sheaves.Proof. Let H 2 Sh Ab S be flabby. For .U ! Y/ 2 S=Y and 2 cov S .U / welet jY 2 cov S=Y .U ! Y/ be the induced covering family as in 3.3.1. Note thatLC . jY I H jY / Š LC .I H/is acyclic. Since cov S=Y .U ! Y/is exhausted by familiesof the form jY , 2 cov S .U / it follows that H jY is flabby.3.3.9 We now consider sheaves of abelian groups F;G 2 Sh Ab S.Proposition 3.12. Assume that the site S has the property that for all U 2 S therestriction ShS ! ShS=U preserves representable sheaves. If Y 2 S, then inD C .Sh Ab S=Y / there is a natural isomorphismRHom ShAb S .F; G/ jY Š RHom ShAb S=Y .F jY ;G jY /:Proof. We choose an injective resolution G ! I and furthermore an injective resolutionI jY ! J . Note that by Lemma 3.11 restriction is exact and therefore G jY ! J isa resolution of G jY . Then we haveRHom ShAb S .F; G/ jY Š Hom ShAb S .F; I / jY(14)Š Hom ShAb S=Y .F jY ;I jY /:Furthermore,RHom ShAb S=Y .F jY ;G jY / Š Hom ShAb S=Y .F jY ;J/

Duality for topological abelian group stacks and T -duality 255and the map I jY ! J inducesRHom ShAb S .F; G/ jY Š Hom ShAb S=Y .F jY ;I jY /! Hom ShAb S=Y .F jY ;J/Š RHom ShAb S=Y .F jY ;G jY /:(15)If the restriction f W ShS ! ShS=Y would preserve injectives, then I jY ! J wouldbe a homotopy equivalence and the marked map would be a quasi-isomorphism.In the generality of the present paper we do not know whether f preserves injectives.Nevertheless we show that our assumption on S implies that the marked map in(15) is a quasi isomorphism for all sheaves F 2 Sh Ab S.3.3.10 We first show a special case.Lemma 3.13. If F is representable, then the marked map in (15) is a quasi isomorphism.Proof. Let .U ! Y/2 S=Y and .A ! Y/2 S. Then on the one hand we haveHom ShAb S=Y .Z.A/ jY ;I jY /.U ! Y/D Hom .Sh S=Y /=.U !Y/ ..A jY / j.U !Y/ ;.I jY / jU !Y /(by 3.2.7)Š Hom Sh S=U .A jU ;I jU /(by 3.3.7):Since A jU is representable by our assumption on S there exists .B ! U/2 S=U suchthat A jU Š B ! U .Since the restriction functor is exact (Lemma 3.8) and the restriction of an injectivesheaf from S to S=Y is still flabby (Lemma 3.11) we see that G jU ! I jU is a flabbyresolution. It follows thatOn the other handHom Sh S=U .A jU ;I jU / Š Hom Sh S=U .B ! U ;I jU /Š I jU .B ! U/Š R.B ! U; G jU /:Hom ShAb S=Y .Z.A/ jY ; J /.U ! Y/Š Hom .Sh S=Y /=.U !Y/ ..A jY / jU !Y ;J jU !Y /Š Hom Sh S=U .A jU ;J jU /Š Hom Sh S=U .B ! U ;J jU /Š J.B ! U/Š R.B ! U; G jU /:

256 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.3.11 We consider the functor which associates to F 2 Sh Ab S the coneC.F/ WD Cone Hom ShAb S=Y .F jY ;I jY / ! Hom ShAb S=Y .F jY ;J/ :In order to show that the marked map in (15) is a quasi isomorphism we must show thatC.F/ is acyclic for all F 2 Sh Ab S.The restriction functor .:::/ jY is exact by Lemma 3.8. For H 2 Sh Ab S=Y thefunctor Hom ShAb S=Y .:::;H/is a right-adjoint. As a contravariant functor it transformscolimits into limits and is left exact. In particular, the functorsHom ShAb S=Y ..:::/ jY ;I jY /;Hom ShAb S=Y ..:::/ jY ;J/transform coproducts of sheaves into products of complexes. It follows that F ! C.F/also transforms coproducts of sheaves into products of complexes. If P 2 Sh Ab S is acoproduct of representable sheaves, then by Lemma 3.13 C.A/ is a product of acycliccomplexes and hence acyclic.We claim that F ! C.F/ transforms short exact sequences of sheaves to shortexact sequences of complexes. Since J is injective and .:::/ jY is exact, the functorF 7! Hom ShAb S=Y .F jY ;J/is a composition of exact functors and thus has this property.Furthermore we have Hom ShAb S=Y .F jY ;I jY / Š Hom ShAb S .F; I / jY . Since I isinjective, the functor F ! Hom ShAb S=Y .F jY ;I jY / has this property, too. This impliesthe claim.We now argue by induction. Let n 2 N and assume that we have already shownthat H i .C.F // Š 0 for all F 2 Sh Ab S and i

Duality for topological abelian group stacks and T -duality 2573.3.12 Note that a site S which has finite products satisfies the assumption of Proposition3.12. In fact, for A 2 S we have by Lemma 3.6 thatA jY Š A Y ! Y :Therefore the restriction functor .:::/ jY preserves representables.Corollary 3.14. Let S be as in Proposition 3.12. For sheaves F;G 2 Sh Ab S and Y 2 Swe haveExt i Sh Ab S .F; G/ jY Š Ext i ShAb S=Y .F jY ;G jY /:Proof. Since .:::/ jY is exact (Lemma 3.8) we haveExt i Sh Ab S .F; G/ jY Š .H i RHom ShAb S .F; G// jYŠ H i .RHom ShAb S .F; G/ jY /Š H i .RHom ShAb S=Y .F jY ;G jY // (by Proposition 3.12)Š Ext i ShAbS=Y .F jY ;G jY /:3.3.13 Let us assume that S has finite fibre products. Fix U 2 S. Then we can definea morphism of sites U W S ! S=U which maps A 2 S to .U A ! U/2 S=U. Oneeasily checks the conditions given in [Tam94, 1.2.2]. This morphism of sites inducesan adjoint pair of functorsU W ShS , ShS=U W U :Let f W S=U ! S be the restriction defined in 3.3.2 (with Y replaced by U ) and letf W ShS=U , ShS W f be the corresponding pair of adjoint functors.Lemma 3.15. We assume that S has finite products. Then we have a canonical isomorphismf Š U .Proof. We first define this isomorphism on representable sheaves. Since every sheafcan be written as a colimit of representable sheaves, U commutes with colimits asa left-adjoint, and f commutes with colimits by Lemma 3.8, the isomorphism thenextends to all sheaves. Let W 2 S and F 2 ShS=U/. Then we haveHom Sh S=U . U W ;F/Š Hom Sh S .W ; U .F //Š U .F /.W /Š F.W U ! U/Š Hom Sh S=U ..W U ! U/;F/ (by Lemma 3.6)Š Hom Sh S=U .f W ;F/This isomorphism is represented by a canonical isomorphismU .W / Š f .W /in ShS=U.

258 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.3.14 Let f W S ! S 0 be a morphism of sites. It induces an adjoint pairf W ShS , ShS 0 W f :Lemma 3.16. For F 2 ShS the functor f has the explicit descriptionProof. In fact, for G 2 ShS 0 we havef .F / WD colim .U !F/2Sh.S=F / f.U/:Hom Sh S 0.colim .U !F/2Sh.S/=F f.U/;G/Š lim .U !F/2Sh.S/=F Hom Sh S 0.f .U /;G/Š lim .U !F/2Sh.S/=F G.f.U//Š lim .U !F/2Sh.S/=F f G.U /Š lim .U !F/2Sh.S/=F Hom Sh S 0.U ;f G/Š Hom Sh S .colim .U !F/2Sh.S/=F U ;f G/Š Hom Sh S .F; f G/:Lemma 3.17. For A 2 S we havef .A/ D f .A/:Proof. Since .id A W A ! A/ 2 S=A is final we havef .A/ Š colim .U !A/2Sh.S/=A f.U/Š colim .U !A/2S=A f.U/Š f .A/:Lemma 3.18. If f W S ! S 0 is fully faithful, then we have for A 2 S that f f A Š A.Proof. We calculate for U 2 S thatf f A.U / Š f f .A/.U / (by Lemma 3.17)Š f .A/.f .U //Š Hom S 0.f .U /; f .A//Š Hom S .U; A/Š A.U /3.3.15 For U 2 S we have an induced morphism of sites U f W S=U ! S 0 =f .U /which maps .A ! U/to .f .A/ ! f.U//.If f is fully faithful, then U f is fully faithful for all U 2 S.Lemma 3.19. For G 2 ShS 0 we have in S=U the identityU f .G jf.U/ / Š .f G/ jU :If we know in addition that S, S 0 have finite products which are preserved by f W S ! S 0 ,then for F 2 S we have in S 0 =f .U / the identityU f F jU Š .f F/ jf.U/ :

Duality for topological abelian group stacks and T -duality 259Proof. Indeed, for .V ! U/2 S=U we have.f G/ jU .V ! U/D G.f .V //D G jf.U/ .f .V / ! f.U//D . U f G jf.U/ /.V ! U/:In order to see the second identity note that we can writeF jU Š colim .A!FjU /2Sh.S=U /=F jUA:Since U f is a left-adjoint it commutes with colimits. Furthermore the restrictionfunctors .:::/ jU and .:::/ f.U/ are exact by Lemma 3.8 and therefore also commutewith colimits. WritingF Š colim .A!F/2Sh.S/=F Awe getF jU Š .colim .A!F/2Sh.S/=F A/ jU Š colim .A!F/2Sh.S/=F A jUŠ colim .A!F/2Sh.S/=F A U ! A (by Lemma 3.6).Using that f preserves products in the isomorphism marked by .Š/ we calculateU f F jU Š U f colim .A!F/2Sh.S/=F A U ! UŠ colim .A!F/2Sh.S/=F U f A U ! UŠ colim .A!F/2Sh.S/=F f.A U/! f.U/ (by Lemma 3.17).Š/Š colim .A!F/2Sh.S/=F f .A/ f.U/! f.U/Š colim .A!F/2Sh.S/=F f .A/ jf.U/Š .colim .A!F/2Sh.S/=F f .A// jf.U/Š .f F/ jf.U/ : (by Lemma 3.16)Lemma 3.20. Assume that S; S 0 have finite products which are preserved by f W S ! S 0 .For F 2 ShS and G 2 ShS 0 we have a natural isomorphismProof. For U 2 S we calculateHom Sh S .F; f G/ Š f Hom Sh S 0.f F;G/:Hom Sh S .F; f G/.U / D Hom Sh S=U .F jU ;.f G/ jU /Š Hom Sh S=U .F jU ; U f .G jf.U/ // (by Lemma 3.19)Š Hom Sh S 0 =f .U /. U f .F jU /; G jf.U/ /Š Hom Sh S 0 =f .U /..f F/ jf.U/ ;G jf.U/ / (by Lemma 3.19)D Hom Sh S 0.f F; G/.f .U //D f Hom Sh S 0.f F; G/.U /

260 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.3.16 We now consider the derived version of Lemma 3.20. Let F 2 Sh Ab S andG 2 Sh Ab S 0 be sheaves of abelian groups.Proposition 3.21. We make the following assumptions:1. The sites S; S 0 have finite products.2. The morphism of sites f W S ! S 0 preserves finite products.3. We assume that f W ShS 0 ! ShS is exact.Then in D C .Sh Ab S/ there is a natural isomorphismR Hom ShAb S .F; f G/ Š f RHom ShAb S 0.f F;G/:Proof. By [Tam94, Proposition 3.6.7] the conditions on the sites and f imply that thefunctor f is exact. It follows that f preserves injectives (see e.g. [Tam94, Proposition3.6.2]). We choose an injective resolution G ! I . Then f G ! f I is aninjective resolution of f G. It follows thatf RHom ShAb S 0.f F;G/ Š f Hom ShAb S 0.f F;I /Š Hom ShAb S .F; f I / Š RHom ShAb S .F; f G/:Proposition 3.21 is very similar in spirit to Proposition 3.12. On the other hand,we can not deduce 3.12 from 3.21 since the functor f W ShS=U ! S in general doesnot preserve products. In fact, if S has fibre products, then .A ! U/ .B ! U/ D.A U B ! U/. Then f ..A ! U/ .B ! U// Š A U B while f.A !U/ f.B ! U/Š A B.3.3.17 Recall the notations W ; W from 3.3.13.Lemma 3.22. Let W 2 S. For every F 2 ShS and A 2 S we haveHom Sh S .W ; F /.A/ Š W W F .A/:Proof. For A 2 S we consider f W S=A ! S and getHom Sh S .W ; F /.A/ Š Hom Sh S=A .W jA ;F jA /Š Hom Sh S=A .f W ;f F/Š Hom Sh S=A . A .W /; f F/ (by Lemma 3.15)Š Hom Sh S .W ; A f .F //Š A .f F /.W / (16)Š f F.A W ! A/Š F.A W/Š W .f F /.A/Š W . W .F //.A/ (by Lemma 3.15):

Duality for topological abelian group stacks and T -duality 2613.3.18 To abbreviate, let us introduce the following notation.Definition 3.23. If S has finite products, then for W 2 S we introduce the functorR W WD W ı W W ShS ! ShS:We denote its restriction to the category of sheaves of abelian groups by the samesymbol. Since W is exact and W is left-exact, it admits a right-derived functorRR W W D C .Sh Ab S/ ! D C .Sh Ab S/.Lemma 3.24. Let F 2 Sh Ab S and W 2 S. Then we have a canonical isomorphismRHom ShAb S .Z.W /; F / Š RR W .F /.Proof. This follows from the isomorphism of functors Hom ShAb S .Z.W /;:::/ ŠR W .:::/from Sh Ab S to Sh Ab S. In fact, we have for F 2 Sh Ab S thatHom ShAb S .Z.W /; F / Š Hom Sh S .W ; F .F //Š W . W .F // (Equation 16) (17)Š R W .F / (Definition 3.23):3.4 Application to sites of topological spaces3.4.1 In this subsection we consider the site S of compactly generated topologicalspaces as in 3.1.2 and some of its sub-sites. We are interested in proving that restrictionto sub-sites preserve Ext i -sheaves.We will further study properties of the functor R W . In particular, we are interestedin results asserting that the higher derived functors R i R W .F /, i 1 vanish undercertain conditions on F and W .Lemma 3.25. If C 2 S is compact and H is a discrete space, then Map.C; H / isdiscrete, andR C .H / D Map.C; H /:Proof. We first show that Map.C; H / is a discrete space in the compact-open topology.Let f 2 Map.C; H /. Since C is compact, the image f.C/ is compact, hence finite.We must show that ¹f º Map.C; H / is open. Let h 1 ;:::;h r be the finite set of valuesof f . The sets f 1 .h i / C are closed and therefore compact and their union is C .The sets ¹h i º H are open. Therefore U i WD ¹g 2 Map.C; H / j g.f 1 .h i // ¹h i ººare open subsets of Map.C; H /. We now see that ¹f ºD T riD1 U i is open.We have by the exponential lawR C .H /.A/ Š H .A C/ Š Hom S .A C;H/Š Hom S .A; Map.C; H // Š Map.C; H /.A/:

262 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.4.2 A sheaf G 2 Sh Ab S is called R W -acyclic if R i R W .G/ Š 0 for i 1.Lemma 3.26. If G is a discrete group, then G is R R n-acyclic.Proof. Let G ! I be an injective resolution. Then we have for A 2 S thatR R n.I /.A/ (16)Š I .A R n /:Therefore R i R R n.G/ is the sheafification of the presheafA 7! H i .I .A R n //:Since G is discrete, the sheaf cohomology of G is homotopy invariant, and thereforeH i .I .A R n // Š H i .A R n I G/ Š Š H i .AI G/:To be precise this can be seen as follows. Let .A/ denote the site of open subsetsof A. It comes with a natural map A W .A/ ! S. The sheaf cohomology functor isthe derived functor of the evaluation functor. In order to indicate on which categorythis evaluation functor is defined we temporarily use subscripts. If I 2 Sh Ab S isinjective, then A I 2 Sh Ab.A/ is still flabby (see [BSS, Lemma 2.32]). This impliesthat H Sh Ab S .AI G/ Š H .A/ .A; A .G//.The diagramA id Aid A 0pr A AA R nis a homotopical isomorphism (in the sense of [KS94, 2.7.4]) A ! A R n over A. Wenow apply [KS94, 2.7.7] which says that the natural mapR.pr A / pr A . A .G// ! R.id A/ id A . A .G//is an isomorphism in D C .Sh Ab .A//. But since G is discrete we get pr A . A G/ Š AR n .G/. If we apply the functor R .A/ .A;:::/to this isomorphism and take cohomologywe get the desired isomorphism marked by Š. 4Now, the sheafification of the presheaf S 3 A ! H i .AI G/ 2 Ab is exactly the ithcohomology sheaf of I which vanishes for i 1.Lemma 3.27. The sheaf R n is R W -acyclic for every compact W 2 S.Proof. Let R n ! I be an injective resolution. Then R i R W .R n / is the sheafificationof the presheafS 3 A 7! H i .Hom ShAb S .Z.W jA /; I jA // Š H i .I .A W //:4 It is tempting to apply a Künneth formula to calculate H .A R n ;G/. But since A is not necessarilycompact it is not clear that the Künneth formula holds.

Duality for topological abelian group stacks and T -duality 263Let Œs 2 H i .I .AW//be represented by s 2 I i .AW/, and a 2 A. Then we mustfind a neighbourhood U A of a such that Œs jU W D 0, i.e. s jU W D dt for somet 2 I i 1 .U W/, where d W I i 1 ! I i is the boundary operator of the resolution.The ring structure of R induces on R n the structure of a sheaf of rings. In order todistinguish this sheaf of rings from the sheaf of groups R n we will use the notation C.Note that R n is in fact a sheaf of C-modules.The forgetful functor resW Sh C-mod S ! Sh Ab S fits into an adjoint pairind W Sh Ab S , Sh C-mod S W res;where ind is given by Sh Ab S 3 V ! V ˝Z C 2 Sh C-mod . Since C is a torsion-freesheaf it is flat. It follows that ind is exact and res preserves injectives.We can now choose an injective resolution R n ! J in Sh C-mod S and assume thatI D res.J /.Since the complex of sheaves I is exact we can find an open covering .V r / r2R ofA W such that s jVr D dt r for some t r 2 I i 1 .V r /. Since W is compact (locallycompact suffices), by [Ste67, Theorem 4.3]) the product topology on A W is thecompactly generated topology used for the product in S. Hence after refining thecovering .V r / we can assume that V r D A r W r for open subsets A r A andW r W for all r 2 R.We define R a WD ¹r 2 R j a 2 A r º. The family .W r / r2Ra is an open coveringof W . Since W is compact we can choose a finite set r 1 ;:::;r k 2 R a such thatW WD .W r1 ;:::;W rk / is still an open covering of W . The subset U WD T kj D1 A r jisan open neighbourhood of a 2 A.Since I i 1 is injective we can choose 5 extensions Qt r 2 I i 1 .A W/such that.Qt r / jVr D t r .We choose a partition of unity . 1 ;:::; v / subordinate to the finite covering W.We take advantage of the fact that I D res.J / which implies that we can multiplysections by continuous functions, and that d commutes with this multiplication. WedefinevXt WD k .Qt rk / jU W 2 I i 1 .U W/:kD1Note that k .s d Qt rk / jU W D 0. In fact we have k .s d Qt rk / j.U W/\Vrk D k .sdt rk / j.U W/\Vrk D 0. Furthermore, there is a neighbourhood Z of the complement of.U W/\ V rk in U W where k vanishes. Therefore the restrictions k .s d Qt rk /vanish on the open covering ¹Z; .U W/\ V rk º of U W , and this implies the5 Let U X be an open subset. Then we have an injection U ! X and hence an injection Z.U / !Z.X/. For an injective sheaf I we get a surjection Hom ShAb S.Z.X/; I / ! Hom ShAb S.Z.U /; I /. In othersymbols, I.X/ ! I.U/ is surjective.

264 U. Bunke, T. Schick, M. Spitzweck, and A. Thomassertion. We getdt DDvXd. k .Qt rk / jU W / Dj D1vX k .d Qt rk / jU W Dj D1D s jU W :vX k d.Qt rk / jU Wj D1vX k s jU WCorollary 3.28. 1. If G is discrete, then we have Ext i Sh Ab S .Z.Rn /; G/ Š 0 for alli 1.2. For every compact W 2 S and n 1 we have Ext i Sh Ab S .Z.W /; Rn / Š 0 for alli 1.Lemma 3.29. If W is a profinite space, then every sheaf F 2 Sh Ab S is R W -acyclic.Consequently, Ext i Sh Ab S .Z.W /; F / Š 0 for i 1.Proof. We first show the following intermediate result which is used in 3.4.3 in orderto finish the proof of Lemma 3.29.Lemma 3.30. If W is a profinite space, then .W I :::/is exact.Proof. A profinite topological space can be written as limit, W Š lim I W n , for an inversesystem of finite spaces .W n / n2I . Let p n W W ! W n denote the projections. First of all,W is compact. Every covering of W admits a finite subcovering. Furthermore, a finitecovering admits a refinement to a covering by pairwise disjoint open subsets of the form¹p 1n.x/º x2W nfor an appropriate n 2 I . This implies the vanishing HLp .W; F / Š 0of the Čech cohomology groups for p 1 and every presheaf F 2 Pr Ab S.Let H q D R q i be the derived functor of the embedding i W Sh Ab S ! Pr Ab S ofsheaves into presheaves. We now consider the Čech cohomology spectral sequence[Tam94, 3.4.4] .E r ;d r / ) R .W;F/ with E p;q2Š HLp .W I H q .F // and use[Tam94, 3.4.3] to the effect that HL0 .W I H q .F // Š 0 for all q 1. Combiningthese two vanishing results we see that the only non-trivial term of the second page ofthe spectral sequence is E 0;02Š HL0 .W I H 0 .F // Š F.W/. Vanishing of R i .W I :::/for i 1 is equivalent to the exactness of .W I :::/.3.4.3 We now prove Lemma 3.29. Let i 1. We use that R i R W .F / is the sheafificationof the presheaf S 3 A 7! R i .A W I F/ 2 Ab. For every sheaf F 2 Sh Ab Swe have by some intermediate steps in (16)j D1.A W I F/Š .W I R A .F //:Let us choose an injective resolution F ! I . Using Lemma 3.30 for the secondisomorphism we getH i .AI R W .I // s7!QsŠ H i .A W I I / Qs7!NsŠ .W I H i R A .I //: (18)

Duality for topological abelian group stacks and T -duality 265Consider a point a 2 A and s 2 H i .AI R W .I //. We must show that there existsa neighbourhood a 2 U A of a such that s jU D 0. Since I is an exact sequenceof sheaves in degree 1 there exists an open covering ¹Y r º r2R of A W such thatQs jYr D 0. After refining this covering we can assume that Y r Š U r V r for suitable opensubsets U r A and V r W . Consider the set R a WD ¹r 2 R j a 2 U r º. Since W iscompact the covering ¹V r º r2Ra of W admits a finite subcovering indexed by Z R a .The set U WD T r2Z U r A is an open neighbourhood of a. By further restriction weget Qs jU Vr D 0 for all r 2 Z. By (18) this means that 0 D s jU jV r2 .W I H i R U .I //.Therefore s jU vanishes locally on W and therefore globally. This implies s jU D 0.3.4.4 Let f W S lc ! S be the inclusion of the full subcategory of locally compacttopological spaces. Since an open subset of a locally compact space is again locallycompact we can define the topology on S lc bycov Slc .A/ WD cov S .A/; A 2 S lc :The compactly generated topology and the product topology on products of locallycompact spaces coincides. The same applies to fibre products. Furthermore, a fibreproduct of locally compact spaces is locally compact. The functor f preserves fibreproducts. In view of the definition of the topology S lc the inclusion functor f W S lc ! Sis a morphism of sites.Lemma 3.31. Restriction to the site S lc commutes with sheafification, i.e.i ] ı p f Š f ı i ] :Proof. (Compare with the proof of Lemma 3.7.) This follows from cov Slc .A/ Šcov S .A/ for all A 2 S and the explicit construction of i ] in terms of the set cov Slc(see [Tam94, Section 3.1]).Lemma 3.32. The restriction f W ShS ! ShS lc is exact.Proof. (Compare with the proof of Lemma 3.8.) The functor f is a right-adjoint andthus commutes with limits. Colimits of presheaves are defined object-wise, i.e. for adiagramC ! Pr S; c 7! F cof presheaves we have. p colim c2C F c /.U / D colim c2C F c .U /:It follows from the explicit description of p f that this functor commutes with colimitsof presheaves. For a diagram of sheaves F W C ! ShS we havecolim c2C F c D i ]p colim c2C F c :By Lemma 3.31 we get f colim c2C F c Š f i ]p colim c2C F c Š i ]p f p colim c2C F c Ši ]p colim c2C p f F c Š colim c2C f F c :

266 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.4.5 We have now verified that f W S lc ! S satisfies the assumptions of Proposition3.21.Corollary 3.33. Let f W S lc ! S be the inclusion of the site of locally compact spaces.For F 2 Sh Ab S lc and G 2 Sh Ab S we havef RHom ShAb S .f F;G/ Š RHom ShAb S lc.F; f G/:In particular we havef Ext k Sh Ab S .f F;G/ Š Ext k Sh Ab S lc.F; f G/for all k 0.In fact, the first assertion implies the second since f is exact.We need this result in the following special case. If G 2 S is a locally compact group,then by abuse of notation we write G for the sheaves of abelian groups represented byG in both categories Sh Ab S and Sh Ab S lc .We have f G D G. By Lemma 3.17 we also have f G Š G.Corollary 3.34. Let G; H 2 S lc be locally compact abelian groups. We havef Ext k Sh Ab S .G;H/ Š Extk Sh Ab S lc.G;H/for all k 0.3.4.6 In some places we will need a second sub-site of S, the site S loc-acyc of locallyacyclic spaces.Definition 3.35. A space U 2 S is called acyclic, if H i .U I H / Š 0 for all discreteabelian groups H and i 1.By Lemma 3.30 all profinite spaces are acyclic. The space R n is another exampleof an acyclic space. In fact, the homotopy invariance used in the proof of 3.26 showsthat the inclusion 0 ! R n induces an isomorphism H i .R n I H / Š H i .¹0ºI H /, and aone-point space is clearly acyclic.Definition 3.36. A space A 2 S is called locally acyclic if it admits an open coveringby acyclic spaces.In general we do not know if the product of two locally acyclic spaces is againlocally acyclic (the Künneth formula needs a compactness assumption). In order toensure the existence of finite products we consider the combination of the conditionslocally acyclic and locally compact.Note that all finite-dimensional manifolds are locally acyclic and locally compact.An open subset of a locally acyclic locally compact space is again locally acyclic. Welet S lc-acyc S be the full subcategory of locally acyclic locally compact spaces. Thetopology on S lc-acyc is given bycov Slc-acyc .A/ WD cov S .A/:Let g W S lc-acyc ! S be the inclusion. The proofs of Lemma 3.31, Lemma 3.32 andCorollary 3.33 apply verbatim.

Duality for topological abelian group stacks and T -duality 267Corollary 3.37. 1. The restriction g W ShS ! ShS lc-acyc is exact.2. For F 2 Sh Ab S lc-acyc and G 2 Sh Ab S we haveandg RHom ShAb S .g F;G/ Š RHom ShAb S lc-acyc.F; g G/g Ext k Sh Ab S .g F;G/ Š Ext k Sh Ab S lc-acyc.F; g G/for all k 0.Corollary 3.38. Let G; H 2 S lc-acyc be locally acyclic abelian groups. We havefor all k 0.g Ext k Sh Ab S .G;H/ Š Extk Sh Ab S lc-acyc.G;H/Note that the product Q NT is compact but not locally acyclic.3.5 Z mult -modules3.5.1 We consider the multiplicative semigroup Z mult of non-zero integers. Everyabelian group G (written multiplicatively) has a tautological action of Z mult by homomorphismsgiven byZ mult G ! G; .n; g/ 7! g n :Definition 3.39. When we consider G with this action we write G.1/.The semigroup Z mult will therefore act on all objects naturally constructed from anabelian group G. We will in particular use the action of Z mult on the group-homologyand group-cohomology of G.If G is a topological group, then Z mult acts by continuous maps. It therefore alsoacts on the cohomology of G as a topological space. Also this action will play a rolelater.The remainder of the present subsection sets up some language related to Z mult -actions.Definition 3.40. A Z mult -module is an abelian group with an action of Z mult by homomorphisms.We will write this action as Z mult G 3 .m; g/ 7! ‰ m .g/ 2 G. Thus in the caseof G.1/ we have ‰ m .g/ D g m .3.5.2 We let Z mult -mod denote the category of Z mult -modules. An equivalent descriptionof this category is as the category of modules under the commutative semigroupring ZŒZ mult . The category Z mult -mod is an abelian tensor category.We have an exact inclusion of categoriesAb ,! .Z mult -mod/;G 7! G.1/

268 U. Bunke, T. Schick, M. Spitzweck, and A. ThomByF W .Z mult -mod/ ! Abwe denote the forgetful functor.Definition 3.41. Let G be an abelian group. For k 2 Z we let G.k/ 2 Z mult -mod denotethe Z mult -module given by the action Z mult G 3 .p; g/ 7! ‰ p .g/ WD g pk 2 G.Observe that for abelian groups V;W we have a natural isomorphism3.5.3 Let V be a Z mult -module.V.k/˝Z W.l/ Š .V ˝Z W /.k C l/: (19)Definition 3.42. We say that V has weight k if there exists an isomorphism V ŠF .V /.k/ of Z mult -modules.If V has weight k, then every sub-quotient of V has weight k. Note that aZ mult -module can have many weights. We have e.g. isomorphisms of Z mult -modules.Z=2Z/.1/ Š .Z=2Z/.k/ for all k 6D 0.3.5.4 Let V 2 Z mult -mod. We say that v 2 V has weight k if it generates a submoduleZ V of weight k. Fork 2 N we let W k W .Z mult -mod/ ! Ab be the functorwhich associates to V 2 Z mult -mod its subgroup of vectors of weight k. Then we havean adjoint pair of functors.k/ W Ab , .Z mult -mod/ W W k :Observe that the functor W k is not exact. Consider for example a prime number p 2 Nand the sequence0 ! Z.1/ ! p Z.1/ ! .Z=pZ/.p/ ! 0:The projection map is indeed Z mult -equivariant since m p m mod p for all m 2 Z.Then0 Š W p .Z.1// ! W p ..Z=pZ/.p// Š Z=pZis not surjective.3.5.5 Let V 2 Ab and V .1/ 2 Z mult -mod. Then we can form the graded tensor algebraT Z .V .1// D Z ˚ V .1/ ˚ V .1/ ˝Z V .1/ ˚ :We see that T k .V .1// has weight k. The elements x ˝ x 2 V .1/ ˝ V .1/ generate ahomogeneous ideal I . Hence the graded algebra ƒ Z .V .1// WD T Z .V .1//=I has theproperty that ƒ k Z.V .1// has weight k.

Duality for topological abelian group stacks and T -duality 2693.5.6 It makes sense to speak of a sheaf or presheaf of Z mult -modules on the site S.We let Sh Zmult -modS and Pr Zmult -mod S denote the corresponding abelian categories ofsheaves and presheaves.Definition 3.43. Let V 2 Sh Zmult -modS (or V 2 Pr Zmult -mod S) and k 2 Z. We say thatV is of weight k if the map V mk m! V vanishes for all m 2 Zmult .We define the functors .k/W Sh Ab S ! Sh Zmult -modS, F W Sh Zmult -modS ! Sh Ab S,and W k W Sh Zmult -modS ! Sh Ab S (and their presheaf versions) object-wise. We alsohave a pair of adjoint functors.k/ W Sh Ab S , Sh Zmult -modS W W k(and the corresponding presheaf version). A sheaf V 2 Sh Zmult -modS of Z mult -moduleshas weight k 2 Z if V Š F .V /.k/ Š W k .V /.k/.4 Admissibility of sheaves represented by topological abeliangroups4.1 Admissible sheaves and groups4.1.1 The main topic of the present paper is a duality theory for abelian group stacks(Picard stacks, see 2.4) on the site S. A Picard stack P 2 PIC.S/ gives rise to thesheaf of objects H 0 .P / and the sheaf of automorphisms of the neutral object H 1 .P /.These are sheaves of abelian groups on S.We will define the notion of a dual Picard stack D.P / (see 5.4). With the intentionto generalize the Pontrjagin duality for locally compact abelian groups to group stackswe study the question under which conditions the natural map P ! D.D.P // is anisomorphism. In Theorem 5.11 we see that this is the case if the sheaves are dualizable(see 5.2) and admissible (see 4.1). Dualizability is a sheaf-theoretic generalization ofthe classical Pontrjagin duality and is satisfied e.g. for the sheaves G for locally compactgroups G 2 S (see 5.3). Admissibility is more exotic and will be defined below (4.1).One of the main results of the present paper asserts that the sheaves G are admissiblefor a large (but not exhaustive) class of locally compact groups G 2 S, and for an evenlarger class if S is replaced by S lc-acyc .The main tool in our computations is a certain double complex, which will beintroduced in Section 4.2. Similar techniques were used by Lawrence Breen in [Bre69],[Bre76], [Bre78].Definition 4.1. We call a sheaf of groups F admissible if Ext i Sh Ab S .F; T/ Š 0 fori D 1; 2. A topological abelian group G 2 S is called admissible if G is an admissiblesheaf.We will also consider the sub-sites S lc-acyc S lc S of locally compact spaces.

270 U. Bunke, T. Schick, M. Spitzweck, and A. ThomDefinition 4.2. A locally compact abelian topological group G 2 S lc is called admissibleon S lc if Ext i Sh Ab S lc.G; T/ Š 0 for i D 1; 2 (and correspondingly for S lc-acyc ).Lemma 4.3. Let f W S lc ! S be the inclusion. If F 2 Sh Ab S lc and f F is admissible,then F is admissible on S lc .Proof. This is an application of Corollary 3.33.Corollary 4.4. If G 2 S lc is admissible, then it is admissible on S lc .This is an application of Corollary 3.34.Lemma 4.5. The class of admissible sheaves is closed under finite products and extensions.Proof. For finite products the assertions follows from the fact that Ext ShAb S .:::;T/commutes with finite products. Given an extension of sheaves0 ! F ! G ! H ! 0such that F and H are admissible, then also G is admissible. This follows immediatelyfrom the long exact sequence obtained by applying Ext Sh Ab S .:::;T/.4.1.2 In this paragraph we formulate one of the main theorems of the present paper.Let G be a locally compact topological abelian group. We first recall some technicalconditions.Definition 4.6. We say that G satisfies the two-three condition, if1. it does not admit Q n2NZ=2Z as a subquotient,2. the multiplication by 3 on the component G 0 of the identity has finite cokernel.Definition 4.7. We say that G is locally topologically divisible if for all primes p 2 Nthe multiplication map p W G ! G has a continuous local section.Using that the class of admissible groups is closed under finite products and extensionswe get the following general theorem.Theorem 4.8. 1. If G is a locally compact abelian group which satisfies the two-threecondition, then it is admissible over S lc-acyc .2. Assume that(a) G satisfies the two-three condition,(b) G has an open subgroup of the form C R n with C compact such that G=C R nis finitely generated(c) the connected component of the identity of G is locally topologically divisible.Then G is admissible over S lc .

Duality for topological abelian group stacks and T -duality 271Proof. By [HM98, Theorem 7.57(i)] the group G has a splitting G Š H R n for somen 2 N 0 , where H has a compact open subgroup U . The quotient G=.U R n / Š H=Uis therefore discrete. Using Lemma 4.5 conclude that G is admissible over S lc if R nand H are so.Admissibility of R n follows from Theorem 4.33 (which we prove later) in conjunctionwith 4.5 and 4.4.Since D WD H=U is discrete, the exact sequence0 ! U ! H ! H=U ! 0has local sections and therefore induces an exact sequence of associated sheaves0 ! U ! H ! D ! 0by Lemma 3.4. If D is finitely generated, then it is admissible by Theorem 4.18,and hence admissible on S lc by Lemma 4.4. Otherwise it is admissible on S lc-acyc byTheorem 4.28.Therefore H is admissible on S lc (or S lc-acyc , respectively) by Lemma 4.5 if U isadmissible over S lc (or S lc-acyc , respectively). The compact group U fits into an exactsequence0 ! U 0 ! U ! P ! 0where P is profinite and U 0 is closed and connected. By assumption U , U 0 and P satisfythe two-three condition. The connected compact group U 0 is admissible on S lc (or onS lc without the assumption that it is locally topologically divisible), by Theorem 4.79,and the profinite P is admissible by Theorem 4.66, and hence admissible on S lc byLemma 4.4.Note that0 ! U 0 ! U ! P ! 0is exact by Lemma 4.39. Now it follows from Lemma 4.5 that U is admissible on S lc(or S lc-acyc , respectively).4.1.3 We conjecture that the assumption that G satisfies the two-three condition is onlytechnical and forced by our technique to prove admissibility of compact groups.In the remainder of this section, we will mainly be concerned with the proof of thestatements used in the proof of Theorem 4.8 above.4.1.4 Our proofs of admissibility for a sheaf F 2 Sh Ab S will usually be based on thefollowing argument.Lemma 4.9. Assume that F 2 Sh Ab S satisfies1. Ext i Sh Ab S .F; Z/ Š 0 for i D 2; 3;2. Ext i Sh Ab S .F; R/ Š 0 for i D 1; 2.Then F is admissible.

272 U. Bunke, T. Schick, M. Spitzweck, and A. ThomProof. We apply the functor Ext Sh Ab S .F;:::/to the sequence0 ! Z ! R ! T ! 0and get the following segments of the long exact sequence!Ext i Sh Ab S .F; R/ ! Exti Sh Ab S .F; T/ ! ExtiC1Sh Ab S.F; Z/ ! :We see that the assumptions on F imply that Ext i Sh Ab S .F; T/ Š 0 for i D 1; 2.4.1.5 A space W 2 S is called profinite if it can be written as the limit of an inversesystem of finite spaces. Lemma 3.29 has as a special case the following theorem.Theorem 4.10. If W 2 S is profinite, then Z.W / is admissible.4.2 A double complex4.2.1 Let H 2 Sh Ab S be a sheaf of groups with underlying sheaf of sets F .H / 2ShS. Applying the linearization functor Z (see 3.2.6) we get again a sheaf of groupsZ.F .H // 2 Sh Ab S. The group structure of H induces on Z.F .H // a ring structure.We denote this sheaf of rings by ZŒH . It is the sheafification of the presheaf p ZŒH which associates to A 2 S the integral group ring p ZŒH .A/ of the group H.A/.We consider the category Sh ZŒH -mod S of sheaves of ZŒH -modules. The trivialaction of the sheaf of groups H on the sheaf Z induces the structure of a sheaf ofZŒH -modules on Z. With the notation introduced below we could (but refrain fromdoing this) write this sheaf of ZŒH -modules as coind.Z/.4.2.2 The forgetful functor resW Sh ZŒH -mod S ! Sh Ab S fits into an adjoint pair offunctorsind W Sh Ab S , Sh ZŒH -mod S W res:Explicitly, the functor ind is given byind.V / WD ZŒH ˝Z V:Since ZŒH is a torsion-free sheaf and therefore a sheaf of flat Z-modules the functorind is exact. Consequently the functor res preserves injectives.4.2.3 We let coinv W Sh ZŒH -mod S ! Sh Ab S denote the coinvariants functor given bySh ZŒH -mod S 3 V 7! coinv.V / WD V ˝ZŒH Z 2 Sh Ab S:This functor fits into the adjoint paircoinv W Sh ZŒH -mod S , Sh Ab S W coindwith the coinduction functor which maps W 2 Sh Ab S to the sheaf of ZŒH -modulesinduced by the trivial action of H in W , formally this can be written ascoind.W / WD Hom ShAb S .Z;W/:

Duality for topological abelian group stacks and T -duality 273Lemma 4.11. Every sheaf F 2 Sh ZŒH -mod S is a quotient of a flat sheaf.Proof. Indeed, the counits of the adjoint pairs .Z.:::/;F / and .ind; res/ induce asurjection ind.Z.F .res.F //// ! F . Explicitly it is given by the composition of thesum and action map (omitting to write some forgetful functors)ZŒH ˝Z ZŒF ! ZŒH ˝ F ! F:Since ZŒH is a sheaf of unital rings this action is surjective. Moreover, for A 2Sh ZŒH -mod S we haveA ˝ZŒH .ZŒH ˝Z ZŒF / Š A ˝Z ZŒF :Since ZŒF is a torsion-free sheaf of abelian groups the operation A ! A ˝ZŒH .ZŒH ˝Z ZŒF / preserves exact sequences in Sh ZŒH -mod S. Therefore ZŒH ˝Z ZŒF is a flat sheaf of ZŒH -modules.Lemma 4.12. The class of flat ZŒH -modules is coinv-acyclic.Proof. Let F be an exact lower bounded homological complex of flat ZŒH -modules.We choose a flat resolution P ! Z in Sh ZŒH -mod S which exists by Lemma 4.11 6 .Since F consists of flat modules the induced mapF ˝ZŒH P ! F ˝ZŒH Z D coinv.F /is a quasi-isomorphism. Since P consists of flat modules, tensoring by P commuteswith taking cohomology, so that we haveH .F ˝ZŒH P / Š H .H .F / ˝ZŒH P / Š 0:Therefore coinv.:::/ maps acyclic complexes of flat ZŒH -modules to acyclic complexesof Z-modules.Corollary 4.13. We can calculate L coinv.Z/ using a flat resolution.4.2.4 We will actually work with a very special flat resolution of Z. The bar constructionon the sheaf of groups H gives a sheaf H of simplicial sets with an action of H .We let C .H / WD C.Z.H // be the sheaf of homological chain complexes associatedto the sheaf of simplicial groups Z.H /. The H -action on H induces an H -action onC .H / and therefore the structure of a sheaf of ZŒH -modules. In order to understandthe structure of C .H / we first consider the presheaf version p C .H / WD C. p Z.H //.In fact, we can writeH i Š H H„Hƒ‚ …;i factorsas a sheaf of H -sets, and thereforep C i .H / Š p ZŒH ˝pZp Z.H„Hƒ‚ …/ (20)i factors6 Note that P is a homological complex, i.e. the differentials have degree 1.

274 U. Bunke, T. Schick, M. Spitzweck, and A. ThomThe cohomology of the complex p C .H / is given by´ pZ;H i . p i D 0;C .H // Š0; i 1;where p Z 2 Pr Ab S denotes the constant presheaf with value Z. Since sheafification i ]is an exact functor and by definition C .H / D i ]p C .H / we get´H i Z; i D 0;.C .H // Š0; i 1:Furthermore,C i .H / Š ZŒH ˝Z Z.H„Hƒ‚ …/ D ind.Z.H„Hƒ‚ …// (21)i factorsi factorsshows that C i .H / is flat. Let us write C WD C .H / D ind.D / withD i WD Z.H„Hƒ‚ …/:i factorsDefinition 4.14. For a sheaf H 2 Sh Ab S we define the complex U coinv.C /.WD U .H / WDIt follows from the construction that U depends functorially on H . In particular,for ˛ W H ! H 0 we have a map of complexes U .˛/W U .H / ! U .H 0 /.4.2.5 The main tool in our proofs of admissibility of a sheaf F is the study of thesheavesR Hom ShZŒH -mod S .Z; coind.W //for W D Z; R. Let us write this in a more complicated way using the special flatresolution C .H / ! Z constructed in 4.2.4. We choose an injective resolutioncoind.W / ! I in Sh ZŒH -mod S. Using that res ı coind D id, and that res.I / isinjective in Sh Ab S,weget 7RHom ShZŒH -mod S .Z; coind.W // Š Hom Sh ZŒH -mod S / .Z;IŠ Hom ShZŒH -mod S .C .H /; I /Š Hom ShZŒH -mod S /; I /.ind.DŠ Hom ShAb S ; res.I //.DD Hom ShAb S ; res.coind.res.I ////.DD Hom ShZŒH -mod S /; coind.res.I ///.ind.D7 Note that D is considered in this calculation as a sequence of sheaves, not as a complex. The differentialsin the intermediate steps involving D are still induced via the isomorphisms from the differentials of C and I .

Duality for topological abelian group stacks and T -duality 275Š Hom ShZŒH -mod S .C ; coind.res.I ///Š Hom ShAb S .coinv.C /; res.I //Š RHom ShAb S .Lcoinv.Z/; W /:4.2.6 In general, the coinvariants functor coinv.:::/D˝ZŒH Z can be written interms of the tensor product in the sense of presheaves composed with a sheafification.Furthermore, C .H / is the sheafification of p C .H /. Using the fact that the tensorproduct of presheaves commutes with sheafificationU i D coinv.C i /Š .C i .H / ˝pZŒH Z/] (Definition 4.14)D .. p C i .H // ] ˝p. p Z/ ] / ]. p ZŒH / ]Š . p C i .H / ˝pp ZŒH p Z/ ] (22)Š . p ZŒH ˝pZp Z.H„Hƒ‚ …/ ˝pp Z/ ] ZŒH (Equation (20))i factorsŠ . p Z.H„Hƒ‚ …// ]i factorsD . p D i / ]withIn particular, we havep D i WD p Z.H„Hƒ‚ …/: (23)i factorsU i D Z.H i /: (24)4.2.7 Let G be a group. Then we can form the standard reduced bar complex for thegroup homology with integer coefficientsG !Z.G n / ! Z.G n 1 / !!Z ! 0:The abelian group Z.G n / (sitting in degree n) is freely generated by the underlying setof G n , and we write the generators in the form Œg 1 j :::jg n . The differential is givenbynXd D . 1/ i d i W Z.G n / ! Z.G n 1 /;whereiD08ˆ< Œg 2 j :::jg n ; i D 0;d i Œg 1 j :::jg n WD Œg 1 j :::jg i g iC1 j :::jg n ; 1 i n 1;ˆ:Œg 1 j :::jg n 1 ; i D n:The cohomology of this complex is the group homology H .GI Z/.

276 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.2.8 For A 2 S the complex p D .A/ (see (23)) is exactly the standard complex (see4.2.7) for the group homology H .H.A/; Z/ of the group H.A/. The cohomologysheaves H .U / are thus the sheafifications of the cohomology presheavesS 3 A 7! H .H.A/; Z/ 2 Ab:4.2.9 In this paragraph we collect some facts about the homology of abelian groups.An abelian group V is the same thing as a Z-module. We define the graded Z-algebraƒ ZV as the quotient of the tensor algebraT Z V WD M n0V ˝Z ˝Z V„ ƒ‚ …n factorsby the graded ideal I T Z V generated by the elements x ˝ x, x 2 V .4.2.10 Let G be an abelian group. We refer to [Bro82, V.6.4] for the following fact.Fact 4.15. There exists a canonical mapmW ƒ i Z G ! H i.GI Z/:It is an isomorphism for i D 0; 1; 2, and it becomes an isomorphism after tensoring withQ for all i 0. IfG is torsion-free, then it is an isomorphism mW ƒ i Z G ! H i .GI Z/for all i 0.4.2.11 Let U D U .H / (see Definition 4.14) for H 2 Sh Ab S. The cohomologysheaves L coinv.Z// Š H .U / are the sheafifications of the presheaves H . p D /.By the fact 4.15 we have H i . p D / Š ƒ i ZH for i D 0; 1; 2 for the presheaves S 3U 7! ƒ i Z H.U/. In particular, we have H 0 .U / Š Z and H 1 .U / Š H . If H isa torsion-free sheaf, then H i .U / Š .ƒ i Z H/] for all i 0. Finally, for an arbitrarysheaf H 2 Sh Ab S we have.ƒ Z H/] ˝Z Q Š H .U / ˝Z Q:4.2.12 Our application of this relies on the study of the two spectral sequences convergingtoExt Sh Ab S .Lcoinv.Z/; Z/ Š Ext Sh ZŒH -mod S .Z; Z/:We choose an injective resolution Z ! I in Sh Ab S. ThenExt Sh Ab S .Lcoinv.Z/; Z/ Š H .Hom ShAb S .U ;I //:The first spectral sequence denoted by .F r ;d r / is obtained by taking the cohomologyin the U -direction first. Its second page is given byF p;q2Š Ext p Sh Ab S .Lq coinv.Z/; Z/:

Duality for topological abelian group stacks and T -duality 277This page contains the object of our interest, namely by 4.2.11 the sheaves of groupsF p;12Š Ext p Sh Ab S.H; Z/:The other spectral sequence .E r ;d r / is obtained by taking the cohomology in theI -direction first. In view of (22) its first page is given byE p;q1Š Ext q Sh Ab S .Z.H p /; Z/;which can be evaluated easily in many cases.Let us note thatH RHom ShZŒH -mod S .Z; Z/ Š Ext Sh ZŒH -mod S .Z; Z/has the structure of a graded ring with multiplication given by the Yoneda product.4.2.13 We now verify Assumption 2 of Lemma 4.9 for all compact groups.Proposition 4.16. Let H 2 S be a compact group. Then we have Ext i Sh Ab S .H ; R/ Š 0for i D 1; 2.As in Definition 4.14 let U WD U .H /. Let R ! I be an injective resolution.Then we get a double complex Hom ShAb S .U ;I /.We first take the cohomology in the I -, and then in the U -direction. We get aspectral sequence with first termE p;q1Š Ext q Sh Ab S .Z.F H p /; R/:It follows from Corollary 3.28, 2., that E p;q1Š 0 for q 1.We consider the complexC .H; R/W 0 ! Map.H; R/ !!Map.H p1 ; R/ ! Map.H p ; R/ !of topological groups which calculates the continuous group cohomology Hcont .H I R/of H with coefficients in R. Now observe that by the exponential law for A 2 SExt 0 Sh Ab S .Z.F H p /; R/.A/Lemma 3.9ŠHom S .H p A; R/ Š Hom S .A; Map.H p ; R//:Hence the complex .E ;01 ;d 1/.A/ is isomorphic to the complexHom S .A; C .H; R// D C .H; R/.A/:Since H is a compact group we have Hcont i .H I R/ Š 0 for i 1. Of importance forus is a particular continuous chain contraction h p W Map.H p ; R/ ! Map.H p 1 ; R/,p 1, which is given by the following explicit formula. If c 2 Map.H p ; R/ is acocycle, then we can define h p .c/ WD b 2 Map.H p 1 ; R/ by the formulaZb.t 1 ;:::;t p 1 / WD . 1/ p c.t 1 ;:::;t p 1 ;t/dt;H

278 U. Bunke, T. Schick, M. Spitzweck, and A. Thomwhere dt is the normalized Haar measure. Then we have db D c. The maps.h p / p>0 induce a chain contraction .h p / p>0 of the complex C .H; R/. ThereforeH i Map.A; C .H; R// Š H i C .H; R/.A/ for i 1, too.The spectral sequence thus degenerates from the second page on. We conclude thatH i Hom ShAb S .U ;I / Š 0; i 1:We now take the cohomology of the double complex Hom i Sh Ab S .U ;I / in the otherorder, first in the U -direction and then in the I -direction. In order to calculate thecohomology of U in degree 2 we use the fact 4.15. We get again a spectral sequencewith second term (for q 2, using 4.2.11)F p;q2Š Ext p Sh Ab S ..ƒq Z H /] ; R/:We know by Corollary 3.28, 2., thatF p;02Š Ext p Sh Ab S.Z; R/ Š 0; p 1(note that we can write Z D Z.¹º/ for a one-point space). Furthermore note thatF 0;12Š Hom ShAb S .H ; R/ Š Hom top-Ab.H; R/ Š 0since there are no continuous homomorphisms H ! R. The second page of thespectral sequence thus has the following structure.2 Hom ShAb S ..ƒ2 Z H /] ; R/1 0 Ext 1 Sh Ab S .H ; R/ Ext2 Sh Ab S .H ; R/0 R 0 0 0 00 1 2 3 4Since the spectral sequence must converge to zero in positive degrees we see thatExt 1 Sh Ab S .H ; R/ Š 0:We claim that Hom ShAb S ..ƒ2 Z H /] ; R/ Š 0. Note that for A 2 S we haveHom ShAb S ..ƒ2 Z H /] ; R/.A/ Š Hom PrAb S .ƒ2 Z H ; R/.A/ Š Hom Pr Ab S=A.ƒ 2 Z H jA ; R jA /:An element 2 Hom PrAb S=A.ƒ 2 Z H jA ; R jA / induces a family of a biadditive (antisymmetric)maps W W H .W / H .W / ! R.W /for .W ! A/ 2 S=A which is compatible with restriction. Restriction to points givescontinuous biadditive maps H H ! R. Since H is compact the only such map isthe constant map to zero. Therefore W vanishes for all .W ! A/. This proves theclaim.Again, since the spectral sequence .F r ;d r / must converge to zero in higher degreeswe see thatF 2;12Š Ext 2 Sh Ab S .H ; R/ Š 0:This finishes the proof of the lemma.

Duality for topological abelian group stacks and T -duality 2794.3 Discrete groups4.3.1 In this subsection we study admissibility of discrete abelian groups. First weshow the easy fact that a finitely generated discrete abelian group is admissible. Inthe second step we try to generalize this result using the representation of an arbitrarydiscrete abelian group as a colimit of its finitely generated subgroups. The functorExt Sh Ab S .:::;T/ does not commute with colimits because of the presence of higherR lim-terms in the spectral sequence (26), below.And in fact, not every discrete abelian group is admissible.Lemma 4.17. For n 2 N we have Ext i Sh Ab S .Z=nZ; Z/ Š 0 for i 2.Proof. We apply the functor Ext Sh Ab S .:::;Z/ to the exact sequenceand get the long exact sequence0 ! Z ! Z ! Z=nZ ! 0Ext i 1Sh Ab S .Z; Z/ ! Exti Sh Ab S .Z=nZ; Z/ ! Exti Sh Ab S .Z; Z/ ! :Since by Theorem 4.10 Ext i Sh Ab S .Z; Z/ Š 0 for i 1, the assertion follows.Theorem 4.18. A finitely generated abelian group is admissible.Proof. The group Z=nZ is admissible, since Assumption 2 of Lemma 4.9 followsfrom Proposition 4.16, while Assumption 1 follows from Lemma 4.17. The group Zis admissible by Theorem 4.10 since we can write Z Š Z.¹º/ for a point ¹º2 S. Afinitely generated abelian group is a finite product of groups of the form Z and Z=nZfor various n 2 N. A finite product of admissible groups is admissible.4.3.2 We now try to extend this result to general discrete abelian groups using colimits.Let I be a filtered category (see [Tam94, 0.3.2] for definitions). The categorySh Ab S is an abelian Grothendieck category (see [Tam94, Theorem I.3.2.1]). The categoryHom Cat .I; Sh Ab S/ is again abelian and a Grothendieck category (see [Tam94,Proposition 0.1.4.3]). We have the adjoint paircolim W Hom Cat .I; Sh Ab S/ , Sh Ab S W C ‹ ;where to F 2 Sh Ab S there is associated the constant functor C F 2 Hom Cat .I; Sh Ab S/with value F by the functor C ‹ . The functor C ‹ also has a right-adjoint lim:C ‹ W Sh Ab S , Hom Cat .I; Sh Ab S/ W lim:This functor is left-exact and admits a right-derived functor R lim. Finally, for F 2Hom Cat .I; Sh Ab S/ we have the functorI Hom ShAb S .F;:::/W Sh Ab S ! Hom Cat .I op ; Sh Ab S/

280 U. Bunke, T. Schick, M. Spitzweck, and A. Thomwhich fits into the adjoint pairZ I˝F W Hom Cat .I op ; Sh Ab S/ ,W Sh Ab S W I Hom ShAb S .F;:::/; (25)where for G 2 Hom Cat .I op ; Sh Ab S/ the symbol R I G ˝ F 2 Sh Ab S denotes the coendof the functor I op I ! Sh Ab , .i; j / 7! G.i/˝ F.j/; correspondingly R Idenotes theend of the appropriate functor. Indeed we have for all A 2 Hom Cat .I op ; Sh Ab S/ andB 2 Sh Ab S a natural isomorphismI ZIHom ShAb S.ZA ˝ F;B/ ŠopI Hom ShAb S .A ˝ F;B/IZIŠop Hom ShAb S .A; I Hom ShAb S .F; B//IŠ Hom HomCat .I op ;Sh Ab S/.A; I Hom ShAb S .F; B//The functor I Hom ShAb S .F;:::/ is therefore also left-exact and admits a rightderivedversion.4.3.3 We say that P 2 Hom Cat .I; Sh Ab S/ is I -free if there exists a collection offlat sheaves .U.l// l2I , U.l/ 2 Sh Ab S, such that P.j/ D L l!jU.l/, and P.i !j/W L l!i U.l/ ! L l!jU.l/ maps the summand U.l/ at l ! i identically to thesummand U.l/ at l ! i ! j .Lemma 4.19. If P 2 Hom Cat .I; Sh Ab S/ is I -free, and if J 2 Sh Ab S is injective, thenI Hom ShAb S .P; J / 2 Hom Cat.I op ; Sh Ab S/ is injective.Proof. We consider an exact sequence.A W 0 ! A 0 ! A 1 ! A 2 ! 0/in Hom Cat .I op ; Sh Ab S/. Then by Equation (25) we haveHom HomCat .I op ;Sh Ab S/.A ; I Hom ShAb S .P; J // Š Hom Sh Ab S Z IA ˝ P;JExactness in Hom Cat .I op ; Sh Ab S/ is defined object-wise [Tam94, Theorem 0.1.3.1].Therefore 0 ! A 0 .i/ ! A 1 .i/ ! A 2 .i/ ! 0 is an exact complex of sheaves for alli 2 I . Since P.j/is flat for all j 2 I the complex A .i/ ˝ P.j/is exact for all pairs.i; j / 2 I I . The complex of sheaves R I A ˝ P is the complex of push-outs alongthe exact sequence of diagramsF.i!j/2Mor.I /.j / ˝ P.i/id˝P.i!j/ FA j 2I.j / ˝ P.j/A:A .i!j/˝idF i2I.i/ ˝ P.i/.A

Duality for topological abelian group stacks and T -duality 281We claim that this R I A ˝ P Š L i2I.i/ ˝ U.i/. Since U.i/ is flat for all i 2 IAthis is a sum of exact sequences and hence exact. Since J is injective we conclude thatHom ShAb S. R I A ˝P;J/is exact. So Hom HomCat .I op ;Sh Ab S/.:::; I Hom ShAb S .P; J // preservesexactness, hence I Hom ShAb S .P; J / is an injective object in Hom Cat.I op ; Sh Ab S/.In order to finish the proof it remains to show the claim. We expand the push-outdiagram by inserting the structure of P .Li!j.j / ˝ LA l!i U.l/ aWDid˝.i!j/ Ll!j.j / ˝ U.l/AbWDA .i!j/˝idL l!i.i/ ˝ U.l/AdWDA .l!i/˝id L l2I.l/ ˝ U.l/.AIt suffices to show that the lower right corner has the universal property. The lowerhorizontal map has a split s W L l2I.l/ ˝ U.l/ ! L A l!i.i/ ˝ U.l/ given by theAinclusion of summands induced by l 7! l ! id l. Two maps l ; r from the lower leftand upper right corner to some sheaf V which satisfy l ı b D r ı a must inducea unique map W L l2I.l/ ˝ U.l/ ! V such that ı d D A l and ı c D r .We have now other choice than to define WD l ı s, and this map has the requiredproperties as can be checked by an easy diagram chaise.Lemma 4.20. Every sheaf F 2 Hom Cat .I; Sh Ab S/ has an I -free resolution.Proof. It suffices to show that there exists a surjection P ! F from an I -free sheaf.We start with the surjection Z.F / ! F and note that Z.F /.i/ is flat for all i 2 I . Thenwe define the I -free sheaf P by P.i/ WD ˚l!i Z.F /.l/, and the surjection P ! Z.F /by .l ! i/ W Z.F /.l/ ! Z.F /.i/ on the summand of P.i/with index .l ! i/.Lemma 4.21. We have a natural isomorphism in D C .Sh Ab S/RHom ShAb S .colim.F /; H / Š R limRI Hom ShAb S .F; H /:Proof. Let F 2 Hom Cat .I; Sh Ab S/. By Lemma 4.20 we can choose an I -free resolutionP of F . Let H 2 Sh Ab S and H ! I be an injective resolution. Then we haveRHom ShAb S .colim F;H/ Š Hom Sh Ab S .colim F;I /:Since the category Sh Ab S is a Grothendieck abelian category [Tam94, Theorem I.3.2.1]the functor colim is exact [Tam94, Theorem 0.3.2.1]. Therefore colimP .F / ! colimFis a quasi-isomorphism. It follows thatHom ShAb S .colim F;I / Š Hom ShAb S .colim P .F /; I /:

282 U. Bunke, T. Schick, M. Spitzweck, and A. ThomBy Lemma 3.8 the restriction functor is exact and therefore commutes with colimits.We getHom ShAb S .colim P .F /; I /.A/ Š Hom ShAb S=A..colim P .F // jA ;I jA /Š Hom ShAb S=A.colim.P .F / jA /; I jA /Š lim I Hom ShAb S=A.P .F / jA ;I jA /Š lim . I Hom ShAb S .P .F /; I /.A//Š .lim I Hom ShAb S .P .F /; I //.A/;where in the last step we use that a limit of sheaves is defined object-wise. In otherwordsHom ShAb S .colim F;I / Š Hom ShAb S .colim P .F /; I / Š lim I Hom ShAb S .P .F /; I /Applying Lemma 4.19 we see that I Hom ShAb S .P .F /; I / 2 Hom Cat .I; Sh Ab S/ isinjective, andRHom ShAb S .colim F;H/ Š lim I Hom ShAb S .P .F /; I /Š R lim I Hom ShAb S .P .F /; I /Š R lim R I Hom ShAb S .F; H /:4.3.4 Lemma 4.21 implies the existence of s spectral sequence with second pageE p;q2Š R p lim R qI Hom ShAb S .F; H / (26)which converges to GrR Hom ShAb S .colim F;H/.Lemma 4.22. Let I be a category and U 2 S. Then we have a commutative diagramD C ..Sh Ab S/ I op R lim/ D C .Sh Ab S/R.U;::: /D C .Ab I op /R limR.U;::: / D C .Ab/.Proof. Since the limit of a diagram of sheaves is defined object-wise we have .U;:::/ılim D lim ı .U;:::/. We show that the two compositions R.U;:::/ı R lim ,R lim ıR.U;:::/are both isomorphic to R..U;:::/ılim/ by showing that lim and .U;:::/preserve injectives.The left-adjoint of the functor limW .Sh Ab S/ I op! Sh Ab S is the constant diagramfunctor C ::: W Sh Ab S ! .Sh Ab S/ I op which is exact. Therefore lim preserves injectives.The left-adjoint of the functor .U;:::/is the functor ˝Z Z.U /. Since Z.U /is a sheaf of flat Z-modules (it is torsion-free) this functor is exact (object-wise andtherefore on diagrams). It follows that .U;:::/preserves injectives.

Duality for topological abelian group stacks and T -duality 2834.3.5 The general remarks on colimits above are true for all sites with finite products(because of the use of Lemma 3.8). In particular we can replace S by the site S lc-acyc oflocally acyclic locally compact spaces.Lemma 4.23. Let F 2 .Sh Ab S lc-acyc / I op . If for all acyclic U 2 S lc-acyc the canonicalmap lim F.U/ ! R lim.F .U // is a quasi-isomorphism of complexes of abelian groups,then lim F ! R limF is a quasi-isomorphism.Proof. We choose an injective resolution F ! I in .Sh Ab S lc-acyc / I op . As in the proofof Lemma 4.22 for each U 2 S lc-acyc the complex I .U / is injective. If we assumethat U is acyclic, the map F.U/ ! I .U / is a quasi-isomorphism in .Ab/ I op , andR lim .F .U // Š lim.I .U //. By assumption.limF /.U / Š lim.F .U // ! lim.I .U // Š .limI /.U / Š .R limF /.U /is a quasi-isomorphism for all acyclic U 2 S lc-acyc . An arbitrary object A 2 S lc-acyc canbe covered by acyclic open subsets. Therefore limF ! limI is an quasi-isomorphismlocally on A for each A 2 S lc-acyc . Hence limF ! R limF is an isomorphism inD C .Sh Ab S lc-acyc / .4.3.6 Let D be a discrete group. Then we let I be the category of all finitely generatedsubgroups of D. This category is filtered. Let F W I ! Ab be the “identity” functor.Then we have a natural isomorphismD Š colim F:By Theorem 4.18 we know that a finitely generated group G is admissible, i.e.R q Hom ShAb S .G; T/ Š 0; q D 1; 2:Note that by Lemma 3.3 we have colimF D D. Using the spectral sequence (26) weget for p D 1; 2 thatR p Hom ShAb S .D; T/ Š Rp lim Hom ShAb S .F ; T/: (27)Let g W S lc-acyc ! S be the inclusion. Since T and D belong to S lc-acyc using Lemma 3.38we also haveg R p Hom ShAb S .D; T/ Š Rp Hom ShAb S lc-acyc.D; T/Š R p (28)lim Hom ShAb S lc-acyc.F ; T/:4.3.7 We now must study the R p lim-term. First we make the index category I slightlysmaller. Let xD D ˝Z Q DW D Q be the image of the natural map i W D ! D Q ,d 7! d ˝ 1. We observe that xD generates D Q . We choose a basis B xD of theQ-vector space D Q and let ZB xD be the Z-lattice generated by B. For a subgroupA D let xA WD i.A/ D Q . We consider the partially ordered (by inclusion) setJ WD ¹A D j A finitely generated and Q xA \ ZB xA and Œ xA W xA \ ZB < 1º:Here ŒG W H denotes the index of a subgroup H in a group G. We still let A denotethe “identity” functor AW J ! Ab.

284 U. Bunke, T. Schick, M. Spitzweck, and A. ThomLemma 4.24. We have D Š colim A.Proof. It suffices to show that J I is cofinal. Let A D be a finitely generatedsubgroup. Choose a finite subset N b B such that the Q-vectorspace Q N b D Qgenerated by N b contains xA and choose representatives b in D of the elements of N b.They generate a finitely generated group U D such that xU D i.U/ D Z N b.Wenowconsider the group G WD hU; Ai. This group is still finitely generated. Similarly, sinceQ xG D Q xU C Q xA D Q N b we haveQ xG \ ZB D Q N b \ ZB D N b xU xG:Moreover, since Q xG D Q N b D Q. xG \ ZB/, Œ xG W xG \ ZB < 1, i.e. G J . On theother hand, by construction A G.On J we define the grading w W J ! N 0 byw.A/ WD jA tors jCrk xA C Œ xA W xA \ ZB for A 2 J:Lemma 4.25. The category J together with the grading w W J ! N 0 is a directcategory in the sense of Definition 5.1.1 in [Hov99].Proof. We must show that A G implies w.A/ w.G/, and that A ¤ G impliesw.A/ < w.G/. First of all we have A tors G tors and therefore jA tors jjG tors j.Moreover we have xA xG, hence rk xA rk xG. Finally, we claim that the canonicalmapxA=. Na \ ZB/ ! xG=. xG \ ZB/is injective. In fact, we have xA \ . xG \ ZB/ D . xA \ xG/ \ ZB D xA \ ZB: It followsthatj xA W xA \ ZBj jxG W xG \ ZBj:Let now A G and w.A/ D w.G/. We want to see that this implies A D G. Firstnote that the inclusion of finite groups A tors ! G tors is an isomorphism since both groupshave the same number of elements. It remains to see that xA ! xG is an isomorphism.We have rk xA D rk xG. Therefore xA \ ZB D Q xA \ ZB D Q xG \ BZ D xG \ BZ.Now the equality Œ xA W xA \ ZB D Œ xG W xG \ ZB implies that xA D xG.4.3.8 The category C.Ab/ has the projective model structure whose weak equivalencesare the quasi-isomorphisms, and whose fibrations are level-wise surjections.By Lemma 4.25 the category J op with the grading w is an inverse category. OnC.Ab/ J opwe consider the inverse model structure whose weak equivalences are thequasi-isomorphisms, and whose cofibrations are the object-wise ones. The fibrationsare characterized by a matching space condition which we will explain in the following.For j 2 J let J j J be the category with non-identity maps all non-identity mapsin J with codomain j . Furthermore consider the functorM j W C.Ab/ J op restriction ! C.Ab/ .J j / op lim ! C.Ab/

Duality for topological abelian group stacks and T -duality 285There is a natural morphism F.j/ ! M j F for all F 2 C.Ab/ J op . The matching spacecondition asserts that a map F ! G in C.Ab/ J op is a fibration if and only ifF.j/ ! G.j / Mj G M j Fis a fibration in C.Ab/, i.e. a level-wise surjection, for all j 2 J . In particular, F isfibrant if F.j/ ! M j F is a level-wise surjection for all j 2 J .For X 2 C.Ab/ J op the fibrant replacement X ! RX induces the morphismlim X ! lim RX Š R lim X:Let now again A denote the identity functor AW J ! Ab for the discrete abeliangroup D.Lemma 4.26. 1. For all U 2 S which are acyclic the diagram of abelian groupsHom ShAb S.A; T/.U / 2 C.Ab/ J op is fibrant.2.lim Hom ShAb S lc-acyc.A; T/ ! R lim Hom ShAb S lc-acyc.A; T/ 2 C.Ab/ J opis a quasi-isomorphism.Proof. The assertion 1. for locally compact acyclic U verifies the assumption of Lemma4.23. Hence 2. follows from 1. We now concentrate on 1. We must show thatis surjective. We haveHom ShAb S .A.j /; T/.U / ! M j Hom ShAb S .A; T/.U / (29)M j Hom ShAb S .A; T/ Š lim Hom Sh Ab S .A jJ j; T/ Š Hom ShAb S .colim A jJ j; T/:The map (29) is induced by the mapcolimA jJj ! F .j /: (30)By Lemma 3.3 we have colim A jJj Š colim A jJj . The map colim A jJj ! A.j / isof course an injection.We finish the argument by the following observation. Let H ! G be an injectivemap of finitely generated groups (we apply this with H WD colimA jJj and G WD F.j/).Then for U 2 Sis a surjection. In fact we haveHom ShAb S .G; T/.U / ! Hom Sh Ab S .H ; T/.U /Hom ShAb S .G; T/.U / Š Hom top-Ab.G; T/.U / (by Lemma 3.5)Š yG.U /

286 U. Bunke, T. Schick, M. Spitzweck, and A. Thom(and a similar equation for H ). Since H ! G is injective, its Pontrjagin dual yG ! yHis surjective. Because of the classification of discrete finitely generated abelian groups,yG and yH both are homeomorphic to finite unions of finite dimensional tori. BecauseU is acyclic, every map U ! yH lifts to yG. Therefore yG.U / ! yH .U / is surjective.Corollary 4.27. We have R p lim Hom ShAb S lc-acyc.A; T/ Š 0 for all p 1.Theorem 4.28. Every discrete abelian group is admissible on S lc-acyc .Proof. This follows from (28) and Corollary 4.27.4.3.9 We now present an example which shows that not every discrete group D isadmissible on S or S lc , using Corollary 4.41 to be established later.Lemma 4.29. Let I be an infinite set, 1 6D n 2 N and D WD ˚I Z=nZ. Then D is notadmissible on S or S lc .Proof. We consider the sequence0 ! Z=nZ ! T n ! T ! 0 (31)which has no global section, and the product of I copies of it0 ! Y IZ=nZ ! Y IT ! Y IT ! 0: (32)This sequence of compact abelian groups does not have local sections. In fact, an opensubset of Q I T always contains a subset of the form U D Q I 0 T V , where I 0 I iscofinite and V Q I nI 0 T is open. A section s W U ! Q IT would consist of sectionsof the sequence (31) at the entries labeled with I 0 .By Lemma 3.4 the sequence of sheaves associated to (32) is not exact. In view ofCorollary 4.41 below, the group Q 4I Z=nZ Š˚I Z=nZ is not admissible on S or S lc .4.4 Admissibility of the groups R n and T nTheorem 4.30. The group T is admissible.Proof. Since T is compact,Assumption 2:of Lemma 4.9 follows from Proposition 4.16.It remains to show the first assumption of Lemma 4.9. As we will see this follows fromthe following result.Lemma 4.31. We have Ext i Sh Ab S .R; Z/ D 0 for i D 1; 2; 3.Let us assume this lemma for the moment. We apply Ext Sh Ab S .:::;Z/ to the exactsequence0 ! Z ! R ! T ! 0

Duality for topological abelian group stacks and T -duality 287and consider the following part of the resulting long exact sequence for i D 2; 3:Ext i 1Sh Ab S .Z; Z/ ! Exti Sh Ab S .T; Z/ ! Exti Sh Ab S .R; Z/ ! Exti Sh Ab S .Z; Z/:The outer terms vanish since RHom ShAb S .Z;F/ Š F for all F 2 Sh Ab S. Thereforeby Lemma 4.31Ext i Sh Ab S .T; Z/ Š Exti Sh Ab S .R; Z/ Š 0;and this is Assumption 1. of 4.9.It remains to prove Lemma 4.31.4.4.1 Proof of Lemma 4.31.We choose an injective resolution Z ! I . The sheafR gives rise to the homological complex U introduced in 4.14. We get the doublecomplex Hom ShAb S .U ;I / as in 4.2.12. As before we discuss the associated twospectral sequences which compute Ext Sh ZŒR-mod S .Z; Z/.4.4.2 We first take the cohomology in the I -, and then in the U -direction. In viewof (24), the first page of the resulting spectral sequence is given byE p;q1Š Ext q Sh Ab S .Z.F Rp /; Z/;where F R denotes the underlying sheaf of sets of R. By Corollary 3.28, 1. we haveE p;q1Š 0 for q 1.We now consider the case q D 0. ForA 2 S we have E p;01.A/ Š .A R p I Z/ Š.AI Z/ for all p, i.e. E p;01Š Z. We can easily calculate the cohomology of thecomplex .E ;01 ;d 1/ which is isomorphic to0 ! Z 0 ! Z id ! Z 0 ! Z id ! Z ! :We getH i .E ;01 ;d 1/ Š´Z; i D 0;0; i 1:The spectral sequence .E r ;d r / thus degenerates at the second term, and´H i Hom ShAb S .U Z; i D 0;;I / Š0; i 1:(33)4.4.3 We now consider the second spectral sequence .Frp;q ;d r / associated to the doublecomplex Hom ShAb S .U ;I / which takes cohomology first in the U and then in theI -direction. Its second page is given byF p;q2Š Ext p Sh Ab S .H q U ; Z/:Since R is torsion-free we can apply 4.2.11 and getF p;q2Š Ext p Sh Ab S ..ƒq Z R/] ; Z/:

288 U. Bunke, T. Schick, M. Spitzweck, and A. ThomIn particular we have ƒ 0 ZR Š Z and thusF p;02Š Ext p Sh Ab S.Z; Z/ Š 0for p 1 (recall that Ext p Sh Ab S .Z;H/ Š 0 for every H 2 Sh Ab S and p 1).Furthermore, since ƒ 1 ZR Š R, by Lemma 3.5 we haveF 0;12Š Hom ShAb S .R; Z/ Š Hom top-Ab.R; Z/ Š 0since Hom top-Ab .R; Z/ Š 0.Here is a picture of the relevant part of the second page.3 Hom ShAb S ..ƒ3 Z R/] ; Z/2 Hom ShAb S ..ƒ2 Z R/] ; Z/ Ext 1 Sh Ab S ..ƒ2 Z R/] ; Z/1 0 Ext 1 Sh Ab S .R; Z/ Ext2 Sh Ab S .R; Z/ Ext3 Sh Ab S .R; Z/0 Z 0 0 0 0 00 1 2 3 4 54.4.4 Let V be an abelian group. Recall the definition of ƒ V from 4.2.9. If V has thestructure of a Q-vector space, then T 1ZV has the structure of a graded Q-vector space,and I T 1ZV is a graded Q-vector subspace. Therefore ƒ1ZV has the structure ofa graded Q-vector space, too.4.4.5 We claim thatfor i 1. Note thatLet A 2 S. An elementF 0;i2Š Hom ShAb S ..ƒi Z R/] ; Z/ Š 0Hom ShAb S ..ƒi Z R/] ; Z/ Š Hom PrAb S .ƒi ZR; Z/: 2 Hom PrAb S .ƒi Z R; Z/.A/ Š Hom Pr Ab S=A.ƒ i Z R jA ; Z jA /induces a homomorphism of groups W W ƒ i ZR.W / ! Z.W / for every .W ! A/ 2S=A. Since ƒ i ZR.W / is a Q-vector space and Z.W / does not contain divisible elementswe see that W D 0. This proves the claim.The claim implies that F 2;12Š Ext 2 Sh Ab S .R; Z/ survives to the limit of the spectralsequence. Because of (33) it must vanish. This proves Lemma 4.31 in the case i D 2.The term F 1;12Š Ext 1 Sh Ab S .R; Z/ also survives to the limit and therefore alsovanishes because of (33). This proves Lemma 4.31 in the case i D 1.

Duality for topological abelian group stacks and T -duality 2894.4.6 Finally, since F 0;33Š 0, we see that d 2 W F 1;22! F 3;12must be an isomorphism,i.e.d 2 W Ext 1 Sh Ab S ..ƒ2 Z R/] ; Z/ ! Ext 3 Sh Ab S .R; Z/:We will finish the proof of Lemma 4.31 for i D 3 by showing that d 2 D 0.4.4.7 We consider the natural action of Z mult on R and hence on R. This turns R intoa sheaf of Z mult -modules of weight 1 (see 3.5.1). It follows that Hom ShAb S / is a .R;Icomplex of sheaves of Z mult -modules of weight 1. Finally we see that Ext i Sh Ab S .R; Z/are sheaves of Z mult -modules of weight 1 for i 0.Now observe (see 3.5.5) that ƒ 2 Z R is a presheaf of Z mult-modules of weight 2. Hence.ƒ 2 Z R/] and thus Ext 1 Sh Ab S ..ƒ2 Z R/] ; Z/ are sheaves of Z mult -modules of weight 2.Since R ! U .R/ DW U is a functor we get an action of Z mult on U and henceon the double complex Hom ShAb S .U ;I /. This implies that the differentials of theassociated spectral sequences commute with the Z mult -actions (see 4.6.11 for moredetails). This in particular applies to d 2 . The equality d 2 D 0 now directly followsfrom the following lemma.Lemma 4.32. Let V;W 2 Sh Ab S be sheaves of Z mult -modules of weights k 6D l.Assume that W has the structure of a sheaf of Q-vector spaces. If d 2 Hom ShAb S.V; W /is Z mult -equivariant, then d D 0.Proof. Let ˛ W Z mult ! End ShAb S.V / and ˇ W Z mult ! End ShAb S.W / denote the actions.Then we have d ı ˛.q/ ˇ.q/ ı d D 0 for all q 2 Z mult . We consider q WD 2.Then .2 k 2 l / ı d D 0. Since W is a sheaf of Q-vector spaces and .2 k 2 l / 6D 0 thisimplies that d D 0.Theorem 4.33. The group R is admissible.Proof. The outer terms of the exact sequence0 ! Z ! R ! T ! 0are admissible by Theorem 4.30 and Theorem 4.18. We now apply Lemma 4.5, stabilityof admissibility under extensions.4.5 Profinite groupsDefinition 4.34. A topological group G is called profinite if there exists a small leftfiltered8 poset 9 I and a system F 2 Ab I such that1. for all i 2 I the group F.i/is finite,2. for all i j the morphism F.i/ ! F.j/is a surjection,3. there exists an isomorphism G Š lim i2I F.i/as topological groups.8 i.e. for every pair i;k 2 I there exists j 2 I with j i and j k9 A poset is considered here as a category.

290 U. Bunke, T. Schick, M. Spitzweck, and A. ThomFor the last statement we consider finite abelian groups as topological groups withthe discrete topology. Note that the homomorphisms G ! F.i/ are surjective for alli 2 I . We call the system F 2 Ab I an inverse system.Lemma 4.35 ([HR63]). The following assertions on a topological abelian group Gare equivalent:1. G is compact and totally disconnected.2. Every neighbourhood U G of the identity contains a compact subgroup Ksuch that G=K is a finite abelian group.3. G is profinite.Lemma 4.36. Let G be a profinite abelian group and n 2 Z. We define the groups K; Qas the kernel and cokernel of the multiplication map by n, i.e. by the exact sequenceThen K and Q are again profinite.0 ! K ! G n ! G ! Q ! 0: (34)Proof. We write G WD lim j 2J G j for an inverse system .G j / j 2J of finite abelian groups.We define the system of finite subgroups .K j / j 2J by the sequences0 ! K j ! G jn! Gj :Since taking kernels commutes with limits the natural projections K ! K j , j 2 Jrepresent K as the limit K Š lim j 2J K j .Since cokernels do not commute with limits we will use a different argument for Q.Since G is compact and the multiplication by n is continuous, nG G is a closedsubgroup. Therefore the group theoretic quotient Q is a topological group in thequotient topology.A quotient of a profinite group is again profinite [HM98], Exercise E.1.13. Here isa solution of this exercise, using the following general structural result about compactabelian groups.Lemma 4.37 ([HR63]). If H is a compact abelian group, then for every open neighbourhood1 2 U H there exists a compact subgroup C U such that H=C ŠT a F for some a 2 N 0 and a finite group F .Since G is compact, its quotient Q is compact, too. This lemma in particularimplies that Q is the limit of the system of these quotients Q=C . In our case, since Gis profinite, it can not have T a as a quotient, i.e given a surjectionG ! Q ! T a Fwe conclude that a D 0. Hence we can write Q as a limit of an inverse system of finitequotients. This implies that Q is profinite.

Duality for topological abelian group stacks and T -duality 2914.5.1 Let0 ! K ! G ! H ! 0be an exact sequence of profinite groups, where K ! G is the inclusion of a closedsubgroup.Lemma 4.38. The sequence of sheavesis exact.0 ! K ! G ! H ! 0Proof. By [Ser02, Proposition 1] every surjection between profinite groups has a section.Hence we can apply Lemma 3.4.In this result one can in fact drop the assumptions that K and G are profinite. In ourbasic example the group K is the connected component G 0 G of the identity of G.Lemma 4.39. Let0 ! K ! G ! H ! 0be an exact sequence of compact abelian groups such that H is profinite. Then thesequence of sheaves0 ! K ! G ! H ! 0is exact.Proof. We can apply [HM98, Theorem 10.35] which says that the projection G ! Hhas a global section. Thus the sequence of sheaves is exact by 3.4 (even as sequenceof presheaves).4.5.2 In order to show that certain discrete groups are not admissible we use thefollowing lemma.Lemma 4.40. Let0 ! K ! G ! H ! 0be an exact sequence of compact abelian groups. If the discrete abelian group yK isadmissible, then sequence of sheavesis exact.0 ! K ! G ! H ! 0Proof. If U is a compact group, then its Pontrjagin dual yU WD Hom top-Ab .U; T/ is adiscrete group. Pontrjagin duality preserves exact sequences. Therefore we have theexact sequence of discrete groups0 ! yH ! yG ! yK ! 0:

292 U. Bunke, T. Schick, M. Spitzweck, and A. ThomThe surjective map of discrete sets yG ! yK has of course a section. Therefore byLemma 3.4 the sequence of sheaves of abelian groups0 ! yH ! yG ! yK ! 0is exact. We apply Hom ShAb S .:::;T/ and get the exact sequence0 ! Hom ShAb S . yK; T/ ! Hom ShAb S . yG; T/! Hom ShAb S . yH ; T/ ! Ext 1 Sh Ab S . yK; T/ ! :By Lemma 3.5 we have Hom ShAb S . yG; T/ Š G etc. Therefore this sequence translatesto0 ! K ! G ! H ! Ext 1 Sh Ab S . yK; T/ ! : (35)By our assumption Ext 1 Sh Ab S . yK; T/ D 0 so that G ! H is surjective.4.5.3 The same argument does apply for the site S lc . Evaluating the surjection G ! Hon H we conclude the following fact.Corollary 4.41. If K G is a closed subgroup of a compact abelian group suchthat yK is admissible or admissible over S lc , then the projection G ! G=K has localsections.4.5.4 If G is an abelian group, then let G tors G denote the subgroup of torsionelements. We call G a torsion group, if G tors D G. IfG tors Š 0, then we say that G istorsion-free. If G is a torsion group and H is torsion-free, then Hom Ab .H; G/ Š 0.A presheaf F 2 Pr S S is called a presheaf of torsion groups if F .A/ is a torsiongroup for every A 2 S. The notion of a torsion sheaf is more complicated.Definition 4.42. A sheaf F 2 Sh Ab S is called a torsion sheaf if for each A 2 S andf 2 F .A/ there exists an open covering .U i / i2I of A such that f jUi 2 F.U i / tors .The following lemma provides equivalent characterizations of torsion sheaves.Lemma 4.43 ([Tam94], (9.1)). Consider a sheaf F 2 Sh Ab S. The following assertionsare equivalent.1. F is a torsion sheaf.2. F is the sheafification of a presheaf of torsion groups.3. The canonical morphism colim n2N . n F/! F is an isomorphism, where . n F/ n2Nis the direct system n F WD ker.F nŠ! F/.Note that a subsheaf or a quotient of a torsion sheaf is again a torsion sheaf.Lemma 4.44. If H is a discrete torsion group, then H is a torsion sheaf.

Duality for topological abelian group stacks and T -duality 293Proof. We write H D colim n2N . n H/, where n H WD ker.H nŠ! H/. By Lemma 3.3we have H D colim n2N n H . Now colim n2N n H is the sheafification of the presheafp colim n2N n H of torsion groups and therefore a torsion sheaf.In general H is not a presheaf of torsion groups. Consider e.g. H WD ˚n2N Z=nZ.Then the element id 2 H .H / is not torsion.Definition 4.45. A sheaf F 2 Sh Ab S is called torsion-free if the group F .A/ is torsionfreefor all A 2 S.A sheaf F is torsion-free if and only if for all n 2 N the map F ! n F is injective.It suffices to test this for all primes.Lemma 4.46. If F 2 Sh Ab S is a torsion sheaf and E 2 Sh Ab S is torsion-free, thenHom ShAb S .F; E/ Š 0.Proof. We haveHom ShAb S .F; E/ Š Hom Sh Ab S .colim n2N. n F/;E/Š lim n2N Hom ShAb S . nF;E/:On the one hand, via the first entry multiplication by nŠ induces on Hom ShAb S . nF;E/the trivial map. On the other hand it induces an injection since E is torsion-free.Therefore Hom ShAb S . nF;E/ Š 0 for all n 2 N. This implies the assertion.4.5.5 If F 2 ShS and C 2 S, then we can form the sheaf R C .F / 2 ShS by theprescription R C .F /.A/ WD F.A C/ for all A 2 S (see 3.3.18). We will show thatR C preserves torsion sheaves provided C is compact.Lemma 4.47. If C 2 S is compact and H 2 Sh Ab S is a torsion sheaf, then R C .H / isa torsion sheaf.Proof. Given s 2 R C .H /.A/ D H.A C/ and a 2 A we must find n 2 N and aneighbourhood U of a such that .ns/ jU C D 0. Since C is compact and A 2 S iscompactly generated by assumption on S, the compactly generated topology (this is thetopology we use here) on the product AC coincides with the product topology ([Ste67,Theorem 4.3]). Since H is torsion there exists an open covering .W i D A i C i / i2Iof A C and a family of non-zero integers .n i / i2I such that .n i s/ jWi D 0.The set of subsets of C¹C i j i 2 I; a 2 A i ºforms an open covering of C . Using the compactness of C we choose a finite seti 1 ;:::;i r 2 I with a 2 A ik such that ¹C ik j k D 1;:::rº is still an open coveringof C . Then we define the open neighbourhood U of a 2 A by U WD T rkD1 A i k. Setn WD Q rkD1 n i k. Then we have .ns/ U C D 0.Lemma 4.48. Let H be a discrete torsion group and G 2 S be a compactly generated 10group. Then the sheaf Hom ShAb S .G;H/ is a torsion sheaf.10 i.e. generated by a compact subset

294 U. Bunke, T. Schick, M. Spitzweck, and A. ThomProof. Let C G be a compact generating set of G. Precomposition with the inclusionC ! G gives an inclusion Hom top-Ab .G; H / ! Map.C; H /. WehaveforA 2 SHom ShAb S .G;H/.A/ Š Hom top-Ab.G; H /.A/ (by Lemma 3.5)Š Hom S .A; Hom top-Ab .G; H // Hom S .A; Map.C; H //D Hom S .A C;H/Š H .A C/Š R C .H /.A/:By Lemma 4.44 the sheaf H is a torsion sheaf. It follows from Lemma 4.47 thatR C .H / is a torsion sheaf. By the calculation above Hom ShAb S .G;H/ is a sub-sheafof the torsion sheaf R C .H / and therefore itself a torsion sheaf.Lemma 4.49. If G is a compact abelian group, thenExt 1 Sh Ab S .G; Z/ Š Hom Sh Ab S .G; T/ Š yG:Moreover, if G is profinite, then yG is a torsion sheaf.Proof. We apply the functor Ext Sh Ab S .G;:::/to0 ! Z ! R ! T ! 0and get the following segment of a long exact sequence!Hom ShAb S .G; R/ ! Hom Sh Ab S .G; T/! Ext 1 Sh Ab S .G; Z/ ! Ext1 Sh Ab S .G; R/ ! :Since G is compact we have 0 D Hom top-Ab .G; R/ Š Hom ShAb S .G; R/ by Lemma 3.5.Furthermore, by Proposition 4.16 we have Ext 1 Sh Ab S .G; R/ D 0. Therefore we getExt 1 Sh Ab S .G; Z/ Š Hom Sh Ab S .G; T/ Š Hom top-Ab.G; T/ Š yG;again by Lemma 3.5. If G is profinite, then (and only then) its dual yG is a discretetorsion group by [HM98, Corollary 8.5]. In this case, by Lemma 4.44 the sheaf yG is atorsion sheaf.4.5.6 Let H be a discrete group. For A 2 S we consider the continuous group cohomologyHcont i .GI Map.A; H //, which is defined as the cohomology of the groupcohomology complex0 ! Map.A; H / ! Hom S .G; Map.A; H // !!Hom S .G i ; Map.A; H // !with the differentials dual to the ones given by 4.2.7. The mapS 3 A 7! Hcont i .GI Map.A; H //defines a presheaf whose sheafification we will denote by H i .G; H /.

Duality for topological abelian group stacks and T -duality 295Lemma 4.50. If G is profinite and H is a discrete group, then H i .G; H / is a torsionsheaf for i 1.Proof. We must show that for each section s 2 Hcont i .GI Map.A; H // and a 2 A thereexists a neighbourhood U of a and a number l 2 Z such that .ls/ jU D 0. Thisadditional locality is important. Note that by the exponential law we haveCcont i .G; Map.A; H // WD Hom S.G i ; Map.A; H // Š Hom S .G i A; H /:Let s 2 Hcont i .GI Map.A; H // and a 2 A. Let s be represented by a cycle Os 2Ccont i .G; Map.A; H //. Note that Os W Gi A ! H is locally constant. The sets ¹Os 1 .h/ jh 2 H º form an open covering of G i A.Since G and therefore G i are compact and A 2 S is compactly generated byassumption on S, the compactly generated topology (this is the topology we use here)on the product G i A coincides with the product topology ([Ste67, Theorem 4.3]).Since G i ¹aº G i A is compact we can choose a finite set h 1 ;:::;h r 2 H suchthat ¹Os 1 .h i / j i D 1;:::;rº covers G i ¹aº. Now there exists an open neighbourhoodU A of a such that G i U S rsD1 Os 1 .h i /.OnG i U the function Os has at mosta finite number of values belonging to the set ¹h 1 ;:::;h r º.Since G is profinite there exists a finite quotient group G ! xG such that Os jG i Uhas a factorization Ns W xG i U ! H . Note that Ns is a cycle in Ccont i . xG;Map.U; H //.Now we use the fact that the higher (i.e. in degree 1) cohomology of a finitegroup with arbitrary coefficients is annihilated by the order of the group. Hence j xGjNs isthe boundary of some Nt 2 Ccont i 1.xG;Map.U; H //. Pre-composing Nt with the projectionG i U ! xG i U we get Ot 2 Ccont i .G; Map.U; H // whose boundary is Os. This showsthat .j xGjs/ jU D 0.4.5.7 Let G be a profinite group. We consider the complex U WD U .G/ as definedin 4.14. Let Z ! I be an injective resolution. Then we get a double complexHom ShAb S .U ;I / as in 4.2.12.Lemma 4.51. For i 1H i Hom ShAb S .U ;I / is a torsion sheaf: (36)Proof. We first take the cohomology in the I -, and then in the U -direction. The firstpage of the resulting spectral sequence is given byE r;q1Š Ext q Sh Ab S .Z.Gr /; Z/:Since the G r are profinite topological spaces, by Lemma 3.29 we get E r;q1Š 0 forq 1. We now consider the case q D 0. ForA 2 S we haveE r;01 .A/ Š Z.A Gr / Š Hom S .G r A; Z/ Š Hom S .G r ; Map.A; Z//for all r 0. The differential of the complex .E ;01 .A/; d 1/ is exactly the differential ofthe complex Ccont i .G; Map.A; Z// considered in 4.5.6. By Lemma 4.50 we conclude thatthe cohomology sheaves E i;02are torsion sheaves. The spectral sequence degenerates atthe second term. We thus have shown that H i Hom ShAb S .U ;I / is a torsion sheaf.

296 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.5.8 For abelian groups V;W let V Z W WD Tor Z 1 .V; W / denote the Tor-product. If Vand W in addition are Z mult -modules then V W is a Z mult -module by the functorialityof the Tor-product.Lemma 4.52. If V;W are Z mult -modules of weight k; l, then V ˝Z W and V Z Ware of weight k C l.Proof. The assertion for the tensor product follows from (19). Let P ! V 0 andQ ! W 0 be projective resolutions of the underlying Z-modules V 0 ;W 0 of V;W .Then P .k/ ! V and Q .l/ ! W are Z mult -equivariant resolutions of V and W .Wehave by (19)V Z W Š H 1 .P .k/ ˝Z Q .l// Š H 1 .P ˝Z Q /.k C l/:4.5.9 The tautological action of Z mult on an abelian group G (we write G.1/ forthis Z mult -module) (see 3.5.1) induces an action of Z mult on the bar complex Z.G / bydiagonal action on the generators, and therefore on the group cohomology H .G.1/I Z/of G.In the following lemma we calculate the cohomology of the group Z=pZ.1/ as aZ mult -module.Lemma 4.53. Let p 2 N be a prime. Then we have8ˆ< Z.0/; i D 0;H i .Z=pZ.1/I Z/ Š 0; i odd,ˆ:Z=pZ.k/; i D 2k 2:Proof. The group cohomology of Z=pZ can be identified as a ring with the ring Z ˚cZ=pZŒc, where c has degree 2. Furthermore, for every group there is a canonicalmap yG ! H 2 .GI Z/ which in the case G Š Z=pZ happens to be an isomorphism.This implies that H 2 .Z=pZ.1/I Z/ has weight 1 as a Z mult -module. Since the cupproduct in group cohomology is natural it is Z mult -equivariant. Therefore, the powerc k generates a module of weight k. Hence H 2k .Z=pZ.1/I Z/ has weight k.4.5.10 For a Z mult -module V (with Z mult -action ‰) let P v k be the operator x 7! ‰v xv k x. Note that this is a commuting family of operators. We let M23 v be the monoidgenerated by P2 v;Pv 3 .Definition 4.54. A Z mult -module V is called a weight 2-3-extension if for all x 2 Vand all v 2 Z mult there is P v 2 M23 v such thatP v x D 0: (37)A sheaf of Z mult -modules is called a weight 2-3-extension, if every section locallysatisfies Equation (37) (with P v depending on the section and the neighborhood).In the tables below we will mark weight 2-3-extensions with attribute weight 2-3.

Duality for topological abelian group stacks and T -duality 297Lemma 4.55. Every Z mult -module of weight 2 or of weight 3 is a weight 2-3-extension.The class of weight 2-3-extensions is closed under extensions.Proof. This is a simple diagram chase, using the fact that for a Z mult -module W ofweight k, ‰ v x D v k x for all k 2 Z mult and x 2 W .Lemma 4.56. Let n 2 N and V be a torsion-free Z-module. For i 2¹0; 2; 3º thecohomology H i ..Z=pZ/ n .1/I V/ is a Z=pZ-module whose weight is given by thefollowing table.Moreover, H 1 ..Z=pZ/ n I V/Š 0.i 0 2 3 4weight 0 1 2 2-3Proof. We first calculate H i ..Z=pZ/ n I Z/ using the Künneth formula and inductionby n 1. The start is Lemma 4.53. Let us assume the assertion for products with lessthan n factors. The cases i D 0; 1; 2 are straightforward. We further getH 3 ..Z=pZ/ n .1/I Z/ Š H 2 .Z=pZ.1/I Z/ Z H 2 ..Z=pZ/ n1 .1/I Z/:By Lemma 4.52 the Z -product of two modules of weight 1 is of weight 2. Similarly,we have an exact sequence0 !4MH j .Z=pZ n 1 .1/I Z/ ˝ H 4 j .Z=pZI Z/ ! H 4 ..Z=pZ/ n .1/I Z/j D0!3MH j .Z=pZ.1/I Z/ Z H 5 j ..Z=pZ/ n 1 .1/I Z/ ! 0:j D2By Lemma 4.55 and induction we conclude that H 4 ..Z=pZ/ n .1/I Z/ is a weight 2-3-extension.We can calculate the cohomology H i .GI V/of a group G in a trivial G-moduleby the standard complex C .GI V/ WD C.Map.G ;V//. If G is finite, then we haveC .GI V/Š C .GI Z/ ˝Z V .IfV is torsion-free, it is a flat Z-module and thereforeH i .C .GI Z/ ˝Z V/ Š H i .C .GI Z// ˝Z V . Applying this to G D .Z=pZ/ n weget the assertion from the special case Z D V .4.5.11 An abelian group G is a Z-module. Let p 2 Z be a prime. If pg D 0 for everyg 2 G, then we say that G is a Z=pZ-module.Definition 4.57. A sheaf F 2 Sh Ab S is a sheaf of Z=pZ-modules if F .A/ is a Z=pZmodulefor all A 2 S.

298 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.5.12 Below we consider profinite abelian groups which are also Z=pZ modules.The following lemma describes their structure.Lemma 4.58. Let p be a prime number. If G is compact and a Z=pZ-module, thenthere exists a set S such that G Š Q SZ=pZ. In particular, G is profinite.Proof. The dual group yG of the compact group G is discrete and also a Z=pZ-module,hence an F p -vector space. Let S yG be an F p -basis. Then we can write yG Š˚SZ=pZ. Pontrjagin duality interchanges sums and products. We getG Š yG Š 3˚SZ=pZ Š Y S1Z=pZ Š Y SZ=pZ:4.5.13 Assume that G is a compact group and a Z=pZ-module. Note that the constructionG ! U WD U .G/ is functorial in G (see 4.6.11 for more details). Thetautological action of Z mult on G induces an action of Z mult on U . We can improveLemma 4.51 as follows.Lemma 4.59. Let Z ! I be an injective resolution. For i 2¹2; 3; 4º the sheavesH i Hom ShAb S .U ;I /are sheaves of Z=pZ-modules and Z mult -modules whose weights are given by thefollowing table.i 0 2 3 4Moreover, H 1 Hom ShAb S .U ;I / Š 0.weight 0 1 2 2-3Proof. We argue with the spectral sequence as in the proof of 4.51. Since E p;q1D 0for q 1, it suffices to show that the sheaves E i;02are sheaves of Z=pZ-modulesand Z mult -modules of weight k, where k corresponds to i as in the table (and with theappropriate modification for i D 4).We have for A 2 SE i;01 .A/ Š Z.A Gi / Š Hom S .G i ; Map.A; Z// D Ccont i .GI Map.A; Z//;where for a topological G-module V the complex C cont .GI V/denotes the continuousgroup cohomology complex. We now consider the presheaf zX i defined byA 7! zX i .A/ WD Hcont i .G; Map.A; Z//:Then by definition E i;02WD X i WD . zX i / ] is the sheafification of zX i .The action of Z mult on G induces an action q 7! Œq on the presheaf zX i , whichdescends to an action of Z mult on the associated sheaf X i .We fix i 2¹0; 1; 2; 3º and let k be the associated weight as in the table. For i 2we must show that for each section s 2 Hcont i .GI Map.A; Z// and a 2 A there exists a

Duality for topological abelian group stacks and T -duality 299neighbourhood U of a such that ..q k Œq/s/ jU D 0 and .ps/ jU D 0 for all q 2 Z mult .For i D 4, instead of ..q k Œq/s/ jU D 0 we must find P v 2 M23 v (depending on Uand s) (see 4.5.10 for notation) such that P v s jU D 0. For i D 1 we must show thats jU D 0.We perform the argument in detail for i D 2; 3. It is a refinement of the proof ofLemma 4.50. The cases i D 1 and i D 4 are very similar.Let s be represented by a cycle Os 2 Ccont i .G; Map.A; Z//. Note that Os W Gi A ! Zis locally constant. The sets ¹Os 1 .h/ j h 2 Zº form an open covering of G i A. SinceG and therefore G i is compact and A 2 S is compactly generated by assumption onS, the compactly generated topology (this is the topology we use here) on the productG i A coincides with the product topology ([Ste67, Theorem 4.3]). This allows thefollowing construction.The set of subsets¹Os 1 .h/ j h 2 Zºforms an open covering of G i A. Using the compactness of the subset G i ¹aºG i A we choose a finite set h 1 ;:::;h r 2 Z such that G i ¹aº S riD1 Os 1 .h i /.There exists a neighbourhood U A of a such that G i U S riD1 Os 1 .h i /.We now use that G is profinite by Lemma 4.58. Since Os.G i U/is a finite set (asubset of ¹h 1 ;:::;h r º) there exists a finite quotient group G ! xG such that Os jG i U hasa factorization Ns W xG i U ! Z. In our case xG Š .Z=pZ/ r for a suitable r 2 N. Notethat Ns is a cycle in Ccont i . xG;Map.U; Z//. Now by Lemma 4.56 we know that .q k Œq/Nsand pNs are the boundaries of some Nt; Nt 1 2 Ccont i 1.xG;Map.U; Z//. Pre-composing Nt; Nt 1with the projection G i U ! xG i U we get Ot; Ot 1 2 Ccont i .G; Map.U; Z// whoseboundaries are .q k Œq/Os and pOs, respectively. This shows that .q k Œqs/ jU D 0and .ps/ jU D 0.4.5.14 We consider a compact group G and form the double complex Hom ShAb S.U ;I /defined in 4.5.7. Taking the cohomology first in the U and then in the I -direction weget a second spectral sequence .Frp;q ;d r / with second pageF p;q2Š Ext p Sh Ab S .H q U ; Z/:In the following we calculate the term F 1;22and show the vanishing of several otherentries.Lemma 4.60. The left lower corner of F p;q2has the following form.3 02 0 Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/1 0 Ext 1 Sh Ab S .G; Z/ Ext2 Sh Ab S .G; Z/ Ext3 Sh Ab S .G; Z/0 Z 0 0 0 0 00 1 2 3 4 5

300 U. Bunke, T. Schick, M. Spitzweck, and A. ThomProof. The proof is similar to the corresponding argument in the proof of 4.16. Using4.2.11 we getF p;q2Š Ext p Sh Ab S ..ƒq Z G/] ; Z/for q D 0; 1; 2. In particular we have ƒ 0 ZG Š Z and thusFurthermore, since ƒ 1 ZG Š G we haveF p;02Š Ext p Sh Ab S.Z; Z/ Š 0:F 0;1Lemma 3.52Š Hom ShAb S .G; Z/ Š Hom top-Ab .G; Z/ Š 0since Hom top-Ab .G; Z/ Š 0 by compactness of G.We claim thatF 0;22Š Hom ShAb S ..ƒ2 Z G/] ; Z/ Š 0:Note that for A 2 S we haveHom ShAb S ..ƒ2 Z G/] ; Z/.A/ Š Hom PrAb S .ƒ2 Z G; Z/.A/ Š Hom Pr Ab S=A.ƒ 2 Z G jA ; Z jA /:An element 2 Hom PrAb S=A.ƒ 2 Z G jA ; Z jA / induces a family of biadditive (antisymmetric)maps W W G.W / G.W / ! Z.W /for .W ! A/ 2 S=A which is compatible with restriction. Restriction to points givesbiadditive maps G G ! Z. Since G is compact the only such map is the constantmap to zero. Therefore W vanishes for all .W ! A/. This proves the claim. Thesame kind of argument shows that Hom ShAb S ..ƒq Z G/] ; Z/.A/ D 0 for q 2.Let us finally show that F 0;32Š Hom ShAb S ..H 3 U / ] ; Z/ Š 0. By 4.15 we have anexact sequence (see (23) for the notation p D )0 ! K ! ƒ 3 Z G ! H 3 . p D / ! C ! 0;where K and C are defined as the kernel and cokernel presheaf, and the middle mapbecomes an isomorphism after tensoring with Q. This means that 0 Š K ˝ Q and0 Š C ˝ Q. Hence, these K and C are presheaves of torsion groups. Let A 2 S. Thenan element s 2 Hom ShAb S ..H 3 U / ] ; Z/.A/ induces by precomposition an element Qs 2Hom ShAb S ..ƒ3 Z G/] ; Z/.A/ Š 0. Therefore s factors over Ns 2 Hom ShAb S .C ] ; Z/.A/.Since C is torsion we conclude that Ns D 0 by Lemma 4.46. The same argument showsthat Hom ShAb S ..H q U / ] ; Z/ Š 0 for q 3.Lemma 4.61. Assume that p is an odd prime number, p ¤ 3. IfG is a compact groupand a Z=pZ-module, then Ext i Sh Ab S .GI Z/ Š 0 for i D 2; 3.If p D 3, then at least Ext 2 Sh Ab S .G; Z/ Š 0.

Duality for topological abelian group stacks and T -duality 301Proof. We consider the spectral sequence .F r ;d r / introduced in 4.5.14. It convergesto the graded sheaves associated to a certain filtrations of the cohomology sheaves ofthe total complex of the double complex Hom ShAb S .U ;I / defined in 4.5.7. In degree2; 3; 4 these cohomology sheaves are sheaves of Z=pZ-modules carrying actions ofZ mult with weights determined in Lemma 4.59.The left lower corner of the second page of the spectral sequence was evaluated inLemma 4.60. Note that Ext i Sh Ab S .G; Z/ is a sheaf of Z=pZ-modules with an action ofZ mult of weight 1 for all i 0. The term F 2;12Š Ext 2 Sh Ab S .G; Z/ survives to the limitof the spectral sequence and is a submodule of a sheaf of Z=pZ-modules of weight 2.On the other hand it has weight 1.A Z=pZ-module V with an action of Z mult which has weights 1 and 2 at the sametime must be trivial 11 . In fact, for every q 2 Z mult we get the identity q 2 q D.q 1/q D 0 in End Z .V /, and this implies that q 1.mod p/ for all q 62 pZ. Fromthis follows p D 2, and this case was excluded.Similarly, the sheaf Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ is a sheaf of Z=pZ-modules of weight 2(see 3.5.5), and Ext 3 Sh Ab S .G; Z/ is a sheaf of Z=pZ-modules of weight 1. Since p isodd, this implies that the differential d 1;32W Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ ! Ext 3 Sh Ab S .G; Z/is trivial. Hence, Ext 3 Sh Ab S .G; Z/ survives to the limit. It is a subsheaf of a sheaf ofZ=pZ-modules which is a weight 2-3-extension. On the other hand it has weight 1.Substituting the weight 1-condition into equation (37), because then P2 v D v v2 D.1 v/v and P3 v D v v3 D .1 v/.1 C v/v this implies that locally every sectionsatisfies .1 v/ n .1 C v/ j v n s D 0 for suitable n; j 2 N (depending on s and theneighborhood) and for all v 2 Z mult . If p>3we can choose v such that .1 v/,.1 C v/, v are simultaneously units, and in this case the equation implies that locallyevery section is zero.We conclude that Ext 3 Sh Ab S .G; Z/ Š 0.Lemma 4.62. Let G be a compact group which satisfies the two-three condition (seeDefinition 4.6). Assume further that1. G is profinite, or2. G is connected and locally topologically divisible (Definition 4.7).Then the sheaves Ext i Sh Ab S .G; Z/ are torsion-free for i D 2; 3.Proof. We must show that for all primes p and i D 2; 3 the maps of sheavesExt i Sh Ab S .G; Z/ ! p Ext i Sh Ab S .G; Z/are injective. The multiplication by p can be induced by the multiplicationp W G ! G:11 Note that in general a Z=pZ-module can very well have different weights. E.g a module of weight khas also weight pk.

302 U. Bunke, T. Schick, M. Spitzweck, and A. ThomWe consider the exact sequence0 ! K ! G p ! G ! C ! 0:The groups K and C are groups of Z=pZ-modules and therefore profinite by 4.58.We claim that the sequence of sheaves0 ! K ! G p ! G ! C ! 0is exact. We first show the claim in the case that G is profinite. Since C is profinite,by Lemma 4.39 and Lemma 3.4 we know that0 ! G=K ! G ! C ! 0is exact. Furthermore G=K is profinite so that the projection G ! G=K has localsections. This implies that G=K Š G=K, and hence the claim.We now discuss the case of a connected locally topologically divisible G. In thiscase C D¹1º. Since by assumption p W G ! G has local sections we can again useLemma 3.4 in order to conclude.We decompose this sequence into two short exact sequences0 ! K ! G ! Q ! 0; 0 ! Q ! G ! C ! 0:Now we apply the functor Ext Sh Ab S .:::;Z/ to the first sequence and study the associatedlong exact sequenceExt i 11Sh Ab S .G; Z/ ! ExtiSh Ab S .K; Z/ ! Exti Sh Ab S .Q; Z/ ! Exti Sh Ab S .G; Z/ ! :Note that Ext 2 Sh Ab S .K; Z/ Š 0 by Lemma 4.61 if p is odd, and by Lemma 4.17 if p D 2(the two-three condition ensures that K in this case is an at most finite product of copiesof Z=2Z.). This implies thatis injective.Next we show thatis injective. For this it suffices to see thatExt 3 Sh Ab S .Q; Z/ ! Ext3 Sh Ab S .G; Z/Ext 2 Sh Ab S .Q; Z/ ! Ext2 Sh Ab S .G; Z/Ext 1 Sh Ab S .G; Z/ ! Ext1 Sh Ab S .K; Z/is surjective. But this follows from the diagramExt 1 Sh Ab S .G; Z/ Ext 1 Sh Ab S .K; Z/ ;yGŠsurjective yKŠ

Duality for topological abelian group stacks and T -duality 303where the vertical maps are the isomorphisms given by Lemma 4.49, and surjectivityof yG ! yK follows from the fact that Pontrjagin duality preserves exact sequences.Next we apply Ext Sh Ab S .:::;Z/ to the sequenceand get the long exact sequence0 ! Q ! G ! C ! 0Ext i Sh Ab S .C ; Z/ ! Exti Sh Ab S .G; Z/ ! Exti Sh Ab S .Q; Z/ ! :Again we use that Ext i Sh Ab S .C ; Z/ Š 0 for i D 2; 3 by Lemma 4.61 for p>3, andby Lemma 4.17 if p 2¹2; 3º (in this case C is an at most finite product of copies ofZ=pZ by the two-three condition) in order to conclude thatExt i Sh Ab S .G; Z/ ! Exti Sh Ab S .Q; Z/is injective for i D 2; 3. Therefore the compositionExt i Sh Ab S .G; Z/ ! Exti Sh Ab S .Q; Z/ ! Exti Sh Ab S .G; Z/is injective for i D 2; 3. This is what we wanted to show.Lemma 4.63. Let G be a compact connected group which satisfies the two-three condition.Then the sheaves Ext i Sh Ab S lc-acyc.G; Z/ are torsion-free for i D 2; 3.Proof. The only point in the proof of Lemma 4.62 where we have used the conditionof local topological divisibility was the exactness of the sequence0 ! K ! G p ! G ! 0of sheaves on S. We show that this sequence is exact with this condition if we considerthe sheaves on S lc-acyc (by restriction). Let us start with the dual sequence0 ! yG p ! yG ! yK ! 0of discrete groups. It gives an exact sequence of sheaves0 ! yG p ! yG ! yK ! 0:We apply Hom ShAb S lc-acyc.:::;T/ and get, using Lemma 3.5, the long exact sequence0 ! K ! G p ! G ! Ext 1 Sh Ab S lc-acyc. yK; T/ ! :By Theorem 4.28 we have Ext 1 Sh Ab S lc-acyc. yK; T/ D 0. Now we can argue as in the proofof 4.62.Lemma 4.64. Let G be a profinite abelian group. Then the following assertions areequivalent.

304 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1. Ext i Sh Ab S .G; Z/ is torsion-free for i D 2; 32. Ext i Sh Ab S .G; Z/ Š 0 for i D 2; 3.Proof. It is clear that 2. implies 1. Therefore let us show that Ext i Sh Ab S .G; Z/ Š 0 fori D 2; 3 under the assumption that we already know that it is torsion-free.We consider the double complex Hom ShAb S .U ;I / introduced in 4.5.7. By Lemma4.51 we know that the cohomology sheaves H i .Hom ShAb S .U ;I // of the associatedtotal complex are torsion sheaves.The spectral sequence .F r ;d r / considered in 4.5.14 calculates the associated gradedsheaves of a certain filtration of H i .Hom ShAb S .U ;I //. The left lower corner of itssecond page was already evaluated in Lemma 4.60.The term F 2;12Š Ext 2 Sh Ab S .G; Z/ survives to the limit of the spectral sequence. Onthe one hand by our assumption it is torsion-free. On the other hand by Lemma 4.51,case i D 2, it is a subsheaf of a torsion sheaf. It follows thatExt 2 Sh Ab S .G; Z/ Š 0:This settles the implication 1: ) 2: in case i D 2 of Lemma 4.64.We now claim that Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ is a torsion sheaf. Let us assume the claimand finish the proof of Lemma 4.64. Since F 3;12Š Ext 3 Sh Ab S .G; Z/ is torsion-free byassumption the differential d 2 must be trivial by Lemma 4.46. Since also F 0;33Š 0 thesheaf Ext 3 Sh Ab S .G; Z/ survives to the limit of the spectral sequence. By Lemma 4.51,case i D 3, it is a subsheaf of a torsion sheaf and therefore itself a torsion sheaf. Itfollows that Ext 3 Sh Ab S .G; Z/ is a torsion sheaf and a torsion-free sheaf at the same time,hence trivial. This is the assertion 1: ) 2: of Lemma 4.64 for i D 3.We now show the claim. We start with some general preparations. If F;H aretwo sheaves on some site, then one forms the presheaf S 3 A 7! .F ˝pZH /.A/ WDF .A/ ˝Z H.A/ 2 Ab. The sheaf F ˝Z H is by definition the sheafification ofF ˝pZ H . We can write the definition of the presheaf ƒ2 ZG in terms of the followingexact sequence of presheaves0 ! K ! p Z.F .G// ˛! G ˝pZ G ! ƒ2 Z G ! 0;where K is by definition the kernel. For W 2 S the map ˛W W p Z.G.W // ! G.W /˝ZG.W / is defined on generators by ˛W .x/ D x ˝ x, x 2 G.W /. Sheafification is anexact functor and thus gives0 ! . p Z.F .G//=K/ ] ! G ˝Z G ! .ƒ 2 Z G/] ! 0:We now apply the functor Hom ShAb S .:::;Z/ and consider the following segment of theassociated long exact sequence! Hom ShAb S .G ˝Z G; Z/ ! Hom ShAb S .. p Z.F .G//=K/ ] ; Z/! Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ ! Ext 1 Sh Ab S .G ˝Z G; Z/ !The following two facts imply the claim:(38)

Duality for topological abelian group stacks and T -duality 3051. Ext 1 Sh Ab S .G ˝Z G; Z/ is torsion.2. Hom ShAb S .. p Z.F .G//=K/ ] ; Z/ vanishes.Let us start with 1. Let Z ! I be an injective resolution. We studyWe haveH 1 Hom ShAb S .G ˝Z G;I /:Hom ShAb S .G ˝Z G;I / Š Hom ShAb S .G; Hom Sh Ab S .G;I //:Let K D Hom ShAb S /. Since .G;I Hom ShAb S .G; Z/ Š Hom top-Ab.G; Z/ Š 0 by thecompactness of G the map d 0 W K 0 ! K 1 is injective. Let d 1 W K 1 ! K 2 be thesecond differential. NowH 1 Hom ShAb S .G ˝Z G;I / Š Hom ShAb S .G; ker.d 1 //=im.d 0 /;where d 0 W Hom Sh Ab S .G;K0 / ! Hom ShAb S .G; ker.d 1 // is induced by d 0 . Since d 0is injective we haveim.d 0 / D Hom Sh Ab S .G; im.d 0 //:Applying Hom ShAb S .G;:::/to the exact sequence0 ! K 0 ! ker.d 1 / ! ker.d 1 /=im.d 0 / ! 0and using ker.d 1 /=im.d 0 / Š Ext 1 Sh Ab S .G; Z/ we getIn particular,0 ! Hom ShAb S .G;K0 / d 0! Hom ShAb S .G; ker.d 1 // !! Hom ShAb S .G; Ext1 Sh Ab S .G; Z// ! Ext1 Sh Ab S .G;K0 / ! :Ext 1 Sh Ab S .G ˝Z G; Z/ Š Hom ShAb S .G; ker.d 1 //=Hom ShAb S .G;K0 / Hom ShAb S .G; Ext1 Sh Ab S .G; Z//:By Lemma 4.49 we know that Ext 1 Sh Ab S .G; Z/ Š yG. Since G is compact and profinite,the group yG is discrete and torsion. It follows by Lemma 4.48 thatHom ShAb S .G; Ext1 Sh Ab S .G; Z//is a torsion sheaf. A subsheaf of a torsion sheaf is again a torsion sheaf. This finishesthe argument for the first fact.We now show the fact 2.Note thatHom ShAb S .. p Z.F .G//=K/ ] ; Z/ Š Hom ShAb S .Z.F .G//=K] ; Z/: (39)

306 U. Bunke, T. Schick, M. Spitzweck, and A. ThomConsider A 2 S and 2 Hom ShAb S .Z.F .G//=K] ; Z/.A/. We must show that eacha 2 A has a neighbourhood U A such that jU D 0. Pre-composing with theprojection Z.F .G// ! Z.F .G//=K ] we get an elementN 2 Hom ShAb S .Z.F .G//; Z/.A/ Š Hom Lemma 3.9Sh S .F .G/; Z/.A/ Š Z.A G/:We are going to show that N D 0 after restriction to a suitable neighbourhood ofa 2 A using the fact that it annihilates K ] . Let us start again on the left-hand side of(39). We haveHom ShAb S .. p Z.F .G//=K/ ] ; Z/.A/ Š Hom PrAb S .. p Z.F .G//=K/; Z/.A/Š Hom PrAb S=A. p Z.F .G jA //=K jA ; Z jA /:For .A G ! A/ 2 S=A the morphism gives rise to a group homomorphismy W Z.F .G.A G///=K.A G/ ! Z.A G/:The symbol F .G.A G// denotes the underlying set of the group G.A G/. Wehave.pr G W A G ! G/ 2 G.A G/ and by the explicit description of the element N givenafter the proof of Lemma 3.9 we see N D y .¹Œpr G º/, where Œpr G 2 Z.F .G.AG///denotes the generator corresponding to pr G 2 G.A G/, and ¹::: º indicates that wetake the class modulo K.A G/.The homomorphism y 2 Hom Ab .ZF .G.A G//=K.A G/;Z.A G// is representedby a homomorphismQ W ZF .G.A G// ! Z.A G/;and we have N D Q.Œpr G /. By definition of K we have the exact sequence0 ! K.AG/ ! Z.F .G.AG/// ! G.AG/˝ZG.AG/ ! ƒ 2 ZG.AG/ ! 0:For x 2 G.A G/ let Œx 2 Z.F .G.A G/// denote the corresponding generator.Since .nx/ ˝ .nx/ D n 2 .x ˝ x/ in G.A G/ ˝Z G.A G/ we have n 2 Œx Œnx 2K.A G/ for n 2 Z. It follows that Q must satisfy the relation Q.n 2 Œx/ D Q.Œnx/for all n 2 Z and x 2 G.A G/. Let us now apply this reasoning to x WD Œpr G .Forevery n 2 Z we have a map A G id AnDW n! A G, and by naturality the map Qrespects this action. Moreover, the element Œnx 2 Z.F .G.A G/// is obtained fromŒx via this action, sinceA Gn A Gcommutes. It follows thatGpr Gn Gpr Gn 2 N D Q.n 2 Œpr G / D Q.Œnpr G / D Q.ZF . xG. n //.Œpr G // D Z. n /. N/ D n N;

Duality for topological abelian group stacks and T -duality 307where we write n WD Z. n /.Fork 2 Z we define the open subset V k WD N 1 .k/ AG. The family .V k / k2K forms an open pairwise disjunct covering of AG. Since Gis compact the compactly generated topology of A G is the product topology ([Ste67,Theorem 4.3]). Since G is compact, we can choose a finite sequence k 1 ;:::;k r 2 Zsuch that G ¹aº S riD1 V k. Furthermore, we can find a neighbourhood U A of asuch that G U S riD1 V k.Note that N jGU has at most finitely many values. Therefore there exists a finitequotient G ! F such that N jU G factors through f W U F ! Z.The action of Z on G is compatible with the corresponding action of Z on F ,and we have an action n W Hom S .U F;Z/ ! Hom S .U F;Z/. The elementf 2 Hom S .U F;Z/ still satisfies n 2 f D n f . We now take n WD jF jC1. Thenn D id so that .n 2 1/f D 0, hence f D 0. This implies N jU G D 0.This finishes the proof of the second fact.4.5.15 The Lemma 4.62 verifies Assumption 1. in 4.64 for a large class of profinitegroups.Corollary 4.65. If G is a profinite group which satisfies the two-three condition 4.6,then we have Ext i Sh Ab S .G; Z/ Š 0 for i D 2; 3.Theorem 4.66. A profinite abelian group which satisfies the two-three condition isadmissible.4.6 Compact connected abelian groups4.6.1 Let G be a compact abelian group. We shall use the following fact shown in[HM98, Corollary 8.5].Fact 4.67. G is connected if and only if yG is torsion-free.4.6.2 For a space X 2 S and F 2 Pr Ab S let L H .XI F/denote the Čech cohomologyof X with coefficients in F . It is defined as follows. To each open covering Uone associates the Čech complex LC .UI F/. The open coverings form a left-filteredcategory whose morphisms are refinements. The Čech complex depends functoriallyon the covering, i.e. if V ! U is a refinement, then we have a functorial chain mapLC .UI F/! LC .VI F/. We defineandLC .XI F/WD colim ULC .UI F/L H .XI F/WD colim U H . LC .UI F//Š H .colim ULC .UI F// Š H . LC .XI F //:

308 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.6.3 Fix a discrete group H . Let p H be the constant presheaf with values H . Notethat then the sheafification p H ] is isomorphic to H as defined in 3.2.5. MoreoverLC .XI p H / Š LC .X; H /: (40)In general Čech cohomology differs from sheaf cohomology. The relation betweenthese two is given by the Čech cohomology spectral sequence .E r ;d r / (see [Tam94,3.4.4]) converging to H .XI F/. Let H .F / WD R i.F/ denote the right derivedfunctors of the inclusion i W Sh Ab S ! Pr Ab S. Then the second page of the spectralsequence is given byE p;q2Š HLp .XI H q .F //:By [Tam94, 3.4.7] the edge homomorphismL H p .XI F/! H p .XI F/is an isomorphism for p D 0; 1 and injective for p D 2.4.6.4 We now observe that Čech cohomology transforms strict inverse limits of compactspaces into colimits of cohomology groups.Lemma 4.68. Let H be a discrete abelian group. If .X i / i2I is an inverse system ofcompact spaces in S such that X D lim i2I X i , thenHLp .XI H / Š colim i2I HLp .X i I H /:Proof. We first show that the system of open coverings of X contains a cofinal systemof coverings which are pulled back from the quotients p i W X ! X i . Let U D .U r / r2Rbe a covering by open subsets. For each r there exists a family I r I and subsetsU r;i X i , i 2 I r such that U r D[ i2Ir pi 1 .U r;i /. The set of open subsets ¹pi1 .U r;i / jr 2 R; i 2 I r º is an open covering of X. Since X is compact we can choosea finite subcovering V WD ¹U r1 ;i 1;:::U rk ;i kº which can naturally be viewed as arefinement of U. Since I is left filtered we can choose j 2 I such that j < i dfor d D 1;:::;k. For j i let p ji W X j ! X i be the structure map of the systemwhich are all surjective by the strictness assumption on the inverse system. ThereforeV 0 WD ¹pj;i 1d.U rd ;i d/ j d D 1;:::;kº is an open covering of X j , and V D pj 1 .V 0 /.Now we observe that LC .p V 0 I p H / Š LC .V 0 I p H /. ThereforeLC .XI H / Š LC .XI p H / (by Equation (40))Š colim ULC .UI p H /D colim i2I colim coverings U of XiLC .UI p H /Š colim i2ILC .X i I p H /Š colim i2ILC .X i I H / (by Equation (40))since one can interchange colimits. Since filtered colimits are exact and thereforecommute with taking cohomology this implies the lemma.

Duality for topological abelian group stacks and T -duality 3094.6.5 Next we show that Čech cohomology has a Künneth formula.Lemma 4.69. Let H be a discrete ring of finite cohomological dimension. Assume thatX; Y are compact. Then there exists a Künneth spectral sequence with second termE 2 p;q WDMiCj Dqwhich converges to L H pCq .X Y I H /.Tor H p . L H i .XI H /; L H j .Y I H //Proof. Since X is compact the topology on X Y is the product topology [Ste67,Theorem 4.3]. Using again the compactness of X and Y we can find a cofinal systemof coverings of X Y of the form p U \ q V for coverings U of X and V of Y ,where p W X Y ! X and q W X Y ! Y denote the projections, and the intersectionof coverings is the covering by the collection of all cross intersections. We haveLC .p U \ q VI p H / Š LC .p UI p H / ˝HLC .q VI p H /:Since the tensor product over H commutes with colimits we getand hence by (40)LC .X Y I p H / Š LC .XI p H / ˝HLC .Y I p H /;LC .X Y I H / Š LC .XI H / ˝HLC .Y I H /:The Künneth spectral sequence is the spectral sequence associated to this doublecomplex of flat (as colimits of free) H -modules.4.6.6 We now recall that sheaf cohomology also transforms strict inverse limits ofspaces into colimits, and that it has a Künneth spectral sequence. The following is aspecialization of [Brd97, II.14.6] to compact spaces.Lemma 4.70. Let .X i / i2I be an inverse system of compact spaces and X D lim i2I X i ,and H let be a discrete abelian group. ThenH .XI H / Š colim i2I H .X i I H /:For simplicity we formulate the Künneth formula for the sheaf Z only. The followingis a specialization of [Brd97, II.18.2] to compact spaces and the sheaf Z.Lemma 4.71. For compact spaces X; Y we have a Künneth spectral sequence withsecond termEp;q 2 DMTorp Z .H i .XI Z/; H j .Y I Z//iCj Dqwhich converges to H pCq .X Y I Z/.

310 U. Bunke, T. Schick, M. Spitzweck, and A. ThomOf course, this formulation is much too complicated since Z has cohomologicaldimension one. In fact, the spectral sequence decomposes into a collection of shortexact sequences.Lemma 4.72. If G is a compact connected abelian group, then H .GI Z/ Š HL .GI Z/.Proof. The Čech cohomology spectral sequence provides the mapL H .GI Z/ ! H .GI Z/:We now use that fact that G is the projective limit of groups isomorphic to T a , a 2 N.Since the compact abelian group G is connected, by Fact 4.67 we know that yG istorsion-free. Since yG is torsion-free it is the filtered colimit of its finitely generatedsubgroups yF .If yF yG is finitely generated, then yF Š Z a for some a 2 N, thereforeF WD yF Š T a . Pontrjagin duality transforms the filtered colimit yG Š colim y F y F intoa strict limit G Š limF y F .Since T a is a manifold it admits a cofinal system of good open coverings where allmultiple intersections are contractible. Therefore we get the isomorphismL H .T a I Z/ ! H .T a I Z/:Since both cohomology theories commute with limits of compact spaces we getL H .GI Z/ ! H .GI Z/:4.6.7 We now use the calculation of cohomology of the underlying space of a connectedcompact abelian group G. By [HM98, Theorem 8.83] we haveL H .G; Z/ Š ƒ Z y G (41)as a graded Hopf algebra. This result uses the two properties 4.68 and 4.69 of Čechcohomology in an essential way.By Lemma 4.72 we also haveH .GI Z/ Š ƒ Z y G:Note that ƒ Z y G is torsion free. In fact, ƒ Z y G Š colim y F ƒ Z y F , where the colimitis taken over all finitely generated subgroups yF of yG. We therefore get a colimit ofinjections of torsion-free abelian groups, which is itself torsion-free.4.6.8 Recall the notation related to Z mult -actions introduced in 3.5.Lemma 4.73. If V;W 2 Ab with W torsion-free and k; l 2 Z, k 6D l, thenHom Zmult -mod.V .k/; W.l// D 0:

Duality for topological abelian group stacks and T -duality 311Proof. Let 2 Hom Ab .V; W / and v 2 V . Then for all m 2 Z mult we havem k .v/ D ‰ m ..v// D .‰ m .v// D .m l v/ D m l .v/;i.e. each w 2 im./ W.l/ satisfies .m k m l /w D 0 for all m 2 Z mult . This set ofequations implies that w D 0, since W is torsion-free.Lemma 4.74. If V;W 2 Sh Ab S with W torsion-free and k; l 2 Z, k 6D l, thenHom ShZmult-modS.V .k/; W.l// D 0:We leave the easy proof of this sheaf version of Lemma 4.73 to the interested reader.Note that a subquotient of a sheaf of Z mult -modules of weight k also has weight k.4.6.9 By S lc S we denote the sub-site of locally compact objects. The restriction tolocally compact spaces becomes necessary because of the use of the Künneth (or basechange) formula below.The first page of the spectral sequence .E r ;d r / introduced in 4.2.12 is given byE q;p1D Ext p Sh Ab S .Z.Gq /; Z/:This sheaf is the sheafification of the presheafS 3 A 7! H p .A G q ; Z/ 2 Ab:In order to calculate this cohomology we use the Künneth formula [Brd97, II.18.2] andthat H .G; Z/ is torsion-free (4.67). We get for A 2 S lc thatH .A G q ; Z/ Š H .A; Z/ ˝Z .ƒ yG/˝Zqfor locally compact A. The sheafification of S lc 3 A ! H i .A; Z/ vanishes for i 1,and gives Z for i D 0. Since sheafification commutes with the tensor product with afixed group we get.E q;1 / jS lcŠ .ƒ yG/˝Zq : (42)4.6.10 We now consider the tautological action of the multiplicative semigroup Z multon G of weight 1. As before we write G.1/ for the group G with this action. Observethat this action is continuous. Applying the duality functor we get an action of Z multon the dual group yG which is also of weight 1. ThereforebG.1/ D yG.1/:The calculation of the cohomology (41) of the topological space G with Z-coefficientsis functorial in G. We conclude that H .G.1/; Z/ Š ƒ Z . yG.1// is a decomposableZ mult -module. By 3.5.5 the groupis of weight p.H p .G q .1/; Z/ Š Œ.ƒ yG.1//˝Zq p

312 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.6.11 We now observe that the spectral sequence .E r ;d r / is functorial in G. To thisend we use the fact that G 7! U .G/ is a covariant functor from groups G 2 S tocomplexes of sheaves on S. IfG 0 ! G 1 is a homomorphism of topological groups inS, then we get an induced map U q .G 0 / ! U q .G 1 /, namely the map Z.G q 0 / ! Z.Gq 1 /.Under the identification made in 4.6.9 the induced mapExt p Sh Ab S lc.Z.G q 1 /; Z/ ! Extp Sh Ab S lc.Z.G q 0/; Z/goes to the mapH p .G q 1 I Z/ ! H p .G q 0 I Z/induced by the pull-backH p .G q 1 I Z/ ! H p .G q 0 I Z/associated to the map of spaces G q 0 ! Gq 1 .4.6.12 The discussion in 4.6.11 and 4.6.10 shows that the Z mult -module structure G.1/induces one on the spectral sequence .E r ;d r /, and we see that E q;p1has weight p(which is the number of factors yG.1/ contributing to this term).We introduce the notation H k WD H k .Hom ShAb S .U ;I // jSlc . Note that H k has afiltration0 D F 1 H k F 0 H k F k H k D H kwhich is preserved by the action of Z mult . The spectral sequence .E r ;d r / converges tothe associated graded sheaf Gr.H k /.Note that Gr p .H k / is a subquotient of E k p;p2. Since a subquotient of a sheaf ofweight p also has weight p (see the our remark after Lemma 4.74) we get the followingconclusion.Corollary 4.75. Gr p .H k / has weight p.4.6.13 We now study some aspects of the second page E q;p2of the spectral sequencein order to show the following lemma.Lemma 4.76. 1. For k 1 we have F 0 H k Š 0.2. For k 2 we have F 1 H k Š 0.Proof. We start with 1. We see that E ;02is the cohomology of the complex E ;01ŠHom ShAb S .U ; Z/. Explicitly, using in the last step the fact that G and therefore G qare connected,E q;01Š Hom ShAb S .U q ; Z/ (24)Š Hom ShAb S .Z.Gq /; Z/ (17)Š R G q.Z/Š Map.G q ; Z/ Š Z (by Lemma 3.25):

Duality for topological abelian group stacks and T -duality 313An inspection of the formulas 4.2.7 for the differential of the complex U shows thatd 1 W E q;01! E qC1;01vanishes for even q, and is the identity for odd q. In other wordsthis complex is isomorphic to0 ! Z ! 0 Z ! id Z ! 0 Z ! id Z ! 0 :This implies that E q;02D 0 for q 1 and therefore the assertion of the lemma.We now turn to 2. We calculateH 1 .G q I Z/ Š Œ.ƒ yG/˝Zq 1 D yG ˚˚ yG :„ ƒ‚ …q summandsBy (42) we getE q;11Š yG q :We see that .E ;11 ;d 1/ is the sheafification of the complex of discrete groupsyG W 0 ! yG ! yG 2 ! yG 3 ! ; (43)where the differential is the dual of the differential of the complex0 G G 2 G 3 induced by the maps given in 4.2.7. Let us describe the differential more explicitly. If 2 yG, and W G G ! G is the multiplication map, then we have D .; / 22G G Š yG yG. We can write @W yG q ! yG qC1 asqC1X@ D . 1/ i @ i :Using the formulas of 4.2.7 we getfor i D 1;:::;q. FurthermoreiD0@ i . 1 ;:::; q / D . 1 ;:::; i ; i ;:::; q /@ 0 . 1 ;:::; q / D .0; 1 ;:::; q /and@ qC1 . 1 ;:::; q / D . 1 ;:::; q ; 0/:Note that 0 D @W yG 0 ! yG 1 . We see that we can write the higher ( 1) degree part ofthe complex (43) in the formK ˝ZyG;where K q D Z q for q 1 and @W Z q ! Z qC1 is given by the same formulas as above.One now shows 12 that´H q Z; q D 1;.K / D0; q 2:12 We leave this as an exercise in combinatorics to the interested reader.

314 U. Bunke, T. Schick, M. Spitzweck, and A. ThomSince yG is torsion-free and hence a flat Z-module we haveH q . yG / Š H q .K ˝ZyG/ Š H q .K / ˝ZyG:Therefore H q . yG / D 0 for q 2. This implies the assertion of the lemma in thecase 2.Lemma 4.77. Let F 0 F 1 F k 1 F k D F be a filtered sheaf of Z mult -modules such that Gr l .F / has weight l, and such that F 0 D F 1 D 0. IfV F is atorsion-free sheaf of weight 1, then V D 0.Proof. Assume that V 6D 0. We show by induction (downwards) that F l \ V 6D 0 forall l 1. The case l D 1 gives the contradiction. Assume that l>1. We consider theexact sequence0 ! F l 1 \ V ! F l \ V ! Gr l .F /:First of all, by induction assumption, the sheaf V \ F l is non-trivial, and a as asubsheaf of V it is torsion-free of weight 1. Since Gr l .F / has weight l 6D 1, the mapV \ F l ! Gr l .F / can not be injective. Otherwise its image would be a torsion-freesheaf of two different weights l and 1, and this is impossible by Lemma 4.74. HenceF l 1 \ V 6D 0.4.6.14 We now show that Lemma 4.64 extends to connected compact groups.Lemma 4.78. Let G be a compact connected abelian group. Then the followingassertions are equivalent.1. Ext i Sh Ab S lc.G; Z/ is torsion-free for i D 2; 3.2. Ext i Sh Ab S lc.G; Z/ Š 0 for i D 2; 3.Proof. The non-trivial direction is that 1. implies 2. Therefore let us assume thatExt i Sh Ab S lc.G; Z/ is torsion-free for i D 2; 3. We now look at the spectral sequence.F r ;d r /. The left lower corner of its second page was calculated in 4.60. We seethat F 1;22D Ext 1 Sh Ab S lc..ƒ 2 Z G/] ; Z/ has weight 2, while Ext 3 Sh Ab S lc.G; Z/ has weight 1.Since Ext 3 Sh Ab S lc.G; Z/ is torsion-free by assumption, it follows from Lemma 4.74 thatthe differential d 1;22W Ext 1 Sh Ab S lc..ƒ 2 Z G/] ; Z/ ! Ext 3 Sh Ab S lc.G; Z/ is trivial.We conclude that Ext 2 Sh Ab S lc.G; Z/ and Ext 3 Sh Ab S lc.G; Z/ survive to the limit of thespectral sequence. We see that Ext i Sh Ab S lc.G; Z/ are torsion-free sub-sheaves of weight1 of H iC1 for i D 2; 3. Using the structure of H iC1 given in 4.75 in conjunction withLemmas 4.76 and 4.77 we get Ext i Sh Ab S lc.G; Z/ D 0 for i D 2; 3.4.6.15 The combination of Lemma 4.62 (resp. Lemma 4.62) and Lemma 4.78 givesthe following result.Theorem 4.79. 1. A compact connected abelian group G which satisfies the two-threecondition is admissible on the site S lc-acyc .2. If G is in addition locally topologically divisible, then it is admissible on S lc .

Duality for topological abelian group stacks and T -duality 3155 Duality of locally compact group stacks5.1 Pontrjagin Duality5.1.1 In this subsection we extend Pontrjagin duality for locally compact groups toabelian group stacks whose sheaves of objects and automorphisms are represented bylocally compact groups. In algebraic geometry a parallel theory has been consideredin [DP].The site S denotes the site of compactly generated spaces as in 3.1.2 or one of itssub-sites S lc , S lc-acyc . The reason for considering these sub-sites lies in the fact thatcertain topological groups are only admissible on these sub-sites (see the Definitions4.1, 4.2 and Theorem 4.8).5.1.2 Let F 2 Sh Ab S.Definition 5.1. We define the dual sheaf of F byD.F / WD Hom ShAb S .F; T/:Definition 5.2. We call F dualizable, if the canonical evaluation morphismis an isomorphism of sheaves.c W F ! D.D.F // (44)Lemma 5.3. If G is a locally compact group which together with its Pontrjagin dualHom top-Ab .G; T/ is contained in S, then G 2 Sh Ab S is dualizable.Proof. By Lemma 3.5 we have isomorphismsandHom ShAb S .G; T/ Š Hom top-Ab.G; T/Hom ShAb S .Hom Sh Ab S .G; T/; T/ Š Hom Sh Ab S .Hom top-Ab.G; T/; T/Š Hom top-Ab .Hom top-Ab .G; T/; T/:The morphism c in (44) is induced by the evaluation mapG ! Hom top-Ab .Hom top-Ab .G; T/; T/which is an isomorphism by the classical Pontrjagin duality of locally compact abeliangroups [Fol95], [HM98].5.1.3 The sheaf of abelian groups T 2 Sh Ab S gives rise to a Picard stack BT 2 PIC.S/as explained in 2.3.3. Recall from 2.5.11 the following alternative description of BT.Let TŒ1 be the complex with the only non-trivial entry TŒ1 1 WD T. Then we haveBT Š ch.TŒ1/ in the notation of 2.12.

316 U. Bunke, T. Schick, M. Spitzweck, and A. Thom5.1.4 Let now P 2 PIC.S/ be a Picard stack. Recall Definition 2.11, where we definethe internal HOM between two Picard stacks.Definition 5.4. We define the dual stack byD.P / WD HOM PIC.S/ .P; BT/:We hope that using the same symbol D for the dual in the case of Picard stacks andthe case of a sheaf of abelian groups will not cause confusion.5.1.5 This definition is compatible with Definition 5.1 of the dual of a sheaf of abeliangroups in the following sense.Lemma 5.5. If F 2 Sh Ab S, then we have a natural isomorphismch.D.F // Š D.BF/:Proof. First observe that by definition D.BF/D HOM PIC.S/ .ch.F Œ1/; ch.TŒ1//. Weuse Lemma 2.18 in order to calculate H i .D.BF//.WehaveandH 1 .D.BF//Š R 1 Hom ShAb S .F Œ1; TŒ1/ Š 0H 0 .D.BF//Š R 0 Hom ShAb S .F Œ1; TŒ1/ Š Hom Sh Ab S .F; T/ Š D.F /:The composition of this isomorphism with the projection D.BF/! ch.H 0 .D.BF ///from the stack D.BF/ onto its sheaf of isomorphism classes (considered as Picardstack) provides the asserted natural isomorphism.5.1.6 A sheaf of groups F 2 Sh Ab S can also be considered as a complex F 2C.Sh Ab S/ with non-trivial entry F 0 WD F . It thus gives rise to a Picard stack ch.F /.Recall the definition of an admissible sheaf 4.1.Lemma 5.6. We have natural isomorphismsH 1 .D.ch.F /// Š D.F /;H 0 .D.ch.F /// Š Ext 1 Sh Ab S .F; T/:In particular, if F is admissible, then D.ch.F // Š B.D.F //.Proof. We use again Lemma 2.18. We haveH 1 .D.ch.F /// Š R 1 Hom ShAb S .F; TŒ1/ Š Hom Sh Ab S .F; T/ Š D.F /:Furthermore,H 0 .D.ch.F /// Š R 0 Hom ShAb S .F; TŒ1/ Š Ext1 Sh Ab S .F; T/:

Duality for topological abelian group stacks and T -duality 3175.1.7 Assume now that F is dualizable and admissible (at this point we only needExt 1 Sh Ab S .F; T/ Š 0). Then we haveD.D.ch.F /// Š D.B.D.F /// (by Lemma 5.6)Š ch.D.D.F // (by Lemma 5.5)Š ch.F /:Similarly, if F is dualizable and D.F / admissible (again we only need the weakercondition that Ext 1 Sh Ab S .D.F /; T/ Š 0), then we have, again by Lemmas 5.5 and 5.6,D.D.BF//Š D.ch.D.F /// Š BD.D.F // Š BF:5.1.8 Let us now formalize this observation. Let P 2 PIC.S/ be a Picard stack.Definition 5.7. We call P dualizable if the natural evaluation morphism P ! D.D.P //is an isomorphism.The discussion of 5.1.7 can now be formulated as follows.Corollary 5.8. 1. If F is dualizable and admissible, then ch.F / is dualizable.2. If F is dualizable and D.F / is admissible, then BF is dualizable.The goal of the present subsection is to extend this kind of result to more generalPicard stacks.5.1.9 Let P 2 PIC.S/.Lemma 5.9. We haveH 1 .D.P // Š D.H 0 .P //and an exact sequence0 ! Ext 1 Sh Ab S .H 0 .P /; T/ ! H 0 .D.P // ! D.H 1 .P // ! Ext 2 Sh Ab S .H 0 .P /; T/:(45)Proof. By Lemma 2.13 we can choose K 2 C.Sh Ab S/ such that P Š ch.K/. WenowgetH 1 .D.P // Š R 1 Hom ShAb S .K; TŒ1/ (by Lemma 2.18)Š R 0 Hom ShAb S .K; T/Š Hom ShAb S .H 0 .K/; T/Š D.H 0 .P //:Again by Lemma 2.18 we haveH 0 .D.P // Š R 0 Hom ShAb S .K; TŒ1/ Š R1 Hom ShAb S .K; T/:

318 U. Bunke, T. Schick, M. Spitzweck, and A. ThomIn order to calculate this sheaf in terms of the cohomology sheaves H i .K/ of K wechoose an injective resolution T ! I . Then we haveRHom ShAb S .K; T/ Š Hom Sh Ab S .K; I /:We must calculate the first total cohomology of this double complex. We first take thecohomology in the K-direction, and then in the I -direction. The second page of theassociated spectral sequence has the following form.1 Ext 0 ShAb S .H 1 .P /; T/ Ext 1 ShAb S .H 1 .P /; T/ Ext 2 ShAb S .H 1 .P /; T/ Ext 3 ShAb S .H 1 .P /; T/0 Ext 0 ShAb S .H 0 .P /; T/ Ext 1 ShAb S .H 0 .P /; T/ Ext 2 ShAb S .H 0 .P /; T/ Ext 3 ShAb S .H 0 .P /; T/0 1 2 3The sequence (45) is exactly the edge sequence for the total degree-1 term.5.1.10 The appearance in (45) of the groups Ext i Sh Ab S .H 0 .P /; T/ for i D 1; 2 was themotivation for the introduction of the notion of an admissible sheaf in 4.1.Corollary 5.10. Let P 2 PIC.S/ be such that H 0 .P / is admissible. Then we haveH 0 .D.P // Š D.H 1 .P //; H 1 .D.P // Š D.H 0 .P //:Theorem 5.11. Let P 2 PIC.S/ and assume that1. H 0 .P / and H 1 .P / are dualizable;2. H 0 .P / and D.H 1 .P // are admissible.Then P is dualizable.Proof. In view of 2.14 it suffices to show that the evaluation map c W P ! D.D.P //induces isomorphismsH i .c/W H i .P / ! H i .D.D.P ///for i D 1; 0. Consider first the case i D 0. Then by Corollary 5.10 we have anisomorphismh 0 W H 0 .D.D.P /// ! D.H 1 .D.F // ! D.D.H 0 .P ///:One now checks that the maph 0 ı H 0 .c/W H 0 .P / ! D.D.H 0 .P ///is the evaluation map (44). Since we assume that H 0 .P / is dualizable this map is anisomorphism. Hence H 0 .c/ is an isomorphism, too.For i D 1 we use the isomorphismh 1 W H 1 .D.D.P /// ! D.H 0 .D.P /// ! D.D.H 1 .P ///and the fact that h 1 ı H 1 .c/W H 1 .P / ! D.D.H 1 .P /// is an isomorphism.

Duality for topological abelian group stacks and T -duality 3195.1.11 If G is a locally compact group which together with its Pontrjagin dual belongsto S, then by 5.3 the sheaf G is dualizable. By Theorem 4.8 we know a large class oflocally compact groups which are admissible on S or at least after restriction to S lc orS lc-acyc .We get the following result.Theorem 5.12. Let G 0 ;G 1 2 S be two locally compact abelian groups. We assumethat their Pontrjagin duals belong to S, and that G 0 and DG 1 are admissible on S. IfP 2 PIC.S/ has H i .P / Š G i for i D 1; 0, then P is dualizable.Let us specialize to the case which is important for the application to T -duality.Note that a group of the form T n R m F for a finitely generated abelian group Fis admissible by Theorem 4.8. This class of groups is closed under forming Pontrjaginduals.Corollary 5.13. If P 2 PIC.S/ has H i .P / Š T n i R m i F i for some finitelygenerated abelian groups F i for i D 1; 0, then P is dualizable.For more general groups one may have to restrict to the sub-site S lc or even toS lc-acyc .5.2 Duality and classification5.2.1 Let A; B 2 Sh Ab S. By Lemma 2.20 we know that the isomorphism classesExt PIC.S/ .A; B/ of Picard stacks with H 0 .P / Š B and H 1 .P / Š A are classified bya characteristic class W Ext PIC.S/ .B; A/ ! Ext 2 Sh Ab S .B; A/: (46)Lemma 5.14. If A is dualizable and D.A/; B are admissible, then there is a naturalisomorphismD W Ext 2 Sh Ab S .B; A/ ! Ext 2 Sh Ab S .D.A/; D.B//such that.D.P// D D..P // for all P 2 Ext PIC.S/ .A; B/: (47)Proof. In order to define D we use the identificationsExt 2 Sh Ab S .B; A/ Š Hom D C .Sh Ab S/.B; AŒ2/;Ext 2 Sh Ab S .D.A/; D.B// Š Hom D C .Sh Ab S/.D.A/; D.B/Œ2/:We choose an injective resolution T ! I. For a complex of sheaves F we defineRD.F / WD Hom ShAb S .F; I/. The map T ! I induces a map D.F / ! RD.F /.

320 U. Bunke, T. Schick, M. Spitzweck, and A. ThomNote that RD descends to a functor between derived categories RD W D b .Sh Ab S/ op !D C .Sh Ab S/. We now consider the following web of maps:Hom D C .Sh Ab S/.B; AŒ2/RDHom D C .Sh Ab S/.RD.AŒ 2/; RD.B//D 0DD.A/!RD.A/Hom D C .Sh Ab S/.D.A/; RD.B/Œ2/uD.B/!RD.B/Hom D C .Sh Ab S/.D.A/; D.B/Œ2/.Since B is admissible the map D.B/ ! RD.B/ is an isomorphism in cohomologyin degree 0; 1; 2 (because D.B/ is concentrated in degree 0 and H k .RD.B// DR k Hom ShAb S.B; T/ D Ext k Sh Ab S .B; T/). Since D.A/ is acyclic in non-zero degree,the map u is an isomorphism. For this, observe that D.B/ ! RD.B/ can be replacedup to quasi-isomorphism by an embedding 0 ! RD.B/ B ! RD.B/ ! Q ! 0such that the quotient is zero in degrees 0; 1; 2. The statement then follows from thelong exact Ext-sequence for Hom ShAb S.D.A/; /, because Ext i Sh Ab S .D.A/; B/ D 0 fori D 0; 1; 2. Therefore the diagram defines the mapD WD u 1 ı D 0 W Hom D C .Sh Ab S/.B; AŒ2/ ! Hom D C .Sh Ab S/.D.A/; D.B/Œ2/:5.2.2 We now show the relation (47). By Lemma 2.13 it suffices to show (47) forP 2 PIC.S/ of the form P D ch.K/ for complexesAs in 2.5.9 we considerK W 0 ! A ! X ! Y ! B ! 0; K W 0 ! X ! Y ! 0:K A W 0 ! X ! Y ! B ! 0with B in degree 0. Then by Definition (12) of the map and the Yoneda map Y , theelement .ch.K// 2 Hom D C .Sh Ab S/.B; AŒ2/ is represented by the composition (see(10))Y.K/W B ! ˇ ˛K A AŒ2:Since RD preserves quasi-isomorphisms we get RD.˛ 1/ D RD.˛/ 1 . It followsthatRD.Y.K// W RD.AŒ2/ RD.˛/ ! 1 RD.K A / RD.ˇ/ ! RD.B/:We read off thatD 0 .Y.K//W D.A/ ! RD.A/ RD.˛/ 1 ! RD.K A /Œ2 RD.ˇ/ ! RD.B/Œ2:

Duality for topological abelian group stacks and T -duality 321Let T ! I 0 ! I 1 ! ::: be the injective resolution I of T. We define J WDker.I 1 ! I 2 / and I WD I 0 . Then we have BT D ch.L/ with LW 0 ! I ! J ! 0with J in degree zero. Note that I is injective. Then by Lemma 2.17 we haveD.P / Š ch.H /; (48)where H WD 0 Hom ShAb S .K; L/. LetQ WD ker.Hom ShAb S .X; I / ˚ Hom Sh Ab S .Y; J / ! Hom Sh Ab S .X; J //:Then H is the complexH W 0 ! Hom ShAb S .Y; I / ! Q ! 0:There is a natural map D.B/ ! Hom ShAb S .Y; I / (induced by T ! I and Y ! B),and a projection Q ! D.A/ induced by A ! X and passage to cohomology. SinceB is admissible the complexH W 0 ! D.B/ ! Hom ShAb S .Y; I / d ! Q ! D.A/ ! 0is exact. Note that ker.d/ D H 1 Hom ShAb S .K; I / and coker.d/ D H 0 Hom ShAb S .K; I /.We get.D.P// .48)D .ch.H // (12) Lemma 2.19D Y.H/ D Y 0 .H/:Explicitly, in view of (11) the map Y 0 .H/ 2 Hom D C .Sh Ab S/.D.A/; D.B/Œ2/ is givenby the compositionY 0 .H/W D.A/ ! 1 ıH D.A/ ! D.B/Œ2;whereH D.A/ W 0 ! D.B/ ! Hom ShAb S .Y; I / ! Q ! 0with Q in degree 0, the map W H D.A/ ! D.A/ is the quasi-isomorphism inducedby the projection Q ! D.A/, and ı W H D.A/ ! D.B/Œ2 is the canonical projection.Since B is admissible we have R 1 Hom ShAb S .B; T/ Š 0 and hence a quasi-isomorphismH D.A/ ! H 0 fitting into the following larger diagram.D.A/H D.A/ W0 D.B/ Hom ShAb S .Y; I / Q 0H 0 D.A/ W0 Hom ShAb S .B; I / Hom Sh Ab S .B; J /˚Hom ShAb S .Y; I / Q0RD.K A /W 0 Hom ShAb S .B; I / Hom Sh Ab S .B; I 1 /˚ Hom2Sh Ab S .K A; I/Hom ShAb S .Y; I /:::

322 U. Bunke, T. Schick, M. Spitzweck, and A. Thom5.2.3 The final step in the verification of (47) follows from a consideration of thediagramD.B/Œ2 ıuıY 0 .H/ RD.B/Œ2 RD.ˇ/Y 0 .H/H D.A/H 0 D.A/RD.K A /RD.Y.K//D.A/D.A/RD.˛/ RD.A/D 0 .Y.K//showing the marked equality in.D.P// D Y 0 .H/ Š D D.Y.K// D D..P //:It remains to show thatD W Ext 2 Sh Ab S .B; A/ ! Ext2 Sh Ab S .D.A/; D.B//is an isomorphism.To this end we look at the following commutative web of mapsR 2 Hom ShAb S.D.A/; D.B//DR 2 Hom ShAb S.B; A/Š R 2 Hom ShAb S.B; D.D.A///ŠR 2 Hom ShAb S.D.A/; RD.B//uD 0Š vŠ z R 2 Hom ShAb S.D.A/ ˝L B;T/ R 2 Hom ShAb S.B; RD.D.A//RDŠR 2 Hom ShAb S.RD.A/; RD.B//Š R 2 Hom ShAb S.RD.A/ ˝L B;T/ŠR 2 Hom ShAb S.B; RD.RD.A///.The horizontal isomorphisms in the two lower rows are given by the derived adjointnessof the tensor product and the internal homomorphisms. The horizontal isomorphismin the first row is induced by the isomorphism A ! D.D.A//. The maps u and v areisomorphisms since we assume that D.A/ is admissible, compare the correspondingargument in the proof of Lemma 5.14. The map z is induced by the canonical mapA ! RD.RD.A//.

Duality for topological abelian group stacks and T -duality 3236 T -duality of twisted torus bundles6.1 Pairs and topological T -duality6.1.1 The goal of this subsection is to introduce the main objects of topological T -dualityand review the structure of the theory. Let B be a topological space.Definition 6.1. A pair .E; H / over B consists of a locally trivial principal T n -bundleE ! B and a gerbe H ! E with band T jE . An isomorphism of pairs is a diagramH H0EB E0Bconsisting of an isomorphism of T n -principal bundles and an isomorphism ofT-banded gerbes . By P.B/ we denote the set of isomorphism classes of pairsover B. Iff W B ! B 0 is a continuous map, then pull-back induces a functorial mapP.f/W P.B 0 / ! P.B/.6.1.2 The functorP W TOP op ! Setshas been introduced in [BS05] for n D 1 and in [BRS] for n 1 in connection withthe study of topological T -duality.The main result of [BS05] in the case n D 1 is the construction of an involutivenatural isomorphism T W P ! P , the T -duality isomorphism, which associates to eachisomorphism class of pairs t 2 P.B/ the class of the T -dual pair Ot WD T.t/2 P.B/.In the higher-dimensional case n 2 the situation is more complicated. First ofall, not every pair t 2 P.B/ admits a T -dual. Furthermore, a T -dual, if it exists, maynot be unique. The results of [BRS] are based on the notion of a T -duality triple. Inthe following we recall the definition of a T -duality triple.6.1.3 As a preparation we recall the following two facts. Let W E ! B be aT n -principal bundle. Associated to the decomposition of the global section functor.E;:::/ as composition .E;:::/ D .B;:::/ı W Sh Ab S=E ! Ab there is adecreasing filtrationF n H .EI Z/ F n 1 H .EI Z/ F 0 H .EI Z/of the cohomology groups H .EI Z/ Š R .EI Z/ and a Serre spectral sequencewith second term E i;j2Š H i .BI R j Z/ which converges to GrH .EI Z/.

324 U. Bunke, T. Schick, M. Spitzweck, and A. Thom6.1.4 The isomorphism classes of gerbes over X with band T jX are classified byH 2 .XI T/ Š H 3 .XI Z/. The class associated to such a gerbe H ! X is calledDixmier–Douady class d.X/ 2 H 3 .XI Z/.If H ! X is a gerbe with band T jX over some space X, then the automorphismsof H (as a gerbe with band T jX ) are classified by H 1 .XI T/ Š H 2 .XI Z/.6.1.5 In the definition of a T -duality triple we furthermore need the following notation.We let y 2 H 1 .TI Z/ be the canonical generator. If pr i W T n ! T is the projectiononto the ith factor, then we set y i WD pr i y 2 H 1 .T n I Z/. Let E ! B be a T n -principal bundle and b 2 B. We consider its fibre E b over b. Choosing a base pointe 2 E b we use the T n -action in order to fix a homeomorphism a e W E ! T n such thata e .et/ D t for all t 2 T n . The classes x i WD a e .y i/ 2 H 1 .E b I Z/ are independentof the choice of the base point. Applying this definition to the bundle yE ! B belowgives the classes Ox i 2 H 1 . yE b I Z/ used in Definition 6.2.Definition 6.2. A T -duality triple t WD ..E; H /; . yE; yH/;u/ over B consists of twopairs .E; H /; . yE; yH/ over B and an isomorphism uW Op yH ! p H of gerbes withband T jEBdefined by the diagramyEp H uOp yH H E ByE p E O B,Op yEyH(49)where all squares are two-cartesian. The following conditions are required:1. The Dixmier–Douady classes of the gerbes satisfy d.H/ 2 F 2 H 3 .EI Z/ andd. yH/ 2 F 2 H 3 . yEI Z/.2. The isomorphism of gerbes u satisfies the condition [BRS, (2.7)] which says thefollowing. If we restrict the diagram (49) to a point b 2 B, then we can trivializethe restrictions of gerbes H jEb , yH j yE bto the fibres E b ; yE b of the T n -bundlesover b such that the induced isomorphism of trivial gerbes u b over E b yE b isclassified by P niD1 pr E bx i [ pr yE Ox i 2 H 2 .E b yE b I Z/.6.1.6 There is a natural notion of an isomorphism of T -duality triples. For a mapf W B 0 ! B and a T -duality triple over B there is a natural construction of a pull-backtriple over B 0 .

Duality for topological abelian group stacks and T -duality 325Definition 6.3. We let Triple.B/ denote the set of isomorphism classes of T -dualitytriples over B. Forf W B 0 ! B we let Triple.f /W Triple.B/ ! Triple.B 0 / be the mapinduced by the pull-back of T -duality triples.6.1.7 In this way we define a functorThis functor comes with specializationsTripleW TOP op ! Sets:s; Os W Triple ! Pgiven by s..E; H /; . yE; yH/;u/ WD .E; H / and Os..E; H /; . yE; yH/;u/ D . yE; yH/.Definition 6.4. A pair .E; H / is called dualizable if there exists a triple t 2 Triple.B/such that s.t/ Š .E; H /. The pair Os.t/ D . yE; yH/is called a T -dual of .E; H /.Thus the choice of a triple t 2 s 1 .E; H / encodes the necessary choices in orderto fix a T -dual. One of the main results of [BRS] is the following characterization ofdualizable pairs.Theorem 6.5. A pair .E; H / is dualizable in the sense of Definition 6.4 if and only ifd.H/ 2 F 2 H 3 .EI Z/.Further results of [BRS, Theorem 2.24] concern the classification of the set of dualsof a given pair .E; H /.6.1.8 For the purpose of the present paper it is more natural to interpret the isomorphismof gerbes uW Op yH ! p H in a T -duality triple ..E; H /; . yE; yH/;u/ in a different,but equivalent manner. To this end we introduce the notion of a dual gerbe.6.1.9 First we recall the definition of the tensor product w W H ˝X H 0 ! X of gerbesuW H ! X and u 0 W H 0 ! X with band T jX over X 2 S. Consider first the fibreproduct of stacks .u; u 0 /W H X H 0 ! X. Let T 2 S=X. An object s 2 H X H 0 .T /is given by a triple .t; t 0 ;/ of objects t 2 H.T/, t 0 2 H 0 .T / and an isomorphism W u.t/ ! u 0 .t 0 /. By the definition of a T jX -banded gerbe the group of automorphismsof t relative to u is the group Aut H.T/=rel.u/ .t/ Š T.T /. We thus have an isomorphismAut H X H 0 =rel..u;u 0 //.s/ Š T.T / T.T /. Similarly, for s 0 ;s 1 2 H X H 0 .T / the setHom H X H 0 =rel..u;u 0 //.s 0 ;s 1 / is a torsor over T.T / T.T /.We now define a prestack H ˝pX H 0 . By definition the groupoid H ˝pX H 0 .T / has thesame objects as H X H 0 .T /, but the morphism sets are factored by the anti-diagonalT.T / antidiag T.T / T.T /, i.e.Hom H ˝pX H 0 =rel.w/ .s 0;s 1 / D Hom H X H 0 =rel..u;u 0 //.s 0 ;s 1 /=antidiag.T.T //:The stack H ˝X H 0 is defined as the stackification of the prestack H ˝pX H 0 . Itisagainagerbe with band T jX . We furthermore have the following relation of Dixmier–Douadyclasses.d.H/ C d.H 0 / D d.H ˝X H 0 /:

326 U. Bunke, T. Schick, M. Spitzweck, and A. Thom6.1.10 The sheaf T jX 2 Sh Ab S=X gives rise to the stack BT jX (see 2.3.6). ThenX BT jX ! X is the trivial T-banded gerbe over X. Let H ! X be a gerbe withband T jX .Definition 6.6. A dual of the gerbe H ! X is a pair .H 0 ! X; / of a gerbe H 0 ! Xand an isomorphism of gerbes W H ˝X H 0 ! X BT jX .Every gerbe H ! X with band T jX admits a preferred dual H op ! X which wecall the opposite gerbe. The underlying stack of H op is H , but we change its structureof a T jX -banded gerbe using the inversion automorphism 1 W T jX ! T jX .If .H0 0 ! X; 0/ and .H1 0 ! X; / are two duals, then there exists a uniqueisomorphism class of isomorphisms H0 0 ! H 1 0 of gerbes such that the induced diagramH ˝X H00 H ˝X H100 1X BT jXcan be filled with a two-isomorphism. Note that if H 0 ! X is underlying gerbe of thedual of H ! X, then we have the relation of Dixmier–Douady classesd.H/ D d.H 0 /:Lemma 6.7. Let yH ! X and H ! X be gerbes with band T jX . There is a naturalbijection between the sets of isomorphism classes of isomorphisms yH ! H of gerbesover X and isomorphism classes of isomorphisms of T jX -banded gerbes yH ˝ H op !X BT jX .Proof. An isomorphism yH ! H induces an isomorphism yH ˝X H op ! H ˝X H op !X BT jX . On the other hand, an isomorphism yH ˝X H op ! X BT jX presentsyH ! X as a dual of H op ! X. Since W H ˝X H op ! X BT jX presents H asa dual of H op we have a preferred isomorphism class of isomorphisms yH ! H , too.6.1.11 In view of Lemma 6.7, in Definition 6.2 of a T -duality triple ..E; H /; . yE; yH/;u/we can consider u as an isomorphismOp yH op ˝X p H ! E ByE BT jEB yE :The condition 6.2 2. can be rephrased as follows. After restriction to a point b 2 B wecan find isomorphismsv W T n BT ! H b ; Ov W T n BT ! yH b : (50)

Duality for topological abelian group stacks and T -duality 327After a choice of ! BT in order to define the map s below we obtain a mapr W T n T n ! BT by the following diagram.yH opb. Ov;v/ H b.T n BT op / .T n BT/s yH opb ˝Eu bb yE bH b Eb yE b BT prBT BTrT n T nThe condition 6.2, 2. is now equivalent to the condition that we can choose the isomorphismsv; Ov in (50) such thatr .z/ DnXpr 1;i x [ pr 2;i x;iD1where z 2 H 2 .BTI Z/ and x 2 H 1 .TI Z/ are the canonical generators, andpr k;i ; W T n T n pr k! T n pr i! T, k D 1; 2, i D 1;:::;n are projections onto thefactors.6.1.12 Important topological invariants of T -duality triples are the Chern classes ofthe underlying T n -principal bundles. For a triple t D ..E; H /; . yE; yH/;u/we definec.t/ WD c.E/;Oc.t/ WD c. yE/:These classes belong to H 2 .BI Z n /.6.2 Torus bundles, torsors and gerbes6.2.1 In this subsection we review various interpretations of the notion of a T n -principalbundle.6.2.2 Let G be a topological group. Let us start with giving a precise definition of aG-principal bundle.Definition 6.8. A G-principal bundle E over a space B consists of a map of spaces W E ! B which admits local sections together with a fibrewise right action E G !E such that the natural mapE G ! E B E;.e; t/ 7! .e; et/is an homeomorphism. An isomorphism of G-principal bundles E ! E 0 is a G-equivariantmap E ! E 0 of spaces over B.

328 U. Bunke, T. Schick, M. Spitzweck, and A. ThomBy Prin B .G/ we denote the category of G-principal bundles over B. The setH 0 .Prin B .G// of isomorphism classes in Prin B .G/ is in one-to-one correspondencewith homotopy classes ŒB; BG of maps from B to the classifying space BG of G. Ifwe fix a universal bundle EG ! BG, then the bijectionŒB; BG ! H 0 .Prin B .G//is given byŒf W B ! BG 7! ŒB f;BG EG ! B:6.2.3 We now specialize to G WD T n . The classifying space BT n of T n has thehomotopy type of the Eilenberg–MacLane space K.Z n ;2/. We thus have a naturalisomorphismH 2 .BI Z n / Def.Š ŒB; K.Z n ; 2/ Š ŒB; BT n Š H 0 .Prin B .T n //:So, T n -principal bundles are classified by the characteristic class c.E/ 2 H 2 .BI Z n /.Using the decompositionH 2 .BI Z n / Š H 2 .BI Z/ ˚ :::H 2 .BI Z/„ ƒ‚ …n summandswe can writec.E/ D .c 1 .E/;:::;c n .E//:Definition 6.9. The class c.E/ is called the Chern class of E. The c i .E/ are called thecomponents of c.E/.In fact, if n D 1, then c.E/ is the classical first Chern class of the T-principalbundle E.6.2.4 Let S be a site and F 2 Sh Ab S be a sheaf of abelian groups.Definition 6.10. An F -torsor T is a sheaf of sets T 2 ShS together with an actionT F ! T such that the natural map T F ! T T is an isomorphism of sheaves.An isomorphism of F -torsors T ! T 0 is an isomorphism of sheaves which commuteswith the action of F .By Tors.F / we denote the category of F -torsors.6.2.5 In 2.3.4 we have introduced the category EXT.Z;F/whose objects are extensionsof sheaves of groupsW W 0 ! F ! W ! w Z ! 0; (51)and whose morphisms are isomorphisms of extensions. We have furthermore definedthe equivalence of categoriesgiven by U.W/ WD w 1 .1/.U W EXT.Z;F/ ! Tors.F /

Duality for topological abelian group stacks and T -duality 3296.2.6 Consider an extension W as in (51) and apply Ext ShAb S.Z;:::/. We get thefollowing piece of the long exact sequence!Hom ShAb S.Z;W/! Hom ShAb S.Z; Z/ı W! Ext 1 Sh Ab S .Z;F/! Ext1 Sh Ab S .Z;W/! :Let 1 2 Hom ShAb S.Z; Z/ be the identity and sete.W/ WD ı W .1/ 2 Ext 1 Sh Ab S .Z;F/:Recall that H 0 .EXT.Z;F// denotes the set of isomorphism classes of the categoryEXT.Z;F/. The following lemma is well-known (see e.g. [Yon60]).Lemma 6.11. The mapinduces a bijectionEXT.Z;F/3 W 7! e.W/ 2 Ext 1 Sh Ab S .Z;F/e W H 0 .EXT.Z;F// ! Ext 1 Sh Ab S .Z;F/:In view of 6.2.5 we have natural bijectionsH 0 .Tors.F // Š U H 0 .EXT.Z;F// Š e Ext 1 Sh Ab S .Z;F/: (52)6.2.7 Let G be an abelian topological group and consider a principal G-bundle Eover B with underlying map W E ! B. By T .E/ WD E ! B 2 ShS=B (see3.2.3) we denote its sheaf of sections. The right action of G on E induces an actionT .E/ G jB ! T .E/.Lemma 6.12. T .E/ is a G jB -torsor.Proof. This follows from the following fact. If X ! B and Y ! B are two maps,thenX B Y ! B Š X ! B Y ! Bin ShS=B. We apply this to the isomorphismE B .B G/ Š E G Š E B Eof spaces over B in order to get the isomorphismT .E/ G jB! T .E/ T .E/:

330 U. Bunke, T. Schick, M. Spitzweck, and A. Thom6.2.8 The construction E 7! T .E/ refines to a functorT W Prin B .G/ ! Tors.G jB /from the category of G-principal bundles Prin B .G/ over B to the category of G jB -torsors Tors.G jB / over B.Lemma 6.13. The functoris an equivalence of categories.T W Prin B .G/ ! Tors.G jB /Proof. It is a consequence of the Yoneda Lemma that T is an isomorphism on the levelof morphism sets. It remains to show that the underlying sheaf T of a G jB -torsor isrepresentable by a G-principal bundle. Since T is locally isomorphic to G jB this istrue locally. The local representing objects can be glued to a global representing object.We can now prolong the chain of bijections (52) toŒB; BG Š H 0 .Prin B .G// Š H 0 .Tors.G jB //Š H 0 .EXT.Z jB ;G jB // Š Ext 1 Sh Ab S=B .Z jB ;G jB / Š H 1 .BI G/:(53)6.2.9 Let S be some site and F 2 S be a sheaf of abelian groups. By Gerbe.F /we denote the two-category of gerbes with band F over S. It is well-known thatisomorphism classes of gerbes with band F are classified by Ext 2 Sh Ab S .ZI F/, i.e. thereis a natural bijectiond W H 0 .Gerbe.F // ! Ext 2 Sh Ab S .ZI F/:6.2.10 Let H ! G be a homomorphism of topological abelian groups with kernelK WD ker.H ! G/. Our main example is R n ! T n with kernel Z n .Definition 6.14. Let T be a space. An H -reduction of a G-principal bundle E D.E ! B/ on T is a diagram.F; /W F E; T Bwhere F ! T is an H -principal bundle, and is H -equivariant, where H acts onE via H ! G. An isomorphism .F; / ! .F 0 ; 0 / of H -reductions over T is anisomorphism f W F ! F 0 of H -principal bundles such that 0 ı f D . Let R E H .T /denote the category of H -reductions of E.

Duality for topological abelian group stacks and T -duality 331Observe that the group of automorphisms of every object in RH E .T / is isomorphicto Map.T; K/ ı (the superscript ı indicates that we take the underlying set). For a mapuW T ! T 0 over B there is a natural pull-back functor RH E .u/W RE H .T 0 / ! RH E .T /.Let S be a site as in 3.1.2 and assume that K; G; H; B belong to S. In this case it iseasy to check thatT 7! RH E .T /is a gerbe with band K jB on S=B.6.2.11 Let W E ! B be a G-principal bundle. Note that the pull-back E ! E hasa canonical section and is therefore trivialized. A trivialized G-bundle has a canonicalH -reduction. In other words, there is a canonical map of stacks over BcanW E ! R E H : (54)Note that an object of E.T / is a map T ! E; to this we assign the pull-back of thecanonical H -reduction of E.6.2.12 The construction Prin B .G/ 3 E 7! RH E 2 Gerbe.K/ is functorial in E andthus induces a natural map of sets of isomorphism classesr W H 0 .Prin B .G// ! H 0 .Gerbe.K//:Lemma 6.15. If H ! G is surjective and has local sections, and H 1 .B; H / ŠH 2 .B; H / Š 0, thenis a bijection.r W H 0 .Prin B .G// ! H 0 .Gerbe.K//Proof. The exact sequence 0 ! K ! H ! G ! 0 induces by 3.4 an exact sequence0 ! K ! H ! G ! 0of sheaves. We consider the following segment of the associated long exact sequencein cohomology:!H 1 .BI H / ! H 1 .BI G/ ı ! H 2 .BI K/ ! H 2 .BI H / ! :By our assumptions ı W H 1 .BI G/ ! H 2 .BI K/ is an isomorphism. One can checkthat the following diagram commutes:H 0 .Prin B .G//rH 0 .Gerbe.K jB //H 1 .BI G/ıd H 2 .BI K/.(55)This implies the result since the vertical maps are isomorphisms.

332 U. Bunke, T. Schick, M. Spitzweck, and A. ThomNote that we can apply this lemma in our main example where H D R n andG D T n . In this case the diagram (55) is the equalityc.E/ D d.R E Z n jB/ (56)in H 2 .BI Z n /.6.3 Pairs and group stacks6.3.1 Let E be a principal T n -bundle over B, or equivalently by (53), an extensionE 2 Sh Ab S=B0 ! T n jB ! E ! Z jB ! 0 (57)of sheaves of abelian groups. Let Qc.E/ 2 Ext 1 Sh Ab S=B .Z jB I T n jB/ be the class of thisextension. Under the isomorphismExt 1 Sh Ab S=B .Z jB I T n jB / Š H 1 .BI T n / Š H 2 .BI Z n /it corresponds to the Chern class c.E/ of the principal T n -bundle introduced in 6.9.6.3.2 We let Q E D Ext PIC.S/ .E; T jB / (see Lemma 2.20 for the notation) denote theset of equivalence classes of Picard stacks P 2 PIC.B/ with isomorphismsBy Lemma 2.20 we have a bijectionH 0 .P / Š ! E; H 1 .P / Š ! T B :Ext 2 Sh Ab S=B .E; T jB / Š Q E:This bijection induces a group structure on Q E which we will use in the discussion oflong exact sequences below. We will not need a description of this group structure interms of the Picard stacks themselves.We apply Ext Sh Ab S=B .:::;T jB / to the sequence (57) and get the following segmentof a long exact sequenceExt 1 Sh Ab S=B .T n jB ; T jB / ˛! Ext 2 Sh Ab S=B .Z jB ; T jB / ! Q E! Ext 2 Sh Ab S=B .T n jB ; T jB / ˇ! Ext 3 Sh Ab S=B .Z jB ; T jB /: (58)The maps ˛; ˇ are given by the left Yoneda product with the classQc.E/ 2 Ext 1 Sh Ab S=B .Z jB I T n jB /:

Duality for topological abelian group stacks and T -duality 3336.3.3 In this paragraph we identify the extension groups in the sequence (58) with sheafcohomology. For a site S with final object and X; Y 2 Sh Ab S we have a local-globalspectral sequence with second termE p;q2Š R p .I Ext q Sh Ab S.X; Y //which converges to Ext pCqSh Ab S.X; Y /.We apply this first to the sheaves Z jB ; T jB 2 Sh Ab S=B. The final object of the siteS=B is idW B ! B. WehaveExt q Sh Ab S=B .Z jB ; T jB / Š 0 for q 1 so that this spectralsequence degenerates at the second page and givesExt p Sh Ab S=B .Z jB ; T jB / Š H p .BI Hom ShAb S=B .Z jB ; T jB //Š H p .BI T jB / Š H p .BI T/ Š H pC1 .BI Z/:Since the group T n is admissible by Theorem 4.30, and by Corollary 3.14 we haveExt q Sh Ab S=B .T n jB ; T jB / Š 0 for q D 1; 2,wegetExt p Sh Ab S=B .T n jB ; T jB / Š H p .BI Hom ShAb S=B .T n jB ; T jB //Š H p .BI Hom ShAb S .T n ; T/ jB /Š H p .BI Z n jB /Š H p .BI Z n /for p D 1; 2. Therefore the sequence (58) has the formH 1 .BI Z n / ˛! H 3 .BI Z/ ! Q EOc! H 2 .BI Z n / ˇ! H 4 .BI Z/: (59)In this picture the maps ˛; ˇ are both given by the cup-product with the Chern classc.E/ 2 H 2 .BI Z n /, i.e. ˛..x i // D P x i [ c i .E/.6.3.4 Recall from 6.1.3 the decreasing filtration .F k H .EI Z// k0 and the spectralsequence associated to the decomposition of functors .EI :::/ D .B;:::/ıp W Sh Ab S=E ! Ab. This spectral sequence converges to GrH .EI Z/. Its secondpage is given by E i;j2Š H i .BI R j p Z/. The edge sequence for F 2 H 3 .EI Z/ has theformker.d 0;22W E 0;22! E 2;12 / d 3! 1;1coker.d 1;12W E 1;12! E 3;02 /2;1! F 2 H 3 .EI Z/ ! .E 2;1 0;22=im.d2 W E 0;22! E 2;1 d2//2! E 4;0We now make this explicit. Since the fibre of p is an n-torus, R 0 p Z Š Z,R 1 p Z Š Z n , R 2 p Z Š ƒ 2 Z n . The differential d 2 can be expressed in terms of theChern class c.E/. We get the following web of exact sequences, where K, A, B aredefined as the appropriate kernels and cokernels.2 :

334 U. Bunke, T. Schick, M. Spitzweck, and A. ThomDKH 1 .BI Z n /H 0 .BI ƒ 2 Z n /s˛H 3 .BI Z/i c.E/H 2 .BI Z n /ˇ(60)K AF 2 H 3 Nˇ.EI Z/ BH 4 .BI Z/0 0Since K as a subgroup of the free abelian group H 0 .B; ƒ 2 Z n / is free we can choosea lift s as indicated.6.3.5 We now define a map (the underlying pair map)upW Q E ! P.B/as follows. Any Picard stack P 2 PIC.S=B/ comes with a natural map of stacksP ! H 0 .P / (where on the right-hand side we consider a sheaf of sets as a stack). IfP 2 Q E , then H 0 .P / Š E has a natural map to Z jB (see (57)). We define the stackG by the pull-back in stacks on S=BG PEE ¹1º jBZ jB .All squares are two-cartesian, and the outer square is the composition of the two innersquares. We have omitted to write the canonical two-isomorphisms. By constructionE is a sheaf of T n -torsors (see 6.2.5), i.e. by Lemma 6.13 a T n -principal bundle, andG ! E is a gerbe with band T. We setup.P / WD .E; G/:

Duality for topological abelian group stacks and T -duality 3356.3.6 Recall Definition 6.4 of a dualizable pair.Lemma 6.16. If P 2 Q E , then up.P / 2 P.B/ is dualizable.Proof. Let .E; H / WD up.E/. In view of Theorem 6.5 we must show that d.H/ 2F 2 H 3 .EI Z/. Fork 2 Z we define G.k/ ! E.k/ by the two-cartesian diagramsG.k/ PE.k/E(61)The group structure of E induces maps ¹kº jBZ jB .W E.k/ B E.m/ ! E.k C m/ (62)On fibres appropriately identified with T n , this map is the usual group structure on T n .Since P is a Picard stack these multiplications are covered by OW G.k/ G.m/ !G.k C m/. The isomorphism class of G.k/ therefore must satisfypr E.k/ G.k/ ˝ pr E.m/ G.m/ Š G.k C m/ 2 Gerbe.E.k/ B E.m//: (63)We now write out this isomorphism in terms of Dixmier–Douady classes d k WDd.G.k// 2 H 3 .E.k/I Z/.We fix a generator of H 1 .TI Z/. This fixes a choice of generators of x i 2 H 1 .T n I Z/,i D 1;:::;nvia pullback along the coordinate projections. Let a W T n T n ! T n bethe group structure. Then we havea .x i / D pr 1 x i C pr 2 x i: (64)where pr i W T n T n ! T n , i D 1; 2, are the projections onto the factors.Let us for simplicity assume that B is connected. Let .E r .k/; d r .k// be the Serrespectral sequence of the composition E.k/ ! B !(see 6.1.7). Then we can identifyE 0;32 .k/ Š ƒ3 Z H 1 .T n I Z/. The class d k has a symbol in E 0;32.k/ which can be writtenas P i

336 U. Bunke, T. Schick, M. Spitzweck, and A. ThomBecause of the presence of mixed terms this is only possible if everything vanishes.This implies that d k 2 F 1 H 3 .E.k/I Z/. We write P i;j x i ^x j ˝u i;j .k/ for its symbolin E 1;22 , where u i;j 2 H 1 .BI Z/. As above equation (63) now impliesXpr 1 .x i ^ x j / ˝ u i;j .k/ C X pr 2 .x l ^ x r / ˝ u l;r .m/i;jl;kD X .pr 1 x a C pr 2 x a/ ^ .pr 1 x b C pr 2 x b/ ˝ u a;b .k C m/:a;bAgain the presence of mixed terms implies that everything vanishes. This shows thatd k 2 F 2 H 3 .E.k/I Z/. The assertion of the lemma is the case k D 1.6.3.7 Let us now combine (60) and (59) into a single diagram. We get the followingweb of horizontal and vertical exact sequences:H 1 .BI Z n /˛H 1 .BI Z n / H 3 .BI Z/KsAhQ EfF 2 H 3 .EI Z/Oc H 2 .BI Z n / BNˇˇ H 4 .BI Z/ H 4 .BI Z/0 0 0.(65)The map f W Q E ! F 2 H 3 .EI Z/ associates to P 2 Q E the Dixmier–Douady classof the gerbe of the pair up.P / 2 P.B/ with underlying T n -bundle E. Here we useLemma 6.16. Surjectivity of f follows by a diagram chase once we have shown thefollowing lemma.Lemma 6.17. The diagram (65) commutes.Proof. We have to check that the left and the right squaresH 3 h.BI Z/ Q EfA F 2 H 3 .EI Z/andOcQ EH 2 .BI Z n /fF 2 H 3 .EI Z/(66)e Bcommute. Let us start with the left square. Let d 2 H 3 .BI Z/. Under the identificationH 3 .BI Z/ Š H 2 .BI T/ Š Ext 2 Sh Ab S=B.ZI T/

Duality for topological abelian group stacks and T -duality 337it corresponds to a group stack P 2 PIC.S=B/ with H 0 .P / Š Z and H 1 .P / Š T.The group stack h.d/ 2 Q E is given by the pull-back (two-cartesian diagram)h.d/ EPZ.In particular we see that the gerbe H ! E of the pair u.h.d// DW .E; H / is given bya pull-backH EG B,where G 2 Gerbe.B/ is a gerbe with Dixmier–Douady class d.G/ D d. The compositionf ı hW H 3 .BI Z/ ! H 3 .EI Z/ is thus given by the pull-back along the mapp W E ! B, i.e. p D f ı h. By construction the composition H 3 .BI Z/ ! A !F 2 H 3 .EI Z/ is a certain factorization of p . This shows that the left square commutes.Now we show that the right square in (66) commutes. We start with an explicitdescription of Oc. Let P 2 Q E . The principal T n -bundle in G.0/ ! E.0/ ! B istrivial (see (61) for notation). Therefore the Serre spectral sequence .E r .0/; d r .0//degenerates at the second term. We already know by Lemma 6.16 that the Dixmier–Douady class of G.0/ ! E.0/ satisfies d 0 2 F 2 H 3 .E.0/I Z/. Its symbol can bewritten as P niD1 x i ˝Oc i .P / for a uniquely determined sequence Oc i .P / 2 H 2 .BI Z/.These classes constitute the components of the class Oc.P/ 2 H 2 .BI Z n /.We write the symbol of f.P/D d 1 as P i x i ˝ a i for a sequence a i 2 H 2 .BI Z/.As in the proof of Lemma 6.16 the equation (63) gives the identitynXnXnXpr 2 x i ˝ a i C pr 1 x i ˝Oc i .P / .pr 1 x i C pr 2 x i/ ˝ a iiD1iD1modulo the image of a second differential d 0;22 .1/. This relation is solved by a i WDOc i .P / and determines the image of the vector a WD .a 1 ;:::;a n / under H 2 .BI Z n / !H 2 .BI Z n /=im.d 0;22.1// DW B uniquely. Note that e ı f.P/is also represented by theimage of the vector a in B. This shows that the right square in (66) commutes.6.4 T -duality triples and group stacks6.4.1 Let R n be the classifying space of T -duality triples introduced in [BRS]. Itcarries a universal T -duality triple t univ WD ..E univ ;H univ /; . yE univ ; yH univ /; u univ /. Letc univ ; Oc univ 2 H 2 .R n I Z n / be the Chern classes of the bundles E univ ! R n , yE univ ! R n .They satisfy the relation c univ [Oc univ D 0. Let E univ be the extension of sheavesiD1

338 U. Bunke, T. Schick, M. Spitzweck, and A. Thomcorresponding to E univ ! R n as in 6.3.1. In [BRS] we have shown that H 3 .R n I Z/ Š 0.The diagram (65) now implies that there is a unique Picard stack P univ 2 Q Euniv withOc.P univ / DOc univ and underlying pair up.P univ / Š .E univ ;H univ / (see 6.3.5).6.4.2 Let us fix a T n -principal bundle E ! B, or the corresponding extension ofsheaves E. Let us furthermore fix a class h 2 F 2 H 3 .EI Z/. In [BRS] we haveintroduced the set Ext.E; h/ of extensions of .E; h/ to a T -duality triple. The maintheorem about this extension set is [BRS, Theorem 2.24].Analogously, in the present paper we can consider the set of extensions of .E; h/to a Picard stack P with underlying T n -bundle E ! B and f.P/ D h, wheref W Q E ! F 2 H 3 .EI Z/ is as in (65). In symbols we can write f1 .h/ for this set.The main goal of the present section is to compare the sets Ext.E; h/ and f1 .h/ Q E . In the following paragraphs we construct maps between these sets.6.4.3 We fix a T n -principal bundle E ! B and let E 2 Sh Ab S=B be the correspondingextensions of sheaves. Let Triple E .B/ denote the set of isomorphism classes of triplest such that c.t/ D c.E/. We first define a mapˆW Triple E .B/ ! Q E :Let t 2 Triple E .B/ be a triple which is classified by a map f t W B ! R n . Pulling backthe group stack P univ 2 PIC.S=R n / we get an element ˆ.t/ WD ft .Puniv/ 2 PIC.S=B/.If t 2 Triple E .B/, then we have ˆ.t/ 2 Q E . We further haveOc.ˆ.t// D ft Oc.P univ / D ft Oc univ DOc.t/: (67)6.4.4 In the next few paragraphs we describe a map‰ W Q E ! Triple E .B/;i.e. a construction of a T -duality triple ‰.P/ 2 Triple E .B/ starting from a Picard stackP 2 Q E .6.4.5 Consider P 2 PIC.S=B/. We have already constructed one pair .E; H /. Thedual D.P / WD HOM PIC.S=B/ .P; BT jB / is a Picard stack with (see Corollary 5.10)H 0 .D.P // Š D.H 1 .P // Š D.T jB / Š Z jB ;H 1 .D.P // Š D.H 0 .P // Š D.E/:In view of the structure (57) of E, the equalities D.T n jB / Š Zn jB , D.Z jB / Š T jB , andExt 1 Sh Ab S=B .T jB ; T jB / Š 0 we have an exact sequence0 ! T jB ! D.E/ ! Z n jB! 0: (68)

Duality for topological abelian group stacks and T -duality 339Using the construction 2.5.12 we can form a quotient D.P / which fits into the sequenceof maps of Picard stacksBT jB ! D.P / ! D.P /;where H 0 .D.P // Š Z jB and H 1 .D.P // Š Z n jB. The fibre productRD.P / ¹1º Z jBdefines a gerbe R ! B with band Z n jB .6.4.6 By Lemma 6.15 there exists a unique isomorphism class yE ! B of aT n -bundle whose Z n jB -gerbe of Rn -reductions R y EZ n is isomorphic to R. We fix suchan isomorphism and obtain a canonical map of stacks can (see (54)) fitting into thediagramyH op zyH D.P /yE can R y EZ nŠRD.P /(69) B ¹1º Z jB .The gerbes yH op ! yE and z yH ! R are defined such that the squares become twocartesian(we omit to write the two-isomorphisms). In this way the Picard stack P 2Q E defines the second pair . yE; yH/ of the triple ‰.P/ D ..E; H /; . yE; yH/;u/ whoseconstruction has to be completed by providing u.Lemma 6.18. We have the equality Oc.P/ DOc.‰.P// in H 2 .BI Z n jB /.Proof. By the definition in 6.1.12 we have Oc.‰.P// D c. yE/. Furthermore, by (56)Ewe have c. yE/ D d.R y / D d.R/. By Lemma 5.14 we have .D.P// D D..P //,Z n jBwhere is the characteristic class (46), andD W Q E Š Ext 2 Sh Ab S=B .E; T jB / ! Ext 2 Sh Ab S=B .D.T jB /; D.E//is as in Lemma 5.14. The map Oc W Q E! H 2 .BI Z/ by its definition fits into the

340 U. Bunke, T. Schick, M. Spitzweck, and A. ThomdiagramQ ET n jB !EExt 2 Sh Ab S=B .T n jB I T jB /DDExt 2 Sh Ab S=B .D.T jB /; D.E// D.E/!D.T n jB / Ext2Sh Ab S=B .D.T jB /I D.T n jB //aOcŠ Ext 2 Sh Ab S=B .Z jB ; Zn jB /ŠH 2 .BI Z n jB /.By construction the class a.P / 2 Ext 2 Sh Ab S=B .Z jB ; Zn jB/ classifies the Picard stackD.E/. This implies that Oc.P/ classifies the gerbe R ! B, i.e. Oc.P/ D c. yE/.6.4.7 It remains to construct the last entryuW H ˝EB yE y H op ! BT jEB yE(70)of the triple ‰.t/ D ..E; H /; . yE; yH/;u/, where we use Picture 6.1.11. Note thatH 1 .P / Š T jB . Construction 2.5.12 gives rise to a sequence of morphisms of PicardstacksBT jB ! P ! xP(we could write xP D E).We have an evaluation ev W P B D.P / ! BT jB . For a pair .r; s/ 2 Z Z weconsider the following diagram with two-cartesian squares.H r zB H y evsP D.P / BT jBu r;sHzry ˝Er ByR sH sP ˝ xP D.P / D.P /wE r B R s xP D.P /¹r; sº jBZ jB Z jB .

Duality for topological abelian group stacks and T -duality 341By an inspection of the definitions one checks that the natural factorization u r;s existsif r D s. Furthermore one checks that.w; u r;s /W H r ˝Er B R sz y H s ! .E r B R s / BT jBis an isomorphism of gerbes with band T jEr B R sif and only if r D s D 1. Bothstatements can be checked already when restricting to a point, and therefore becomeclear when considering the argument in the proof of Lemma 6.19.We define the map (70) byuH H y ˝EB yE op H z1y ˝E1 ByR 1H 1u 1;1BT jBE ByE.id;can/ E1 ByR 1(note that R 1 D R, E 1 D E and H 1 D H ).Lemma 6.19. The triple ‰.P/ D ..E; H /; . yE; yH/;u/ constructed above is a T -duality triple.Proof. It remains to show that the isomorphism of gerbes u satisfies the condition 6.2,2 in the version of 6.1.11.By naturality of the construction ‰ in the base B and the fact that condition 6.2, 2can be checked at a single point b 2 B, we can assume without loss of generality thatB is a point. We can further assume that E D Z T n and P D BT Z T n .Inthis caseD.P / D D.BT/ D.Z/ D.T n / Š Z BT BZ n :We haveH z yH Š .BT T n / .BT BZ n /:The restriction of the evaluation map H z yH ! H ˝ zyH ! BT is the composition.BT T n / .BT BZ n / Š BT .BT/ T n BZ nididev! BT BT BT † ! BT:We are interested in the contribution ev W T n BZ n ! BT n .It suffices to see that in the case n D 1 we haveev .z/ D x ˝ y;where x 2 H 1 .TI Z/, y 2 H 1 .BZI Z/, and z 2 H 2 .BT/ Š Z are the canonicalgenerators. In fact this implies via the Künneth formula that ev .z/ D P niD1 x i ˝ y i ,

342 U. Bunke, T. Schick, M. Spitzweck, and A. Thomwhere x i WD p i .x/, y i WD q i .y/ for the projections onto the components p i W T n !T and q i W BZ n Š .BZ/ n ! BZ. Finally we use that can .y i / D x i , wherecanW T n ! BZ n is the canonical map (54) from a second copy of the torus to itsgerbe of R n -reductions (after identification of this gerbe with BZ n ).This finishes the construction of ‰ which started in 6.4.4.6.4.8 In [BRS, 2.11] we have seen that the group H 3 .BI Z/ acts on Triple.B/ preservingthe subsets Triple E .B/ Triple.B/ for every T n -bundle E ! B. We will recallthe description of the action in the proof of Lemma 6.20 below. By (65) it also acts onQ E .Lemma 6.20. The map ‰ is H 3 .BI Z/-equivariant.The proof requires some preparations.6.4.9 Note that we have a canonical isomorphism T Š D.Z/. In order to work withcanonical identifications we are going to use D.Z/ instead of T.The isomorphisms classes of gerbes with band D.Z/ over a space B 2 S areclassified byH 2 .BI D.Z/ jB / Š Ext 2 Sh Ab S=B .Z jB ;D.Z/ jB/: (71)The latter group also classifies Picard stacks P with fixed isomorphisms H 0 .P / Š Z jBand H 1 .P / Š D.Z jB /.Given P the gerbe G ! B can be reconstructed as a pull-backG P¹1º jBZ jB .If we want to stress the dependence of G on P we will write G.P /.Recall that an object in P.T/ as a stack over S consists of a map T ! B and anobject of P.T ! B/. In a similar manner we interpret morphisms.For example, the stack ¹1º jBis the space B.6.4.10 Let P 2 Ext PIC.S=B/ .Z jB ;D.Z/ jB / be a Picard stack P with a fixed isomorphismsH 0 .P / Š Z jB and H 1 .P / Š D.Z jB /. Let W Ext PIC.S=B/ .Z jB ;D.Z/ jB / !Ext 2 Sh Ab S=B .Z jB ;D.Z/ jB/ be the characteristic class.Lemma 6.21. .D.P// Š .P/.Proof. First of all note that we have canonical isomorphismsH 0 .D.P // Š D.H 1 .P / Š D.D.Z jB // Š Z jBandH 1 .D.P // Š D.H 0 .P // Š D.Z jB / Š D.Z/ jB :

Duality for topological abelian group stacks and T -duality 343Therefore we can consider D.P / 2 Ext PIC.S=B/ .Z jB ;D.Z/ jB / in a canonical way, and.P/ and .D.P// belong to the same group.Note thatd.G.P// D .P /; d.G.D.P /// D .D.P//under the isomorphism (71). It suffices to show that d.G.P// D d.G.D.P ///.In fact, as in 6.4.7 we have the following factorization of the evaluation map:G.P / ˝BG .D.P //P ˝ZjB D.P / P ZjB D.P / P B D.P /evT jBdiag¹1º jBZ jB Z jBZ jB Z jB .This represents G.D.P // as the dual gerbe G.P / op of G.P / in the sense of Definition6.6. The relation d.G.P// D d.G.D.P /// follows.6.4.11 We now start the actual proof of Lemma 6.21.In order to see that ‰ is H 3 .BI Z/-equivariant we will first describe the action ofH 3 .BI Z/ on the sets of isomorphism classes of T -duality triples with fixed underlyingT n -bundles E and yE on the one hand, and on the set of isomorphism classes of Picardstacks Q E , on the other.Consider g 2 H 3 .BI Z/. It classifies the isomorphism class of a gerbe G ! Bwith band T jB .Ift is represented by ..E; H /; . yE; yH/;u/, then g C t is representedby..E; H ˝ G/;. yE; yH ˝O G/;u ˝ id r G/;where the maps ; O;r are as in (49).Note the isomorphism H 3 .BI Z/ Š H 2 .BI T/ Š Ext 2 Sh Ab S=B .Z jB ; T jB /. Thereforethe class g also classifies an isomorphism class of Picard stacks with H 0 . zG/ Š Z jBand H 1 . zG/ Š T jB . From zG we can derive the gerbe G by the pull-backG zG ¹1º jBZ jB .6.4.12 Recall that we consider an extension E 2 Sh Ab S=B of the form0 ! T n jB ! E ! Z jB ! 0:We consider a Picard stack P 2 Ext PIC.S=B/ .E;D.Z/ jB /.Let furthermore zG 2 Ext PIC.S=B/ .Z jB ;D.Z/ jB /. Then we defineP ˝EzG 2 Ext PIC.S=B/ .E;D.Z/ jB /

344 U. Bunke, T. Schick, M. Spitzweck, and A. Thomby the diagramBT jBCBT jB B BT jBP ˝EzG P ZjBzG P B zG(72)Z jBdiagZ jB B Z jB .The right lower square is cartesian, and in the left upper square we take the fibre-wisequotient by the anti-diagonal action of BT jB .6.4.13 We have D.P / 2 Ext PIC.S=B/ .Z;D.E//, where0 ! BT jB ! D.E/ ! D.E/ ! 0:We define D.P / to be the quotient of D.P / by BT jB in the sense of 2.5.12 so thatD.P / ! D.P / is a gerbe with band T.We define D.P / ˝D.P /D. zG/ by the diagramBT jBCBT jB B BT jBD.P / ˝D.P /D. zG/ D.P / ZjB D. zG/ D.P / B D. zG/(73)Z jBdiagZ jB B Z jB .Again, the right lower square is cartesian, and in the left upper square we take thefibre-wise quotient by the anti-diagonal action of BT jB .Proposition 6.22. We have an equivalence of Picard stacksD.P ˝EzG/ Š D.P / ˝D.P /D. zG/:Proof. The diagram (73) defines D.P / ˝D.P /D. zG/ by forming the pull-back to thediagonal and then taking the quotient of the anti-diagonal BT jB -action in the fibre.One can obtain this diagram by dualizing (72) and interchanging the order of pull-backand quotient.

Duality for topological abelian group stacks and T -duality 3456.4.14 We can now finish the argument that ‰ is H 3 .BI Z/-equivariant. We let‰.P/ D ..E; H /; . yE; yH/;u/ and ‰.g C P/ D ..E 0 ;H 0 /; . yE 0 ; yH 0 /; u 0 /. Note thatg C P D P ˝EzG. An inspection of the construction of the first entry of ‰ shows thatE 0 Š E and H 0 Š H ˝E prE ! BG. Proposition 6.22 shows that D.P ˝EzG/ ŠD.P /. This implies that yE 0 Š yE. Furthermore, if we restrict D.P ˝EzG/ along¹1º jB! Z jB we get by Proposition 6.22 and the proof of Lemma 6.21 a diagram ofstacks over B:yH op ˝ yE Gop z y H ˝RyERG opD.P ˝EzG/Š D.P / ˝D.P /D. zG/yEcan RyERD.P ˝EzG/Š D.P /¹1º jBZ jB .The map u 0 is induced by the evaluation .P ˝EzG/ D.P ˝EzG/ ! BT. Withthe given identifications using the “duality” between (73) and (72) we see that thisevaluation the induced by the product of the evaluations.P D.P // . zG D. zG// ! BT BT ! BT:After restriction to ¹1º jBwe see that u 0 D u ˝ v, where v W G ˝ G op ! BT is thecanonical pairing. This finishes the proof of the equivariance of ‰.Theorem 6.23. The maps ‰ and ˆ are inverse to each other.Proof. We first show the assertion under the additional assumption that H 3 .BI Z/ D 0.In this case an element P 2 Q E is determined uniquely by the class Oc.P/ 2 H 2 .BI Z n /.Similarly, a T -duality triple t 2 Triple.B/ with c.t/ D c.E/ is uniquely determined bythe class Oc.t/ D c. yE/. Since for P 2 Q E we have Oc.ˆı‰.P// (67)Lemma 6.18D Oc.‰.P// DLemma 6.18Oc.P/and Oc.‰ıˆ.t// D Oc.ˆ.t// (67)D Oc.t/this implies that ˆı‰ jQE D id QE and‰ ı ˆjs 1 .E/ D id js 1 .E/, where s W Triple ! P is as in 6.1.7. Note that H 3 .R n I Z/ D0. Therefore‰.ˆ.t univ // D t univ ; ˆ.‰.P univ // Š P univ :Now consider a general space B 2 S. We first show that ‰ ı ˆ D id. Lett 2 s 1 .E/ be classified by the map f t W B ! R n , i.e. ft tuniv D t. Then we have‰.ˆ.t// D ‰.ft Puniv/ D ft ‰.Puniv/ D ft tuniv D t, i.e.We consider the group‰ ı ˆ D id: (74) E WD .im.˛/ C im.s//=im.˛/ H 3 .BI Z/=im.˛/:

346 U. Bunke, T. Schick, M. Spitzweck, and A. ThomThis group is exactly the group ker. /=C in the notation of [BRS, Theorem 2.24(3)].It follows from (65) that the action of H 3 .BI Z/ on Q E induces an action of E onQ E which preserves the fibres of f . In [BRS, Theorem 2.24(3)] we have shown that italso acts on Triple E .B/ and preserves the subsets Ext.E; h/.Let us fix a Picard stack P 2 Q E , and let Oc WD Oc.P/ and h 2 H 3 .EI Z/ be suchthat f.P/ 2 Ext.E; h/. From (65) we see that the group E acts simply transitivelyon the setA E; Oc WD ¹Q 2 f1 .h/ jOc.Q/ DOcº:By [BRS, Theorem 2.24(3)] it also acts simply transitively on the setB E; Oc WD ¹t 2 Ext.E; h/ jOc.t/ DOcº:By Lemma 6.18 we have ‰.A E; Oc / B E; Oc . By Lemma 6.20 the map ‰ is E -equivariant. Hence it must induce a bijection between A E; Oc and B E; Oc . Ifwelet Oc runover all possible choices (solutions of Oc [c.E/ D 0) we see that ‰ W Q E ! Triple E .B/is a bijection. In view of (74) we now also get ˆ ı ‰ D id.Acknowledgment. We thank Tony Pantev for the crucial suggestion which initiatedthe work on this paper.References[Brd97] Glen E. Bredon, Sheaf theory, Grad. Texts in Math. 170, Springer-Verlag, New York1997.[Bre69] Lawrence Breen, Extensions of abelian sheaves and Eilenberg-MacLane algebras, Invent.Math. 9 (1969/1970), 15–44.[Bre76] Lawrence Breen, Extensions du groupe additif sur le site parfait, in Algebraic surfaces(Orsay, 1976–78), Lecture Notes in Math. 868, Springer-Verlag, Berlin 1981, 238–262.[Bre78] Lawrence Breen, Extensions du groupe additif, Inst. Hautes Études Sci. Publ. Math.48 (1978), 39–125.[Bro82] Kenneth S. Brown, Cohomology of groups, Grad. Texts in Math. 87, Springer-Verlag,New York 1982.[BRS] Ulrich Bunke, Philipp Rumpf, and Thomas Schick, The topology of T -duality forT n -bundles, Rev. Math. Phys. 18 (2006), 1103–1154.[BS05] Ulrich Bunke and Thomas Schick, On the topology of T -duality, Rev. Math. Phys. 17(2005), 77–112.[BSS] Ulrich Bunke, Thomas Schick, and Markus Spitzweck, Sheaf theory for stacks inmanifolds and twisted cohomology for S 1 -gerbes, Algebr. Geom. Topol. 7 (2007),1007–1062.[Del] P. Deligne, La formule de dualité globale, in Théorie des topos et cohomologie étaledes schémas Tome 3, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964(SGA 4), Lecture Notes in Math. 305, Springer-Verlag, Berlin 1973, 481–587

Duality for topological abelian group stacks and T -duality 347[DP] Ron Donagi and Tony Pantev, Torus fibrations, gerbes, and duality, Mem. Amer. Math.Soc. 193, no. 901, (2008).[Fol95] Gerald B. Folland, A course in abstract harmonic analysis, Stud. Adv. Math. CRCPress, Boca Raton, FL, 1995.[HM98] Karl H. Hofmann and Sidney A. Morris, The structure of compact groups, de GruyterStud. Math. 25, Walter de Gruyter, Berlin 1998.[Hov99] Mark Hovey, Model categories, Math. Surveys Monogr. 63, Amer. Math. Soc., Providence,RI, 1999.[HR63] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis, Vol. I: Structureof topological groups, Integration theory, group representations, Grundlehren Math.Wiss. 115, Springer-Verlag, Berlin 1963.[KS94] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren Math.Wiss. 292, Springer-Verlag, Berlin 1994.[MR05] Varghese Mathai and Jonathan Rosenberg, T -duality for torus bundles with H -fluxesvia noncommutative topology, Comm. Math. Phys. 253 (2005), 705–721.[MR06] Varghese Mathai and Jonathan Rosenberg, T -duality for torus bundles with H -fluxesvia noncommutative topology. II. The high-dimensional case and the T -duality group,Adv. Theor. Math. Phys. 10 (2006), 123–158.[Sch07] Ansgar Schneider, PhD thesis, Georg-August-Universität Göttingen, 2007.[Ser02] Jean-Pierre Serre, Galois cohomology, Springer Monogr. Math., Springer-Verlag,Berlin 2002.[Ste67] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14(1967), 133–152.[Tam94] Günter Tamme, Introduction to étale cohomology, Universitext, Springer-Verlag,Berlin 1994.[Vis05] Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, inFundamental algebraic geometry, Math. Surveys Monogr. 123, Amer. Math. Soc.,Providence, RI, 2005, 1–104.[Yon60] Nobuo Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960),507–576.

Deformations of gerbes on smooth manifoldsPaul Bressler , Alexander Gorokhovsky , Ryszard Nest, and Boris Tsygan 1 IntroductionIn [5] we obtained a classification of formal deformations of a gerbe on a manifold (C 1or complex-analytic) in terms of Maurer–Cartan elements of the differential graded Liealgebra (DGLA) of Hochschild cochains twisted by the cohomology class of the gerbe.In the present paper we develop a different approach to the derivation of this classificationin the setting of C 1 manifolds, based on the differential-geometric approachof [4].The main result of the present paper is the following theorem which we prove inSection 8.Theorem 1. Suppose that X is a C 1 manifold and S is an algebroid stack on X whichis a twisted form of O X . Then, there is an equivalence of 2-groupoid valued functorsof commutative Artin C-algebrasDef X .S/ Š MC 2 .g DR .J X / ŒS /:Notations in the statement of Theorem 1 and the rest of the paper are as follows.We consider a paracompact C 1 -manifold X with the structure sheaf O X of complexvalued smooth functions. Let S be a twisted form of O X , as defined in Section 4.5.Twisted forms of O X are in bijective correspondence with OX -gerbes and are classifiedup to equivalence by H 2 .XI O / Š H 3 .XI Z/.One can formulate the formal deformation theory of algebroid stacks ([17], [16])which leads to the 2-groupoid valued functor Def X .S/ of commutative Artin C-algebras.We discuss deformations of algebroid stacks in Section 6. It is natural to expectthat the deformation theory of algebroid pre-stacks is “controlled” by a suitably constructeddifferential graded Lie algebra (DGLA) well-defined up to isomorphism in thederived category of DGLA. The content of Theorem 1 can be stated as the existence ofsuch a DGLA, namely g.J X / ŒS , which “controls” the formal deformation theory ofthe algebroid stack S in the following sense.To a nilpotent DGLA g which satisfies g i D 0 for i< 1 one can associate itsDeligne 2-groupoid which we denote MC 2 .g/, see [11], [10] and references therein.We review this construction in Section 3. Then Theorem 1 asserts equivalence of the2-groupoids Def X .S/ and MC 2 .g DR .J X / ŒS /. Supported by the Ellentuck Fund Partially supported by NSF grant DMS-0400342 Partially supported by NSF grant DMS-0605030

350 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe DGLA g.J X / ŒS is defined as the ŒS-twist of the DGLAg DR .J X / WD .XI DR. xC .J X //Œ1/:Here, J X is the sheaf of infinite jets of functions on X, considered as a sheaf of topologicalO X -algebras with the canonical flat connection r can . The shifted normalizedHochschild complex xC .J X /Œ1 is understood to comprise locally defined O X -linearcontinuous Hochschild cochains. It is a sheaf of DGLA under the Gerstenhaber bracketand the Hochschild differential ı. The canonical flat connection on J X induces one, alsodenoted r can ,on xC .J X //Œ1. The flat connection r can commutes with the differentialı and acts by derivations of the Gerstenhaber bracket. Therefore, the de Rham complexDR. xC .J X //Œ1/ WD X ˝ xC .J X //Œ1/ equipped with the differential r can C ı andthe Lie bracket induced by the Gerstenhaber bracket is a sheaf of DGLA on X givingrise to the DGLA g.J X / of global sections.The sheaf of abelian Lie algebras J X =O X acts by derivations of degree 1 on thegraded Lie algebra xC .J X /Œ1 via the adjoint action. Moreover, this action commuteswith the Hochschild differential. Therefore, the (abelian) graded Lie algebra X ˝J X =O X acts by derivations on the graded Lie algebra X ˝ xC .J X //Œ1. We denotethe action of the form ! 2 X ˝ J X=O X by ! . Consider now the subsheaf of closedforms . X ˝J X=O X / cl which is by definition the kernel of r can . .X k ˝J X=O X / cl actsby derivations of degree k 1 and this action commutes with the differential r can C ı.Therefore, for ! 2 .XI . 2 ˝ J X =O X / cl / one can define the !-twist g.J X / ! asthe DGLA with the same underlying graded Lie algebra structure as g.J X / and thedifferential given by r can Cı C ! . The isomorphism class of this DGLA depends onlyon the cohomology class of ! in H 2 ..XI X ˝ J X=O X /; r can /.More precisely, for ˇ 2 .XI X 1 ˝ J X=O X / the DGLA g DR .J X / ! andg DR .J X / !Cr canˇ are canonically isomorphic with the isomorphism depending onlyon the equivalence class ˇ C Im.r can /.As we remarked before a twisted form S of O X is determined up to equivalence by itsclass in H 2 .XI O /. The composition O ! O =C log ! O=C j 1 ! DR.J=O/ inducesthe map H 2 .XI O / ! H 2 .XI DR.J=O// Š H 2 ..XI X ˝J X=O X /; r can /.We denote by ŒS 2 H 2 ..XI X ˝ J X=O X /; r can / the image of the class of S. Bythe remarks above we have the well-defined up to a canonical isomorphism DGLAg DR .J X / ŒS .The rest of this paper is organized as follows. In Section 2 we review some preliminaryfacts. In Section 3 we review the construction of Deligne 2-groupoid, its relationwith the deformation theory and its cosimplicial analogues. In Section 4 we review thenotion of algebroid stacks. Next we define matrix algebras associated with a descentdatum in Section 5. In Section 6 we define the deformations of algebroid stacks andrelate them to the cosimplicial DGLA of Hochschild cochains on matrix algebras. InSection 7 we establish quasiisomorphism of the DGLA controlling the deformationsof twisted forms of O X with a simpler cosimplicial DGLA. Finally, the proof the mainresult of this paper, Theorem 1, is given in Section 8

Deformations of gerbes on smooth manifolds 3512 Preliminaries2.1 Simplicial notions2.1.1 The category of simplices. For n D 0;1;2;:::we denote by Œn the categorywith objects 0;:::;nfreely generated by the graph0 ! 1 !!n:For 0 p q n we denote by .pq/ the unique morphism p ! q.We denote by the full subcategory of Cat with objects the categories Œn forn D 0;1;2;:::.For 0 i n C 1 we denote by @ i D @ n i W Œn ! Œn C 1 the i th face map, i.e. theunique map whose image does not contain the object i 2 Œn C 1.For 0 i n 1 we denote by s i D s n i W Œn ! Œn 1 the i th degeneracy map,i.e. the unique surjective map such that s i .i/ D s i .i C 1/.2.1.2 Simplicial and cosimplicial objects. Suppose that C is a category. By definition,a simplicial object in C (respectively, a cosimplicial object in C) is a functor op ! C (respectively, a functor ! C). Morphisms of (co)simplicial objects arenatural transformations of functors.For a simplicial (respectively, cosimplicial) object F we denote the object F .Œn/ 2C by F n (respectively, F n ).2.2 Cosimplicial vector spaces. Let V be a cosimplicial vector space. We denoteby C .V / the associated complex with component C n .V / D V n and the differential@ n W C n .V / ! C nC1 .V / defined by @ n D P i . 1/i @ n i , where @n iis the map inducedby the i th face map Œn ! Œn C 1. We denote cohomology of this complex by H .V /.The complex C .V / contains a normalized subcomplex xC .V /. Here xC n .V / DfV 2 V n j si nv D 0g, where sn iW Œn ! Œn 1 is the i th degeneracy map. Recall thatthe inclusion xC .V / ! C .V / is a quasiisomorphism.Starting from a cosimplicial vector space V one can construct a new cosimplicialvector space yV as follows. For every W Œn ! set yV D V .n/ . Suppose givenanother simplex W Œm ! and morphism W Œm ! Œn such that D ı , i.e., is a morphism of simplices ! . The morphism .0n/ factors uniquely into 0 !.0/ ! .m/ ! n, which, under , gives the factorization of .0n/W .0/ ! .n/(in ) into.0/ f ! .0/ g ! .m/ ! h .n/; (2.1)where g D .0m/. The map .m/ ! .n/ induces the map W yV ! yV : (2.2)Set now yV n D Q yV . The maps (2.2) endow yV with the structure of a cosimplicialvector space. We then have the following well-knownŒn! result:

352 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganLemma 2.1. H .V / Š H . yV/:Proof. We construct morphisms of complexes inducing the isomorphisms in cohomology.We will use the following notations. If f 2 yV n and W Œn ! we will denoteby f ./ 2 yV its component in yV .ForW Œn ! we denote by j Œjl W Œl j ! its truncation: j Œjl .i/ D .i C j/, j Œjl .i; k/ D ..i C j/ .k C j//. For 1 W Œn 1 ! and 2 W Œn 2 ! with 1 .n 1 / D 2 .0/ define their concatenationƒ D 1 2 W Œn 1 C n 2 ! by the following formulas.´ 1 .i/ if i n 1 ;ƒ.i/ D 2 .i n 1 / if i n 1 ;8ˆ< 1 .ik/ if i;k n 1 ;ƒ.ik/ D 2 ..i n 1 /.k n 1 // if i;k n 1 ;ˆ: 2 .0 .k n 1 // ı .in 1 / if i n 1 k:This operation is associative. Finally we will identify in our notations W Œ1 ! withthe morphism .01/ in .The morphism C .V / ! C . yV/is constructed as follows. Let W Œn ! be asimplex in and define k by .k/ D Œ k , k D 0;1;:::;n. Let ‡./W Œn ! .n/be a morphism in defined byThen define the map W V ! yV by the formula.‡.//.k/ D .kn/. k /: (2.3)..v//./ D ‡./ v for v 2 V n :This is a map of cosimplicial vector spaces, and therefore it induces a morphism ofcomplexes.The morphism W C . yV/! C .V / is defined by the formula.f / D . 1/ n.nC1/2X0i k kC1. 1/ i 0CCi n 1f.@ 0 i 0 @ 1 i 1@ n 1i n 1/ for f 2 yV nwhen n>0, and .f / is V 0 component of f if n D 0.The morphism ı is homotopic to Id with the homotopy hW C . yV/! C given by the formula1 . yV/hf ./ DXn 1Xj D0 0i k kC1. 1/ i 0CCi j 1f.@ 0 i 0@ j 1i j 1 ‡.j Œ0j / j Œj .n 1/ /for f 2 yV n when n>0, and h.f / D 0 if n D 0.The composition ı W C .V / ! C .V / preserves the normalized subcomplexxC .V / and acts as the identity on it. Therefore ı induces the identity map oncohomology. It follows that and are quasiisomorphisms inverse to each other.

Deformations of gerbes on smooth manifolds 3532.3 Covers. A cover (open cover) of a space X is a collection U of open subsets ofX such that S U 2U U D X.2.3.1 The nerve of a cover. Let N 0 U D `U 2UU . There is a canonical augmentationmap`U 2U 0 W N 0 U.U ,!X/ ! X:LetN p U D N 0 U X X N 0 Ube the .p C 1/-fold fiber product.The assignment N UW 3 Œp 7! N p U extends to a simplicial space called thenerve of the cover U. The effect of the face map @ n i(respectively, the degeneracymap si n) will be denoted by d i D dn i (respectively, & i D &i n ) and is given by theprojection along the i th factor (respectively, the diagonal embedding on the i th factor).Therefore for every morphism f W Œp ! Œq in we have a morphism N q U ! N p Uwhich we denote by f . We will denote by f the operation .f / of pull-back alongf ;ifF is a sheaf on N p U then f F is a sheaf on N q U.For 0 i n C 1 we denote by pr n i W N nU ! N 0 U the projection onto the i thfactor. For 0 j m, 0 i j n the map pr i0pr imW N n U ! .N 0 U/ mcan be factored uniquely as a composition of a map N n U ! N m U and the canonicalimbedding N m U ! .N 0 U/ m . We denote this map N n U ! N m U by pr n i 0 :::i m.The augmentation map 0 extends to a morphism W N U ! X where the latter isregarded as a constant simplicial space. Its component of degree n n W N n U ! X isgiven by the formula n D 0 ı pr n i. Here 0 i n C 1 is arbitrary.2.3.2 Čech complex. Let F be a sheaf of abelian groups on X. One defines a cosimplicialgroup LC .U; F / D .NUI F /, with the cosimplicial structure inducedby the simplicial structure of N U. The associated complex is the Čech complex ofthe cover U with coefficients in F . The differential L @ in this complex is given byP . 1/ i .d i / .2.3.3 Refinement. Suppose that U and V are two covers of X. A morphism of covers W U ! V is a map of sets W U ! V with the property U .U / for all U 2 U.A morphism W U ! V induces the map NW N U ! N V of simplicial spaceswhich commutes with respective augmentations to X. The map N 0 is determined bythe commutativity ofU N 0 UN 0 .U / N 0 V.It is clear that the map N 0 commutes with the respective augmentations (i.e. is a map ofspaces over X) and, consequently induces maps N n D .N 0 / X nC1 which commutewith all structure maps.

354 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan2.3.4 The category of covers. Let Cov.X/ 0 denote the set of open covers of X. ForU; V 2 Cov.X/ 0 we denote by Cov.X/ 1 .U; V/ the set of morphisms U ! V. LetCov.X/ denote the category with objects Cov.X/ 0 and morphisms Cov.X/ 1 . Theconstruction of 2.3.1 is a functorN W Cov.X/ ! Top op =X:3 Deligne 2-groupoid and its cosimplicial analoguesWe begin this section by recalling the definition of Deligne 2-groupoid and its relationwith the deformation theory. We then describe the cosimplicial analogues of Deligne2-groupoids and establish some of their properties.3.1 Deligne 2-groupoid. In this subsection we review the construction of Deligne2-groupoid of a nilpotent differential graded algebra (DGLA). We follow [11], [10] andreferences therein.Suppose that g is a nilpotent DGLA such that g i D 0 for i< 1.A Maurer–Cartan element of g is an element 2 g 1 satisfyingd C 1 Œ; D 0: (3.1)2We denote by MC 2 .g/ 0 the set of Maurer–Cartan elements of g.The unipotent group exp g 0 acts on the set of Maurer–Cartan elements of g by thegauge equivalences. This action is given by the formula.exp X/ D 1XiD0.ad X/ i.dX C Œ; X/.i C 1/ŠIf exp X is a gauge equivalence between two Maurer–Cartan elements 1 and 2 D.exp X/ 1 thend C ad 2 D Ad exp X.dC ad 1 /: (3.2)We denote by MC 2 .g/ 1 . 1 ; 2 / the set of gauge equivalences between 1 , 2 . ThecompositionMC 2 .g/ 1 . 2 ; 3 / MC 2 .g/ 1 . 1 ; 2 / ! MC 2 .g/ 1 . 1 ; 3 /is given by the product in the group exp g 0 .If 2 MC 2 .g/ 0 we can define a Lie bracket Œ; on g 1 byŒa; b D Œa; db C Œ; b: (3.3)With this bracket g 1 becomes a nilpotent Lie algebra. We denote by exp g 1the corresponding unipotent group, and by exp the corresponding exponential mapg 1 ! exp g 1 .If 1 , 2 are two Maurer–Cartan elements, then the group exp g 1

Deformations of gerbes on smooth manifolds 355acts on MC 2 .g/ 1 . 1 ; 2 /. Let exp t 2 exp g 1 and let exp X 2 MC 2 .g/ 1 . 1 ; 2 /.Then.exp t/ .exp X/ D exp.dt C Œ; t/ expX 2 exp g 0 :Such an element exp t is called a 2-morphism between exp X and .exp t/.exp X/.Wedenote by MC 2 .g/ 2 .exp X; exp Y/the set of 2-morphisms between exp X and exp Y .This set is endowed with a vertical composition given by the product in the groupexp g 1 .Let 1 ; 2 ; 3 2 MC 2 .g/ 0 . Let exp X 12 ; exp Y 12 2 MC 2 .g/ 1 . 1 ; 2 / and exp X 23 ,exp Y 23 2 MC 2 .g/ 1 . 2 ; 3 /. Then one defines the horizontal compositionas follows. Let˝W MC 2 .g/ 2 .exp X 23 ; exp Y 23 / MC 2 .g/ 2 .exp X 12 ; exp Y 12 /! MC 2 .g/ 2 .exp X 23 exp X 12 ; exp X 23 exp Y 12 /exp 2t 12 2 MC 2 .g/ 2 .exp X 12 ; exp Y 12 /; exp 3t 23 2 MC 2 .g/ 2 .exp X 23 ; exp Y 23 /:Thenexp 3t 23 ˝ exp 2t 12 D exp 3t 23 exp 3.e ad X 23.t 12 //To summarize, the data described above forms a 2-groupoid which we denote byMC 2 .g/ as follows:1. the set of objects is MC 2 .g/ 0 ,2. the groupoid of morphisms MC 2 .g/. 1 ; 2 /, i 2 MC 2 .g/ 0 consists of• objects, i.e., 1-morphisms in MC 2 .g/ are given by MC 2 .g/ 1 . 1 ; 2 / – thegauge transformations between 1 and 2 ,• morphisms between exp X, expY 2 MC 2 .g/ 1 . 1 ; 2 / which are given byMC 2 .g/ 2 .exp X; exp Y/.A morphism of nilpotent DGLA W g ! h induces a functor W MC 2 .g/ !MC 2 .g/.We have the following important result ([12], [11] and references therein).Theorem 3.1. Suppose that W g ! h is a quasi-isomorphism of DGLA and let m be anilpotent commutative ring. Then the induced map W MC 2 .g ˝ m/ ! MC 2 .h ˝ m/is an equivalence of 2-groupoids.3.2 Deformations and Deligne 2-groupoid. Let k be an algebraically closed field ofcharacteristic zero.

356 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan3.2.1 Hochschild cochains. Suppose that A is a k-vector space. The k-vector spaceC n .A/ of Hochschild cochains of degree n 0 is defined byC n .A/ WD Hom k .A˝n ; A/:The graded vector space g.A/ WD C .A/Œ1 has a canonical structure of a graded Liealgebra under the Gerstenhaber bracket denoted by Œ;below. Namely, C .A/Œ1 iscanonically isomorphic to the (graded) Lie algebra of derivations of the free associativeco-algebra generated by AŒ1.Suppose in addition that A is equipped with a bilinear operation W A ˝ A ! A,i.e. 2 C 2 .A/ D g 1 .A/. The condition Œ; D 0 is equivalent to the associativityof .Suppose that A is an associative k-algebra with the product . For a 2 g.A/ letı.a/ D Œ; a. Thus, ı is a derivation of the graded Lie algebra g.A/. The associativityof implies that ı 2 D 0, i.e. ı defines a differential on g.A/ called the Hochschilddifferential.For a unital algebra the subspace of normalized cochains xC n .A/ C n .A/ is definedbyxC n .A/ WD Hom k ..A=k 1/˝n ; A/:The subspace xC .A/Œ1 is closed under the Gerstenhaber bracket and the action of theHochschild differential and the inclusion xC .A/Œ1 ,! C .A/Œ1 is a quasi-isomorphismof DGLA.Suppose in addition that R is a commutative Artin k-algebra with the nilpotentmaximal ideal m R The DGLA g.A/ ˝k m R is nilpotent and satisfies g i .A/ ˝k m R D0 for i < 1. Therefore, the Deligne 2-groupoid MC 2 .g.A/ ˝k m R / is defined.Moreover, it is clear that the assignment R 7! MC 2 .g.A/ ˝k m R / extends to a functoron the category of commutative Artin algebras.3.2.2 Star products. Suppose that A is an associative unital k-algebra. Let m denotethe product on A.Let R be a commutative Artin k-algebra with maximal ideal m R . There is a canonicalisomorphism R=m R Š k.An (R-)star product on A is an associative R-bilinear product on A ˝k R such thatthe canonical isomorphism of k-vector spaces .A ˝k R/ ˝R k Š A is an isomorphismof algebras. Thus, a star product is an R-deformation of A.The 2-category of R-star products on A, denoted Def.A/.R/, is defined as thesubcategory of the 2-category Alg 2 Rof R-algebras (see 4.1.1) with• Objects: R-star products on A,• 1-morphisms W m 1 ! m 2 between the star products i those R-algebra homomorphisms W .A ˝k R;m 1 / ! .A ˝k R;m 2 / which reduce to the identity mapmodulo m R , i.e. ˝R k D Id A ,

Deformations of gerbes on smooth manifolds 357• 2-morphisms b W ! , where ; W m 1 ! m 2 are two 1-morphisms, areelements b 2 1 C A ˝k m R A ˝k R such that m 2 ..a/; b/ D m 2 .b; .a//for all a 2 A ˝k R.It follows easily from the above definition and the nilpotency of m R that Def.A/.R/is a 2-groupoid.Note that Def.A/.R/ is non-empty: it contains the trivial deformation, i.e. the starproduct, still denoted m, which is the R-bilinear extension of the product on A.It is clear that the assignment R 7! Def.A/.R/ extends to a functor on the categoryof commutative Artin k-algebras.3.2.3 Star products and the Deligne 2-groupoid. We continue in notations introducedabove. In particular, we are considering an associative unital k-algebra A. Theproduct m 2 C 2 .A/ determines a cochain, still denoted m 2 g 1 .A/ ˝k R, hence theHochschild differential ı D Œm; in g.A/˝k R for any commutativeArtin k-algebra R.Suppose that m 0 is an R-star product on A. Since .m 0 / WD m 0 m D 0 mod m Rwe have .m 0 / 2 g 1 .A/ ˝k m R . Moreover, the associativity of m 0 implies that .m 0 /satisfies the Maurer–Cartan equation, i.e. .m 0 / 2 MC 2 .g.A/ ˝k m R / 0 .It is easy to see that the assignment m 0 7! .m 0 / extends to a functorDef.A/.R/ ! MC 2 .g.A/ ˝k m R /: (3.4)The following proposition is well known (cf. [9], [11], [10]).Proposition 3.2. The functor (3.4) is an isomorphism of 2-groupoids.3.2.4 Star products on sheaves of algebras. The above considerations generalize tosheaves of algebras in a straightforward way.Suppose that A is a sheaf of k-algebras on a space X. Let mW A˝k A ! A denotethe product.An R-star product on A is a structure of a sheaf of an associative algebras on A˝k Rwhich reduces to modulo the maximal ideal m R . The 2-category (groupoid) of Rstar products on A, denoted Def.A/.R/ is defined just as in the case of algebras; weleave the details to the reader.The sheaf of Hochschild cochains of degree n is defined byC n .A/ WD Hom.A˝n ; A/:We have the sheaf of DGLA g.A/ WD C .A/Œ1, and hence the nilpotent DGLA.XI g.A/ ˝k m R / for every commutative Artin k-algebra R concentrated in degrees 1. Therefore, the 2-groupoid MC 2 ..XI g.A/ ˝k m R / is defined.The canonical functor Def.A/.R/ ! MC 2 ..XI g.A/ ˝k m R / defined just as inthe case of algebras is an isomorphism of 2-groupoids.

358 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan3.3 G-stacks. Suppose that G W Œn ! G n is a cosimplicial DGLA. We assume thateach G n is a nilpotent DGLA. We denote its component of degree i by G n;i and assumethat G n;i D 0 for i< 1.Definition 3.3. A G-stack is a triple D . 0 ; 1 ; 2 /, where• 0 2 MC 2 .G 0 / 0 ,• 1 2 MC 2 .G 1 / 1 .@ 0 0 0 ;@ 0 1 0 /,satisfying the conditions 1 0 1 D Id;• 2 2 MC 2 .G 2 / 2 .@ 1 2 . 1 / ı @ 1 0 . 1 /; @ 1 1 . 1 //satisfying the conditions@ 2 2 2 ı .Id ˝ @ 2 0 2/ D @ 2 1 2 ı .@ 2 3 2 ˝ Id/; and s 2 0 2 D s 2 1 2 D Id: (3.5)Let Stack.G/ 0 denote the set of G-stacks.Definition 3.4. For 1 ; 2 2 Stack.G/ 0 a 1-morphism ÇW 1 ! 2 is a pair Ç D.Ç 1 ; Ç 2 /, where Ç 1 2 MC 2 .G 0 / 1 . 0 1 ;0 2 /, Ç2 2 MC 2 .G 1 / 2 . 1 2 ı @0 0 .Ç1 /; @ 0 1 .Ç1 / ı 1 1 /,satisfying.Id ˝ 1 2 / ı .@1 2 Ç2 ˝ Id/ ı .Id ˝ @ 1 0 Ç2 / D @ 1 1 Ç2 ı .2 2 ˝ Id/;s0 1 Ç2 D Id:(3.6)Let Stack.G/ 1 . 1 ; 2 / denote the set of 1-morphisms 1 ! 2 .Composition of 1-morphisms ÇW 1 ! 2 and ÈW 2 ! 3 is given by .È 1 ı Ç 1 ;.Ç 2 ˝ Id/ ı .Id ˝ È 2 //.Definition 3.5. For Ç 1 ; Ç 2 2 Stack.G/ 1 . 1 ; 2 / a 2-morphism W Ç 1 ! Ç 2 is a 2-morphism 2 MC 2 .G 0 / 2 .Ç 1 1 ; Ç1 2/ which satisfiesÇ 2 2 ı .Id ˝ @0 0 / D .@0 1 ˝ Id/ ı Ç2 1 : (3.7)Let Stack.G/ 2 .Ç 1 ; Ç 2 / denote the set of 2-morphisms.Compositions of 2-morphisms are given by the compositions in MC 2 .G 0 / 2 .For 1 ; 2 2 Stack.G/ 0 , we have the groupoid Stack.G/. 1 ; 2 / with the set ofobjects Stack.G/ 1 . 1 ; 2 / and the set of morphisms Stack.G/ 2 .Ç 1 ; Ç 2 / under verticalcomposition.Every morphism of cosimplicial DGLA induces in an obvious manner a functor W Stack.G ˝ m/ ! Stack.H ˝ m/We have the following cosimplicial analogue of Theorem 3.1:Theorem 3.6. Suppose that W G ! H is a quasi-isomorphism of cosimplicial DGLAand m is a commutative nilpotent ring. Then the induced map W Stack.G ˝ m/ !Stack.H ˝ m/ is an equivalence.

Deformations of gerbes on smooth manifolds 359Proof. The proof can be obtained by applying Theorem 3.1 repeatedly.Let 1 , 2 2 Stack.G ˝ m/ 0 , and let Ç 1 , Ç 2 be two 1-morphisms between 1and 2 . Note that W MC 2 .G 0 ˝ m/ 2 .Ç 1 1 ; Ç1 2 / ! MC2 .H 0 ˝ m/ 2 . Ç 1 1 ; Ç 1 2 / isa bijection by Theorem 3.1. Injectivity of the map W Stack.G ˝ m/ 2 .Ç 1 ; Ç 2 / !Stack.H ˝ m/ 2 . Ç 1 ; Ç 2 / follows immediately.For the surjectivity, note that an element of Stack.H˝m/ 2 . Ç 1 ; Ç 2 / is necessarilygiven by for some 2 MC 2 .G 0˝m/.Ç 1 1 ; Ç1 2 / 2 and the following identity is satisfiedin MC 2 .H 1 ˝ m/. . 1 2 ı @0 0 Ç1 1 /; .@ 0 1 Ç1 2 ı 1 1 // 2: .Ç 2 2 ı .@1 0 ˝ Id// D ..Id ˝ @ 1 1 / ı Ç2 1 /:Since W MC 2 .G 1 ˝ m/. 1 2 ı @0 0 Ç1 1 ;@0 1 Ç1 2 ı 1 1 / 2 ! MC 2 .H 1 ˝ m/. . 1 2 ı @0 0 Ç1 1 /; .@ 0 1 Ç1 2 ı 1 1 // 2 is bijective, and in particular injective, Ç 2 2 ı.@1 0 ˝Id/ D .Id˝@1 1 /ıÇ2 1 ,and defines an element in Stack.G ˝ m/ 2 .Ç 1 ; Ç 2 /.Next, let 1 , 2 2 Stack.G ˝ m/ 0 , and let Ç be a 1-morphisms between 1and 2 . We show that there exists È 2 Stack.G ˝ m/ 1 . 1 ; 2 / such that È isisomorphic to Ç. Indeed, by Theorem 3.1, there exists È 1 2 MC 2 .G 0 ˝ m/ 1 . 1 ; 2 /such that MC 2 .H 0˝m/ 2 . È 1 ; Ç/ ¤;. Let 2 MC 2 .H 0˝m/ 2 . È 1 ; Ç/. Define 2MC 2 .H 1 ˝m/ 2 . . 1 2 ı@0 0 È1 /; .@ 0 1 È1 ı 1 1 // by D .@0 1 ˝Id/ 1 ıÇ 2 ı.Id ˝@ 0 0 /.It is easy to verify that the following identities holds:.Id ˝ 1 2 / ı .@1 2 ˝ Id/ ı .Id ˝ @1 0 / D @1 1 ı . 2 2 ˝ Id/;s0 1 D Id:By bijectivity of on MC 2 there exists a unique È 2 2 MC 2 .H 1˝m/ 2 .2 1ı@0 0 È1 ;@ 0 1 È1 ı1 1/ such that È 2 D . Moreover, as before, injectivity of implies that theconditions (3.6) are satisfied. Therefore È D .È 1 ; È 2 / defines a 1-morphism 1 ! 2and is a 2-morphism È ! Ç.Now, let 2 Stack.H ˝ m/ 0 . We construct 2 Stack.G ˝ m/ 0 in such a way thatStack.H˝m/ 1 . ;/ ¤ ¿. By Theorem 3.1 there exists 0 2 MC 2 .G 0 ˝m/ 0 suchthat MC 2 .H 0 ˝ m/ 1 . 0 ; 0 / ¤ ¿. Let Ç 1 2 MC 2 .H 0 ˝ m/ 1 . ;/. ApplyingTheorem 3.1 again we obtain that there exists 2 MC 2 .G 1 ˝ m/ 1 .@ 0 0 0 ;@ 0 1 0 / suchthat there exists 2 MC 2 .H 1 ˝ m/ 2 . 1 ı @ 0 0 Ç1 ;@ 0 1 Ç1 ı /. Then s0 1 is then a2-morphism Ç 1 ! Ç 1 ı .s0 1/, which induces a 2-morphism W .s1 0 / 1 ! Id.As a next step set 1 D ı .@ 0 0 .s1 0 // 1 2 MC 2 .G 1 ˝ m/ 1 .@ 0 0 0 ;@ 0 1 0 /, Ç 2 D˝.@ 0 0 / 1 2 MC 2 .H 1˝m/ 2 . 1 ı@ 0 0 Ç1 ;@ 0 1 Ç1 ı 1 /. It is easy to see that s0 11 D Id,s0 1Ç2 D Id.We then conclude that there exists a unique 2 such that.Id ˝ 2 / ı .@ 1 2 Ç2 ˝ Id/ ı .Id ˝ @ 1 0 Ç2 / D @ 1 1 Ç2 ı . 2 ˝ Id/:Such a 2 necessarily satisfies the conditions@ 2 2 2 ı .Id ˝ @ 2 0 2 / D @ 2 1 2 ı .@ 2 3 2 ˝ Id/;s0 2 2 D s1 2 2 D Id:

360 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganTherefore D . 0 ; 1 ; 2 / 2 Stack.G ˝ m/, and Ç D .Ç 1 ; Ç 2 / defines a 1-morphism ! .3.4 Acyclicity and strictnessDefinition 3.7. A G-stack . 0 ; 1 ; 2 / is called strict if @ 0 0 0 D @ 0 1 0 , 1 D Id and 2 D Id.Let Stack str .G/ 0 denote the subset of strict G-stacks.Lemma 3.8. Stack str .G/ 0 D MC 2 .ker.G 0 G 1 // 0 .Definition 3.9. For elements 1 ; 2 in Stack str .G/ 0 ,a1-morphism Ç D .Ç 1 ; Ç 2 / 2Stack.G/ 1 . 1 ; 2 / is called strict if @ 0 0 .Ç1 / D @ 0 1 .Ç1 / and Ç 2 D Id.For 1 ; 2 2 Stack str .G/ 0 we denote by Stack str .G/ 1 . 1 ; 2 / the subset of strictmorphisms.Lemma 3.10. For 1 ; 2 2 Stack str .G/ 0 one hasStack str .G/ 1 . 1 ; 2 / D MC 2 .ker.G 0 G 1 // 1 :For 1 ; 2 2 Stack str .G/ 0 let Stack str .G/. 1 ; 2 / denote the full subcategory ofStack.G/. 1 ; 2 / with objects Stack str .G/ 1 . 1 ; 2 /.Thus, we have the 2-groupoids Stack.G/ and Stack str .G/ and an embedding of thelatter into the former which is fully faithful on the respective groupoids of 1-morphisms.Lemma 3.11. Stack str .G/ D MC 2 .ker.G 0 G 1 //.Suppose that G is a cosimplicial DGLA. For each n and i we have the vectorspace G n;i , namely the degree i component of G n . The assignment n 7! G n;i is acosimplicial vector space G ;i .We will consider the following acyclicity condition on the cosimplicial DGLA G:H p .G ;i / D 0 for all i 2 Z and for p ¤ 0. (3.8)Theorem 3.12. Suppose that G is a cosimplicial DGLA which satisfies the condition(3.8), and m a commutative nilpotent ring. Then, the functor W Stack str .G ˝ m/ !Stack.G ˝ m// is an equivalence.Proof. As we already noted before, it is immediate from the definitions that if 1 , 2 2 Stack str .G ˝ m/ 0 , and Ç 1 ; Ç 2 2 Stack str .G ˝ m/ 1 . 1 ; 2 /, then the mappingW Stack str .G ˝ m/ 2 .Ç 1 ; Ç 2 / ! Stack str .G ˝ m/ 2 .Ç 1 ; Ç 2 / is a bijection.Let now 1 ; 2 2 Stack str .G ˝ m/ 0 , Ç D .Ç 1 ; Ç 2 / 2 Stack.G ˝ m/ 1 . 1 ; 2 /.Weshow that there exists È 2 Stack str .G/ 1 . 1 ; 2 / and a 2-morphism W Ç ! È. LetÇ 2 D exp @ 00 20 g, where g 2 .G 1 ˝ m/. Then @ 1 0 g @1 1 g C @1 2 g D 0 mod m2 .As a consequence of the acyclicity condition there exists a 2 .G 0 ˝ m/ such that@ 0 0 a @0 1 a D g mod m2 . Set 1 D exp 0 a. Define then È 1 D . 1 Ç 1 ;.@ 02 1 1 ˝ Id/ ı

Deformations of gerbes on smooth manifolds 361Ç 2 ı .Id ˝ @ 0 0 1/ 1 / 2 Stack.G ˝ m/ 1 . 1 ; 2 /. Note that 1 defines a 2-morphismÇ ! È 1 . Note also that È 2 1 2 exp @ 0 0 2 0 .G ˝m 2 /. Proceeding inductively one constructsa sequence È k 2 Stack.G ˝ m/ 1 . 1 ; 2 / such that È 2 k 2 exp.G ˝ mkC1 / and 2-morphisms k W Ç ! È k , kC1 D k mod m k . Since m is nilpotent, for k large enoughwe have È k 2 Stack str .G/ 1 . 1 ; 2 /.Assume now that 2 Stack.G ˝ m/ 0 . We will construct 2 Stack str .G ˝ m/ 0and a 1-morphism ÇW ! .We begin by constructing 2 Stack.G ˝m/ 0 such that 2 D Id and a 1-morphismÈW ! . Wehave: 2 D exp @ 11 @ 0 1 c, c 2 .G 2 ˝ m/. In view of the equations (3.5)0c satisfies the identities@ 2 0 c @2 1 c C @2 2 c @2 3 c D 0 mod m2 ; and s0 2 c D s2 1c D 0: (3.9)By the acyclicity of the normalized complex we can find b 2 G 1 ˝ m such that@ 1 0 b @1 1 b C @1 2 b D c mod m2 , s0 1b D 0. Let 1 D exp @ 01 . b/. Define then0 1 D . 0 1 ;1 1 ;2 1 / where 0 1 D 0 , 1 1 D 1 1 , and 2 1is such that.Id ˝ 2 / ı .@ 1 2 1 ˝ Id/ ı .Id ˝ @ 1 0 1/ D @ 1 1 1 ı . 2 1 ˝ Id/:Note that 2 1 D Id mod m2 and .Id; 1 / is a 1-morphism ! 1 . As beforewe can construct a sequence k such that 2 kD Id mod mkC1 , and 1-morphisms.Id; k /W ! k , kC1 D k mod m k . As before we conclude that this gives thedesired construction of . The rest of the proof, i.e. the construction of is completelyanalogous.Corollary 3.13. Suppose that G is a cosimplicial DGLA which satisfies the condition(3.8). Then there is a canonical equivalence:Stack.G ˝ m/ Š MC 2 .ker.G 0 G 1 / ˝ m/:Proof. Combine Lemma 3.11 with Theorem 3.12.4 Algebroid stacksIn this section we review the notions of algebroid stack and twisted form. We alsodefine the notion of descent datum and relate it with algebroid stacks.4.1 Algebroids and algebroid stacks4.1.1 Algebroids. For a category C we denote by iC the subcategory of isomorphismsin C; equivalently, iC is the maximal subgroupoid in C.Suppose that R is a commutative k-algebra.Definition 4.1. An R-algebroid is a nonempty R-linear category C such that thegroupoid iC is connected

362 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganLet Algd R denote the 2-category of R-algebroids (full 2-subcategory of the 2-categoryof R-linear categories).Suppose that A is an R-algebra. The R-linear category with one object and morphismsA is an R-algebroid denoted A C .Suppose that C is an R-algebroid and L is an object of C. Let A D End C .L/. Thefunctor A C ! C which sends the unique object of A C to L is an equivalence.Let Alg 2 Rdenote the 2-category with• objects: R-algebras,• 1-morphisms: homomorphism of R-algebras,• 2-morphisms ! , where ; W A ! B are two 1-morphisms: elementsb 2 B such that .a/ b D b .a/ for all a 2 A.It is clear that the 1- and the 2- morphisms in Alg 2 Ras defined above induce 1- and2-morphisms of the corresponding algebroids under the assignment A 7! A C . Thestructure of a 2-category on Alg 2 R(i.e. composition of 1- and 2-morphisms) is determinedby the requirement that the assignment A 7! A C extends to an embedding./ C W Alg 2 R ! Algd R .Suppose that R ! S is a morphism of commutative k-algebras. The assignmentA ! A ˝R S extends to a functor ./ ˝R S W Alg 2 R ! Alg2 S .4.1.2 Algebroid stacks. We refer the reader to [1] and [20] for basic definitions. Wewill use the notion of fibered category interchangeably with that of a pseudo-functor.A prestack C on a space X is a category fibered over the category of open subsetsof X, equivalently, a pseudo-functor U 7! C.U /, satisfying the following additionalrequirement. For an open subset U of X and two objects A; B 2 C.U / we have thepresheaf Hom C .A; B/ on U defined by U V 7! Hom C.V / .Aj V ;Bj B /. The fiberedcategory C is a prestack if for any U , A; B 2 C.U /, the presheaf Hom C .A; B/ is asheaf. A prestack is a stack if, in addition, it satisfies the condition of effective descentfor objects. For a prestack C we denote the associated stack by zC.Definition 4.2. A stack in R-linear categories C on X is an R-algebroid stack if it islocally nonempty and locally connected, i.e. satisfies1. any point x 2 X has a neighborhood U such that C.U / is nonempty;2. for any U X, x 2 U , A; B 2 C.U / there exits a neighborhood V U of xand an isomorphism Aj V Š Bj V .Remark 4.3. Equivalently, the stack associated to the substack of isomorphisms iC isa gerbe.Example 4.4. Suppose that A is a sheaf of R-algebras on X. The assignment X U 7! A.U / C extends in an obvious way to a prestack in R-algebroids denoted A C .

Deformations of gerbes on smooth manifolds 363The associated stack A C is canonically equivalent to the stack of locally free A op -modules of rank one. The canonical morphism A C ! A C sends the unique (locallydefined) object of A C to the free module of rank one.1-morphisms and 2-morphisms of R-algebroid stacks are those of stacks in R-linearcategories. We denote the 2-category of R-algebroid stacks by AlgStack R .X/.4.2 Descent data4.2.1 Convolution dataDefinition 4.5. An R-linear convolution datum is a triple .U; A 01 ; A 012 / consistingof• a cover U 2 Cov.X/,• a sheaf A 01 of R-modules A 01 on N 1 U,• a morphismA 012 W .pr 2 01 / A 01 ˝R .pr 2 12 / A 01 ! .pr 2 02 / A 01 (4.1)of R-modulessubject to the associativity condition expressed by the commutativity of the diagram.pr 3 01 / A 01 ˝R .pr 3 12 / A 01 ˝R .pr 3 23 / A 01.pr 3 012 / .A 012/˝Id .pr302/ A 01 ˝R .pr 3 23 / A 01Id˝.pr 3 123 / .A 012.pr 3 023 / .A 012/.pr 3 01 / A 01 ˝R .pr 3 .pr 313 / 013A / .A 012/01 .pr303/ A 01 .For a convolution datum .U; A 01 ; A 012 / we denote by A the pair .A 01 ; A 012 / andabbreviate the convolution datum by .U; A/.For a convolution datum .U; A/ let• A WD .pr 0 00 / A 01 ; A is a sheaf of R-modules on N 0 U,• A p iWD .pr p i / A; thus for every p we get sheaves A p i , 0 i p on N pU.The identities pr 0 01 ı pr0 000 D pr0 12 ı pr0 000 D pr 02 ı pr0 000 D pr0 00imply that thepull-back of A 012 to N 0 U by pr 0 000gives the pairing.pr 0 000 / .A 012 /W A ˝R A ! A: (4.2)The associativity condition implies that the pairing (4.2) endows A with a structure ofa sheaf of associative R-algebras on N 0 U.

364 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe sheaf A p iis endowed with the associative R-algebra structure induced by thaton A. We denote by A p ii the Ap i˝R .A p i /op -module A p i, with the module structuregiven by the left and right multiplication.The identities pr 2 01 ı pr1 001 D pr0 00 ı pr1 0 ,pr2 12 ı pr1 001 D pr2 02 ı pr1 001D Id implythat the pull-back of A 012 to N 1 U by pr 1 001gives the pairing.pr 1 001 / .A 012 /W A 1 0 ˝R A 01 ! A 01 : (4.3)The associativity condition implies that the pairing (4.3) endows A 01 with a structureof a A 1 0 -module. Similarly, the pull-back of A 012 to N 1 U by pr 1 011 endows A 01 witha structure of a .A 1 1 /op -module. Together, the two module structures define a structureof a A 1 0 ˝R .A 1 1 /op -module on A 01 .The map (4.1) factors through the map.pr 2 01 / A 01 ˝A2 1.pr 2 12 / A 01 ! .pr 2 02 / A 01 : (4.4)Definition 4.6. A unit for a convolution datum A is a morphism of R-modulessuch that the compositions1W R ! AA 011˝Id! A10 ˝R A 01.pr 1 001 / .A 012 /! A 01andA 01Id˝1! A 01 ˝R A 1 1.pr 1 011 / .A 012 /! A 01are equal to the respective identity morphisms.4.2.2 Descent dataDefinition 4.7. A descent datum on X is an R-linear convolution datum .U; A/ on Xwith a unit which satisfies the following additional conditions:1. A 01 is locally free of rank one as a A 1 0 -module and as a .A1 1 /op -module;2. the map (4.4) is an isomorphism.4.2.3 1-morphisms. Suppose given convolution data .U; A/ and .U; B/ as in Definition4.5.Definition 4.8. A 1-morphism of convolution data W .U; A/ ! .U; B/

Deformations of gerbes on smooth manifolds 365is a morphism of R-modules 01 W A 01 ! B 01 such that the diagram.pr 2 01 / A 01 ˝R .pr 2 12 / A 01A 012.pr 2 02 / A 01is commutative..pr 2 01 / . 01 /˝.pr 2 12 / . 01 /.pr 2 02 / . 01 / (4.5).pr 2 01 / B 01 ˝R .pr 2 12 / B 012B 01 .pr 2 02 / B 01The 1-morphism induces a morphism of R-algebras WD .pr 0 00 / . 01 /W A ! Bon N 0 U as well as morphisms p iWD .pr p i / ./W A p i! B p ion N p U. The morphism 01 is compatible with the morphism of algebras 0 ˝ op1 W A1 0 ˝R .A 1 1 /op ! B0 1 ˝R.B1 1/op and the respective module structures.A 1-morphism of descent data is a 1-morphism of the underlying convolution datawhich preserves respective units.4.2.4 2-morphisms. Suppose that we are given descent data .U; A/ and .U; B/ asin 4.2.2 and two 1-morphismsas in 4.2.3.A 2-morphism; W .U; A/ ! .U; B/b W !is a section b 2 .N 0 UI B/ such that the diagramA 01b˝ 01 B 1 0 ˝R B 0101˝bB 01 ˝R B11 B 01is commutative.4.2.5 The 2-category of descent data. FixacoverU of X.Suppose that we are given descent data .U; A/, .U; B/, .U; C/ and 1-morphisms W .U; A/ ! .U; B/ and W .U; B/ ! .U; C/. The map 01 ı 01 W A 01 ! C 01 isa 1-morphism of descent data ı W .U; A/ ! .U; C/, the composition of and .Let .i/ W .U; A/ ! .U; B/, i D 1; 2; 3, be1-morphisms and let b .j / W .j / ! .j C1/ , j D 1; 2, be2-morphisms. The section b .2/ b .1/ 2 .N 0 UI B/ defines a2-morphism, denoted b .2/ b .1/ W .1/ ! .3/ , the vertical composition of b .1/ and b .2/ .

366 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganSuppose that .i/ W .U; A/ ! .U; B/ and .i/ W .U; B/ ! .U; C/, i D 1; 2,are 1-morphisms and b W .1/ ! .2/ and c W .1/ ! .2/ are 2-morphisms. Thesection c .1/ .b/ 2 .N 0 UI C/ defines a 2-morphism, denoted c ˝ b, the horizontalcomposition of b and c.We leave it to the reader to check that with the compositions defined above descentdata, 1-morphisms and 2-morphisms form a 2-category, denoted Desc R .U/.4.2.6 Fibered category of descent data. Suppose that W V ! U is a morphism ofcovers and .U; A/ is a descent datum. Let A 01 D .N 1/ A 01 , A 012 D .N 2/ .A 012 /.Then, .V; A / is a descent datum. The assignment .U; A/ 7! .V; A / extends to afunctor, denoted W Desc R .U/ ! Desc R .V/.The assignment Cov.X/ op 3 U ! Desc R .U/, ! is (pseudo-)functor. LetDesc R .X/ denote the corresponding 2-category fibered in R-linear 2-categories overCov.X/ with object pairs .U; A/ with U 2 Cov.X/ and .U; A/ 2 Desc R .U/; amorphism .U 0 ; A 0 / ! .U; A/ in Desc R .X/ is a pair .; /, where W U 0 ! U is amorphism in Cov.X/ and W .U 0 ; A 0 / ! .U; A/ D .U 0 ; A /.4.3 Trivializations4.3.1 Definition of a trivializationDefinition 4.9. A trivialization of an algebroid stack C on X is an object in C.X/.Suppose that C is an algebroid stack on X and L 2 C.X/ is a trivialization. Theobject L determines a morphism End C .L/ C ! C.Lemma 4.10. The induced morphism D EndC .L/ C ! C is an equivalence.Remark 4.11. Suppose that C is an R-algebroid stack on X. Then, there exists a coverU of X such that the stack 0 C admits a trivialization.4.3.2 The 2-category of trivializations. Let Triv R .X/ denote the 2-category with• objects: the triples .C; U;L/ where C is an R-algebroid stack on X, U is anopen cover of X such that 0 C.N 0U/ is nonempty and L is a trivialization of 0 C,• 1-morphisms: .C 0 ; U 0 ;L/! .C; U;L/are pairs .F; / where W U 0 ! U is amorphism of covers, F W C 0 ! C is a functor such that .N 0 / F.L 0 / D L,• 2-morphisms .F; / ! .G; /, where .F; /; .G; /W .C 0 ; U 0 ;L/! .C; U;L/:the morphisms of functors F ! G.

Deformations of gerbes on smooth manifolds 367The assignment .C; U;L/ 7! C extends in an obvious way to a functor Triv R .X/ !AlgStack R .X/.The assignment .C; U;L/7! U extends to a functor Triv R .X/ ! Cov.X/ makingTriv R .X/ a fibered 2-category over Cov.X/. For U 2 Cov.X/ we denote the fiberover U by Triv R .X/.U /.4.3.3 Algebroid stacks from descent data. Consider .U; A/ 2 Desc R .U/.The sheaf of algebras A on N 0 U gives rise to the algebroid stack A C . The sheafA 01 defines an equivalence 01 WD ./ ˝A1 0A 01 W .pr 1 0 / AC ! .pr 1 1 / AC :The convolution map A 012 defines an isomorphism of functors 012 W .pr 2 01 / . 01 / ı .pr 2 12 / . 01 / ! .pr 2 02 / . 01 /:We leave it to the reader to verify that the triple . A C ; 01 ; 012 / constitutes a descentdatum for an algebroid stack on X which we denote by St.U; A/.By construction there is a canonical equivalence A C ! 0 St.U; A/ which endows0 St.U; A/ with a canonical trivialization 1.The assignment .U; A/ 7! .St.U; A/; U; 1/ extends to a cartesian functorStW Desc R .X/ ! Triv R .X/:4.3.4 Descent data from trivializations. Consider .C; U;L/ 2 Triv R .X/. Since 0 ı pr 1 0 D 0 ı pr 1 1 D 1 we have canonical identifications .pr 1 0 / 0 C Š .pr1 1 / 0 C Š 1 C. The object L 2 0 C.N 0U/ gives rise to the objects .pr 1 0 / L and .pr 1 1 / L in 1 C.N 0U/. Let A 01 D Hom 1 C ..pr1 1 / L; .pr 1 0 / L/. Thus, A 01 is a sheaf of R-modules on N 1 U.The object L 2 0 C.N 0U/ gives rise to the objects .pr 2 0 / L, .pr 2 1 / L and .pr 2 2 / Lin 2 C.N 2U/. There are canonical isomorphismsThe composition of morphismsgives rise to the map.pr 2 ij / A 01 Š Hom 2 C ..pr2 i / L; .pr 2 j / L/:Hom 2 C ..pr2 1 / L; .pr 2 0 / L/ ˝R Hom 2 C ..pr2 2 / L; .pr 2 1 / L/! Hom 2 C ..pr2 2 / L; .pr 2 0 / L/A 012 W .pr 2 01 / A 01 ˝R .pr 2 12 / A 01 ! .pr 2 02 / A 02 :Since pr 1 i ıpr0 00 D Id there is a canonical isomorphism A WD .pr0 00 / A 01 Š End.L/which supplies A with the unit section 1W R ! 1 End.L/ ! A.

368 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe pair .U; A/, together with the section 1 is a decent datum which we denotedd.C; U;L/.The assignment .U; A/ 7! dd.C; U;L/extends to a cartesian functorddW Triv R .X/ ! Desc R .X/:Lemma 4.12. The functors St and dd are mutually quasi-inverse equivalences.4.4 Base change. For an R-linear category C and homomorphism of algebras R ! Swe denote by C ˝R S the category with the same objects as C and morphisms definedby Hom C˝RS.A; B/ D Hom C .A; B/ ˝R S.For an R-algebra A the categories .A ˝R S/ C and A C ˝R S are canonicallyisomorphic.For a prestack C in R-linear categories we denote by C ˝R S the prestack associatedto the fibered category U 7! C.U / ˝R S.For U X, A; B 2 C.U /, there is an isomorphism of sheaves Hom C˝RS .A; B/ DHom C .A; B/ ˝R S.Lemma 4.13. Suppose that A is a sheaf of R-algebras and C is an R-algebroid stack.1. . A C ˝R S/z is an algebroid stack equivalent to E .A ˝R S/ C .2. B C ˝R S is an algebroid stack.Proof. Suppose that A is a sheaf of R-algebras. There is a canonical isomorphismof prestacks .A S/ C Š A C ˝R S which induces the canonical equivalenceE.A ˝R S/ C Š AC D˝R S.The canonical functor A C ! A C induces the functor A C ˝R S ! A C ˝R S,hence the functor AC D˝R S ! . A C ˝R S/z.The map A ! A ˝R S induces the functor AC ! .A E ˝R S/ C which factorsthrough the functor AC˝R S ! .A E ˝R S/ C . From this we obtain the functor. A C ˝R S/z!E .A ˝R C .We leave it to the reader to check that the two constructions are mutually inverseequivalences. It follows that . A C ˝R S/z is an algebroid stack equivalent toE.A ˝R S/ C .Suppose that C is an R-algebroid stack. Let U be a cover such that 0 C.N 0U/ isnonempty. Let L be an object in 0 C.N 0U/; put A WD End 0 C .L/. The equivalenceA C ! 0 C induces the equivalence . A C ˝R S/z!. 0 C ˝R S/z. Since the formeris an algebroid stack so is the latter. There is a canonical equivalence . 0 C ˝R S/zŠ 0 . B C ˝R S/; since the former is an algebroid stack so is the latter. Since the propertyof being an algebroid stack is local, the stack B C ˝R S is an algebroid stack.

Deformations of gerbes on smooth manifolds 3694.5 Twisted forms. Suppose that A is a sheaf of R-algebras on X. We will call anR-algebroid stack locally equivalent to A opC a twisted form of A.Suppose that S is twisted form of O X . Then, the substack iS is an OX -gerbe.The assignment S 7! iS extends to an equivalence between the 2-groupoid of twistedforms of O X (a subcategory of AlgStack C .X/) and the 2-groupoid of OX -gerbes.Let S be a twisted form of O X . Then for any U X, A 2 S.U / the canonicalmap O U ! End S .A/ is an isomorphism. Consequently, if U isacoverofX and Lis a trivialization of 0 S, then there is a canonical isomorphism of sheaves of algebrasO N0 U ! End 0 S .L/.Conversely, suppose that .U; A/ is a C-descent datum. If the sheaf of algebrasA is isomorphic to O N0 U then such an isomorphism is unique since the latter has nonon-trivial automorphisms. Thus, we may and will identify A with O N0 U. Hence, A 01is a line bundle on N 1 U and the convolution map A 012 is a morphism of line bundles.The stack which corresponds to .U; A/ (as in 4.3.3) is a twisted form of O X .Isomorphism classes of twisted forms of O X are classified by H 2 .XI OX /. Werecall the construction presently. Suppose that the twisted form S of O X is representedby the descent datum .U; A/. Assume in addition that the line bundle A 01 on N 1 U istrivialized. Then we can consider A 012 as an element in .N 2 UI O /. The associativitycondition implies that A 012 is a cocycle in LC 2 .UI O /. The class of this cocycle inHL2 .UI O / does not depend on the choice of trivializations of the line bundle A 01 andyields a class in H 2 .XI OX /.We can write this class using the de Rham complex for jets. We refer to Section 7.1for the notations and a brief review.The composition O ! O =C log ! O=C j 1 ! DR.J=O/ induces the mapH 2 .XI O / ! H 2 .XI DR.J=O// Š H 2 ..XI X ˝J X=O X /; r can /. Here the latterisomorphism follows from the fact that the sheaf X ˝ J X=O X is soft. We denote byŒS 2 H 2 ..XI X ˝ J X=O X /; r can / the image of the class of S in H 2 .XI O /.InLemma 7.13 (see also Lemma 7.15) we will construct an explicit representative for ŒS.5 DGLA of local cochains on matrix algebrasIn this section we define matrix algebras from a descent datum and use them to constructa cosimplicial DGLA of local cochains. We also establish the acyclicity of thiscosimplicial DGLA.5.1 Definition of matrix algebras5.1.1 Matrix entries. Suppose that .U; A/ is an R-descent datum. Let A 10 WD A 01 , where D pr 1 10 W N 1U ! N 1 U is the transposition of the factors. The pairings.pr 1 100 / .A 012 /W A 10 ˝R A 1 0 ! A 10 and .pr 1 110 / .A 012 /W A 1 1 ˝R A 10 ! A 10 ofsheaves on N 1 U endow A 10 with a structure of a A 1 1 ˝ .A1 0 /op -module.

370 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe identities pr 2 01 ı pr1 010 D Id, pr2 12 ı pr1 010 D and pr2 02 ı pr1 010 D pr0 00 ı pr1 0imply that the pull-back of A 012 by pr 1 010gives the pairing.pr 1 010 / .A 012 /W A 01 ˝R A 10 ! A 1 00 : (5.1)which is a morphism of A 1 0 ˝K .A 1 0 /op -modules. Similarly, we have the pairing.pr 1 101 / .A 012 /W A 10 ˝R A 01 ! A 1 11 : (5.2)The pairings (4.2), (4.3), (5.1) and (5.2), A ij ˝R A jk ! A ik are morphisms ofA 1 i ˝R .A 1 k /op -modules which, as a consequence of associativity, factor through mapsA ij A ˝A1 jk ! A ik (5.3)jinduced by A ij k D .pr 1 ij k / .A 012 /; here i;j;k D 0; 1. Define now for every p 0the sheaves A p ij , 0 i;j p, onN pU by A p ij D .prp ij / A 01 . Define also A p ij k D.pr p ij k / .A 012 /. We immediately obtain for every p the morphismsA p ij k W Ap ij ˝A p jA p jk ! Ap ik : (5.4)5.1.2 Matrix algebras. Let Mat.A/ 0 D A; thus, Mat.A/ 0 is a sheaf of algebras onN 0 U.Forp D 1;2;:::let Mat.A/ p denote the sheaf on N p U defined bypMMat.A/ p D A p ij :i;jD0The maps (5.4) define the pairingMat.A/ p ˝ Mat.A/ p ! Mat.A/ pwhich endows the sheaf Mat.A/ p with a structure of an associative algebra by virtueof the associativity condition. The unit section 1 is given by 1 D P piD0 1 ii, where 1 iiis the image of the unit section of A p ii .5.1.3 Combinatorial restriction. The algebras Mat.A/ p , p D 0;1;:::, do not forma cosimplicial sheaf of algebras on N U in the usual sense. They are, however, relatedby combinatorial restriction which we describe presently.For a morphism f W Œp ! Œq in define a sheaf on N q U bypMf ] Mat.A/ q D A q f.i/f.j/ :i;jD0Note that f ] Mat.A/ q inherits a structure of an algebra.Recall from Section 2.3.1 that the morphism f induces the map f W N q U ! N p Uand that f denotes the pull-back along f . We will also use f to denote the canonicalisomorphism of algebrasf W f Mat.A/ p ! f ] Mat.A/ q (5.5)induced by the isomorphisms f A p ij Š Aq f.i/f.j/ .

Deformations of gerbes on smooth manifolds 3715.1.4 Refinement. Suppose that W V ! U is a morphism of covers. For p D0;1;:::there is a natural isomorphism.N p / Mat.A/ p Š Mat.A / pof sheaves of algebras on N p V. The above isomorphisms are obviously compatiblewith combinatorial restriction.5.2 Local cochains on matrix algebras5.2.1 Local cochains. A sheaf of matrix algebras is a sheaf of algebras B togetherwith a decompositionpMB Di;jD0as a sheaf of vector spaces which satisfies B ij B jk D B ik .To a matrix algebra B one can associate to the DGLA of local cochains defined asfollows. For n D 0 let C 0 .B/ loc D L B ii B D C 0 .B/. Forn>0let C n .B/ locdenote the subsheaf of C n .B/ whose stalks consist of multilinear maps D such thatfor any collection of s ik j k2 B ik j k1. D.s i1 j 1 ˝˝s in j n/ D 0 unless j k D i kC1 for all k D 1;:::;n 1,2. D.s i0 i 1 ˝ s i1 i 2 ˝˝s in 1 i n/ 2 B i0 i n.For I D .i 0 ;:::;i n / 2 Œp nC1 letB ijC I .B/ loc WD Hom k .˝n 1j D0 B i j i j C1; B i0 i n/:The restriction maps along the embeddings ˝n 1j D0 B i j i j C1,! B˝n induce an isomorphismC n .B/ loc !˚I 2Œp nC1C I .B/ loc .The sheaf C .B/ loc Œ1 is a subDGLA of C .B/Œ1 and the inclusion map is a quasiisomorphism.For a matrix algebra B on X we denote by Def.B/ loc .R/ the subgroupoid ofDef.B/.R/ with objects R-star products which respect the decomposition given by.B˝kR/ ij D B ij ˝kR and 1- and 2-morphisms defined accordingly. The compositionDef.B/ loc .R/ ! Def.B/.R/ ! MC 2 ..XI C .B/Œ1/ ˝k m R /takes values in MC 2 ..XI C .B/ loc Œ1/ ˝k m R / and establishes an isomorphism of2-groupoids Def.B/ loc .R/ Š MC 2 ..XI C .B/ loc Œ1/ ˝k m R /.5.2.2 Combinatorial restriction of local cochains. Suppose given a matrix algebraB D L qi;jD0 B ij is a sheaf of matrix k-algebras.

372 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe DGLA C .B/ loc Œ1 has additional variance not exhibited by C .B/Œ1. Namely,for f W Œp ! Œq – a morphism in – there is a natural map of DGLAf ] W C .B/ loc Œ1 ! C .f ] B/ loc Œ1 (5.6)defined as follows. Let f ij W .f ] B/] ij ! B f.i/f.j/ denote the tautological isomorphism.For each multi-index I D .i 0 ;:::;i n / 2 Œp nC1 letLet f nf I]WD ˝n 1j D0 f i j i j C1]W ˝n 1j D0 .f ] B/ ij i j C1!˝n 1iD0 B f.i j /f .i j C1 /:] WD ˚I 2† pnC1f In. The map (5.6) is defined as restriction along f] ] .Lemma 5.1. The map (5.6) is a morphism of DGLAf ] W C .B/ loc Œ1 ! C .f ] B/ loc Œ1:If follows that combinatorial restriction of local cochains induces the functorMC 2 .f ] /W MC 2 ..XI C .B/ loc Œ1/˝km R / ! MC 2 ..XI C .f ] B/ loc Œ1/˝km R /:Combinatorial restriction with respect to f induces the functorf ] W Def.B/ loc .R/ ! Def.f ] B/ loc .R/:It is clear that the following diagram commutes:Def.B/ loc .R/f ]Def.f ] B/ loc .R/MC 2 ..XI C .B/ loc Œ1/ ˝k m R / MC2 .f ]/ MC 2 ..XI C .f ] B/ loc Œ1/ ˝k m R /.5.2.3 Cosimplicial DGLA from descent datum. Suppose that .U; A/ is a descentdatum for a twisted sheaf of algebras as in 4.2.2. Then, for each p D 0;1;::: wehave the matrix algebra Mat.A/ p as defined in 5.1, and therefore the DGLA of localcochains C .Mat.A/ p / loc Œ1 defined in 5.2.1. For each morphism f W Œp ! Œq thereis a morphism of DGLAf ] W C .Mat.A/ q / loc Œ1 ! C .f ] Mat.A/ q / loc Œ1and an isomorphism of DGLAC .f ] Mat.A/ q / loc Œ1 Š f C .Mat.A/ p / loc Œ1induced by the isomorphism f W f Mat.A/ p ! f ] Mat.A/ q from the equation (5.5).These induce the morphisms of the DGLA of global sections.N q UI C .Mat.A/ q / loc Œ1/.N p UI C .Mat.A/fp / loc Œ1/]f .N q UI f C .Mat.A/ p / loc Œ1/.

Deformations of gerbes on smooth manifolds 373For W Œn ! letG.A/ D .N .n/ UI .0n/ C .Mat.A/ .0/ / loc Œ1/:Suppose given another simplex W Œm ! and morphism W Œm ! Œn suchthat D ı (i.e. is a morphism of simplices ! ). The morphism .0n/ factorsuniquely into 0 ! .0/ ! .m/ ! n, which, under , gives the factorization of.0n/W .0/ ! .n/ (in ) intowhere g D .0m/. The mapis the composition.0/ f ! .0/ g ! .m/ h ! .n/; (5.7) W G.A/ ! G.A/ .N .m/ UI g C .Mat.A/ .0/ / loc Œ1/h ! .N .n/ UI h g C .Mat.A/ .0/ / loc Œ1/f ] ! .N .n/ UI h g f C .Mat.A/ .0/ / loc Œ1/:Suppose given yet another simplex, W Œl ! , and a morphism of simplicesW ! , i.e. a morphism W Œl ! Œm such that D ı . Then, the composition ı W G.A/ ! G.A/ coincides with the map . ı / .For n D 0;1;2;:::letG.A/ n DYG.A/ : (5.8)Œn !A morphism W Œm ! Œn in induces the map of DGLA W G.A/ m ! G.A/ n .The assignment 3 Œn 7! G.A/ n , 7! defines the cosimplicial DGLA denotedby G.A/.5.3 AcyclicityTheorem 5.2. The cosimplicial DGLA G.A/ is acyclic, i.e. it satisfies the condition(3.8).The rest of the section is devoted to the proof of Theorem 5.2. We fix a degree ofHochschild cochains k.For W Œn ! let c D .N .n/ UI .0n/ C k .Mat.A/ .0/ / loc /. For a morphism W ! we have the map W c ! c defined as in 5.2.3.Let .C ;@/denote the corresponding cochain complex whose definition we recallbelow. For n D 0;1;::: let C n D Q Œn !c . The differential @ n W C n ! C nC1 isdefined by the formula @ n D P nC1iD0 . 1/i .@ n i / .

374 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan5.3.1 Decomposition of local cochains. As was noted in 5.2.1, for n; q D 0;1;:::there is a direct sum decompositionC k .Mat.A/ q / loc DMC I .Mat.A/ q / loc : (5.9)I 2Œq kC1In what follows we will interpret a multi-index I D .i 0 ;:::;i n / 2 Œq kC1 as a mapI Wf0;:::;kg!Œq. For I as above let s.I/ DjIm.I /j 1. The map I factorsuniquely into the compositionf0;:::;kg I 0 ! Œs.I / m.I / ! Œqwhere the second map is a morphism in (i.e. is order preserving). Then, the isomorphismsm.I / A I 0 .i/I 0 .j / Š A I.i/I.j/ induce the isomorphismm.I / C I 0 .Mat.A/ s.I/ / loc ! C I .Mat.A/ q / loc :Therefore, the decomposition (5.9) may be rewritten as follows:C k .Mat.A/ q / loc D M Me C I .Mat.A/ p / loc (5.10)ewhere the summation is over injective (monotone) maps e W Œs.e/ ! Œq and surjectivemaps I Wf0;:::;kg!Œs.e/. Note that, for e, I as above, there is an isomorphisme C I .Mat.A/ p / loc Š C eBI .e ] Mat.A/ s.e/ / loc :5.3.2 Filtrations. Let F C k .Mat.A/ q / loc denote the decreasing filtration defined byF s C k .Mat.A/ q / loc DM Me C I .Mat.A/ p / locIeWs.e/sNote that F s C k .Mat.A/ q / loc D 0 for s>kand Gr s C k .Mat.A/ q / loc D 0 for s0.I

Deformations of gerbes on smooth manifolds 375For D D .D / 2 F s C n , D 2 c and W Œn1 ! leth n .D/ D X I;ee D I ewhere e W Œs ! .0/ is an injective map, I Wf0;:::;kg!Œs is a surjection, and D I eis the I -component of D e in the sense of the decomposition (5.9). Put h n .D/ D.h n .D/ / 2 C n 1 . It is clear that h n .D/ 2 F s C n 1 . Thus the assignment D 7!h n .D/ defines a filtered map h n W C n ! C n 1 for all n 1. Hence, for any s we havethe map Gr s h n W Gr s C n ! Gr s C n 1 .Next, we calculate the effect of the face maps @ n i 1 W Œn 1 ! Œn. First of all,note that, for 1 i n 1, .@ n i 1 / D Id. The effect of .@ n 0 1 / is the map on globalsections induced by.01/ ] W .1n/ C k .Mat.A/ .1/ / loc ! .0n/ C k .Mat.A/ .0/ / locand .@ n n 1 / D .n 1; n/ . Thus, for D D .D / 2 F s C n ,wehave:X..@ n i 1 / h n .D// D .@ n i1 / e D Ị ı@ n i1 / eOn the other hand,8ˆ0,h nC1 ı @ n C @ n 1 ı h n D Id.Proof of Theorem 5.2. Consider for a fixed k the complex of Hochschild degree kcochains C . It is shown in Proposition 5.5 that this complex admits a finite filtrationF C such that GrC is acyclic in positive degrees. Therefore C is also acyclic inpositive degree. As this holds for every k, the condition (3.8) is satisfied.

376 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan6 Deformations of algebroid stacksIn this section we define a 2-groupoid of deformations of an algebroid stack. We alsodefine 2-groupoids of deformations and star products of a descent datum and relate itwith the 2-groupoid G-stacks of an appropriate cosimplicial DGLA.6.1 Deformations of linear stacksDefinition 6.1. Let B be a prestack on X in R-linear categories. We say that B is flat iffor any U X, A; B 2 B.U / the sheaf Hom B .A; B/ is flat (as a sheaf of R-modules).Suppose now that C is a stack in k-linear categories on X and R is a commutativeArtin k-algebra. We denote by Def.C/.R/ the 2-category with• objects: pairs .B;$/, where B is a stack in R-linear categories flat over R and$ W B ˝R k ! C is an equivalence of stacks in k-linear categories;• 1-morphisms: a 1-morphism .B .1/ ;$ .1/ / ! .B .2/ ;$ .2/ / is a pair .F; / whereF W B .1/ ! B .2/ is a R-linear functor and W $ .2/ ı .F ˝R k/ ! $ .1/ is anisomorphism of functors;• 2-morphisms: a 2-morphism .F 0 ; 0 / ! .F 00 ; 00 / is a morphism of R-linearfunctors W F 0 ! F 00 such that 00 ı .Id $ .2/ ˝ . ˝R k// D 0 .The 2-category Def.C/.R/ is a 2-groupoid.Lemma 6.2. Suppose that B is a flat R-linear stack on X such that B ˝R k is analgebroid stack. Then, B is an algebroid stack.Proof. Let x 2 X. Suppose that for any neighborhood U of x the category B.U / isempty. Then, the same is true about B ˝R k.U/ which contradicts the assumption thatBB ˝R k is an algebroid stack. Therefore, B is locally nonempty.Suppose that U is an open subset and A; B are two objects in B.U /. Let xAand xB be their respective images in B ˝R k.U/. We have: Hom . AB˝Rk.U/xA; xB/ D.UI Hom B .A; B/ ˝R k/. Replacing U by a subset we may assume that there is anisomorphism x W xA ! xB.The short exact sequencegives rise to the sequence0 ! m R ! R ! k ! 00 ! Hom B .A; B/ ˝R m R ! Hom B .A; B/ ! Hom B .A; B/ ˝R k ! 0of sheaves on U which is exact due to flatness of Hom B .A; B/. The surjectivity of themap Hom B .A; B/ ! Hom B .A; B/ ˝R k implies that for any x 2 U there exists aneighborhood x 2 V U and 2 .V I Hom B .Aj V ;Bj V // D Hom B.V / .Aj V ;Bj V /such that xj V is the image of . Since x is an isomorphism and m R is nilpotent itfollows that is an isomorphism.

Deformations of gerbes on smooth manifolds 3776.2 Deformations of descent data. Suppose that .U; A/ is an k-descent datum andR is a commutative Artin k-algebra.We denote by Def 0 .U; A/.R/ the 2-category with• objects: R-deformations of .U; A/; such a gadget is a flat R-descent datum.U; B/ together with an isomorphism of k-descent data W .U; B ˝R k// !.U; A/;• 1-morphisms: a 1-morphism of deformations .U; B .1/ ; .1/ / ! .U; B .2/ ; .2/ /is a pair .;a/, where W .U; B .1/ / ! .U; B .2/ / is a 1-morphism of R-descentdata and a W .2/ ı . ˝R k/ ! .1/ is 2-isomorphism;• 2-morphisms: a 2-morphism . 0 ;a 0 / ! . 00 ;a 00 / is a 2-morphism b W 0 ! 00such that a 00 ı .Id .2/ ˝ .b ˝R k// D a 0 .Suppose that .;a/ is a 1-morphism. It is immediate from the definition above thatthe morphism of k-descent data ˝R k is an isomorphism. Since R is an extension ofk by a nilpotent ideal the morphism is an isomorphism. Similarly, any 2-morphismis an isomorphism, i.e. Def 0 .U; A/.R/ is a 2-groupoid.The assignment R 7! Def 0 .U; A/.R/ is fibered 2-category in 2-groupoids over thecategory of commutative Artin k-algebras ([3]).6.2.1 Star products. An (R-)star product on .U; A/ is a deformation .U; B;/of.U; A/ such that B 01 D A 01 ˝k R and W B 01 ˝R k ! A 01 is the canonicalisomorphism. In other words, a star product is a structure of an R-descent datum on.U; A ˝k R/ such that the natural map.U; A ˝k R/ ! .U; A/is a morphism of such.We denote by Def.U; A/.R/ the full 2-subcategory of Def 0 .U; A/.R/ whose objectsare star products.The assignment R 7! Def.U; A/.R/ is fibered 2-category in 2-groupoids over thecategory of commutative Artin k-algebras ([3]) and the inclusions Def.U; A/.R/ !Def 0 .U; A/.R/ extend to a morphism of fibered 2-categories.Proposition 6.3. Suppose that .U; A/ is a C-linear descent datum with A D O N0 U.Then, the embedding Def.U; A/.R/ ! Def 0 .U; A/.R/ is an equivalence.6.2.2 Deformations and G-stacks. Suppose that .U; A/ is a k-descent datum and.U; B/ is an R-star product on .U; A/. Then, for every p D 0;1;:::the matrix algebraMat.B/ p is a flat R-deformation of the matrix algebra Mat.A/ p . The identificationB 01 D A 01 ˝k R gives rise to the identification Mat.B/ p D Mat.A/ p ˝k R ofthe underlying sheaves of R-modules. Using this identification we obtain the Maurer–Cartan element p 2 .N p UI C 2 .Mat.A/ p / loc ˝k m R /. Moreover, the equation (5.5)

378 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsyganimplies that for a morphism f W Œp ! Œq in we have f p D f ] q . Therefore thecollection p defines an element in Stack str .G.A/ ˝k m R / 0 . The considerations in5.2.1 and 5.2.2 imply that this construction extends to an isomorphism of 2-groupoidsDef.U; A/.R/ ! Stack str .G.A/ ˝k m R /: (6.1)Combining (6.1) with the embeddingwe obtain the functorStack str .G.A/ ˝k m R / ! Stack.G.A/ ˝k m R / (6.2)Def.U; A/.R/ ! Stack.G.A/ ˝k m R /: (6.3)The naturality properties of (6.3) with respect to base change imply that (6.3) extendsto morphism of functors on the category of commutative Artin algebras.Combining this with the results of Theorems 5.2 and 3.12 implies the following:Proposition 6.4. The functor (6.3) is an equivalence.Proof. By Theorem 5.2 the DGLA G.A/ ˝k m R satisfies the assumptions of Theorem3.12. The latter says that the inclusion (6.2) is an equivalence. Since (6.1) is anisomorphism, the composition (6.3) is an equivalence as claimed.7 JetsIn this section we use constructions involving the infinite jets to simplify the cosimplicialDGLA governing the deformations of a descent datum.7.1 Infinite jets of a vector bundle. Let M be a smooth manifold, and E a locally-freeO M -module of finite rank.Let i W M M ! M , i D 1; 2, denote the projection on the i th factor. Denoteby M W M ! M M the diagonal embedding and let M W O M M ! O M be theinduced map. Let I M WD ker.M /.LetJ k .E/ WD . 1 / O M M =I kC1M ˝2 1O 1M 2 E ;JM k WD Jk .O M /. It is clear from the above definition that JM k is, in a natural way, asheaf of commutative algebras and J k .E/ is a sheaf of JM k -modules. If moreover E isa sheaf of algebras, J k .E/ will canonically be a sheaf of algebras as well. We regardJ k .E/ as O M -modules via the pull-back map 1 W O M ! . 1 / O M MFor 0 k l the inclusion I lC1M! IkC1 M induces the surjective map Jl .E/ !J k .E/. The sheaves J k .E/, k D 0;1;:::together with the maps just defined form aninverse system. Define J.E/ WD limJ k .E/. Thus, J.E/ carries a natural topology.We denote by p E W J.E/ ! E the canonical projection. In the case when E D O Mwe denote by p the corresponding projection p W J M ! O M .Byj k W E ! J k .E/ we

Deformations of gerbes on smooth manifolds 379denote the map e 7! 1 ˝ e, and j 1 WD limj k . In the case E D O M we also have thecanonical embedding O M ! J M given by f 7! f j 1 .1/.Letd 1 W O M M ˝ 2 O M2 1 E ! 1 1 1 M ˝1 1OO M MM˝ 12 O 1M 2 Edenote the exterior derivative along the first factor. It satisfiesd 1 .I kC1M ˝2 1O 1M 2 E/ 1 1 1 M ˝1 1OIk M M ˝2 1O 1M 2 Efor each k and, therefore, induces the mapd .k/1W J k .E/ ! 1 M=P ˝O MJ k 1 .E/:The maps d .k/1for different values of k are compatible with the maps J l .E/ ! J k .E/giving rise to the canonical flat connectionr can W J.E/ ! 1 M ˝ J.E/:Here and below we use notation ./ ˝ J.E/ for lim./ ˝OM J k .E/.Since r can is flat we obtain the complex of sheaves DR.J.E// D . M ˝J.E/; rcan /.When E D O M embedding O M ! J M induces embedding of de Rham complexDR.O/ D . M;d/into DR.J/. We denote the quotient by DR.J=O/. All the complexesabove are complexes of soft sheaves. We have the following:Proposition 7.1. The (hyper)cohomology H i .M; DR.J.E/// Š H i ..M I M ˝J.E//; r can / is 0 if i>0. The map j 1 W E ! J.E/ induces the isomorphism between.E/ and H 0 .M; DR.J.E/// Š H 0 ..M I M ˝ J.E//; r can /7.2 Jets of line bundles. Let, as before, M be a smooth manifold, J M be the sheaf ofinfinite jets of smooth functions on M and p W J M ! O M be the canonical projection.Set J M;0 D ker p. Note that J M;0 is a sheaf of O M modules and therefore is soft.Suppose now that L is a line bundle on M . Let Isom 0 .L ˝ J M ; J.L// denote thesheaf of local J M -module isomorphisms L ˝ J M ! J.L/ such that the followingdiagram is commutative:L ˝ J M J.L/Id˝pp L L.It is easy to see that the canonical map J M ! End JM.L˝J M / is an isomorphism.For 2 J M;0 the exponential series exp./ converges. The compositionJ M;0exp! J M ! End JM.L ˝ J M /

380 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygandefines an isomorphism of sheaves of groupsexpW J M;0 ! Aut 0 .L ˝ J M /;where Aut 0 .L ˝ J M / is the sheaf of groups of (locally defined) J M -linear automorphismsof L ˝ J M making the diagramL ˝ J M L ˝ J MId˝pId˝p Lcommutative.Lemma 7.2. The sheaf Isom 0 .L ˝ J M ; J.L// is a torsor under the sheaf of groupsexp J M;0 .Proof. Since L is locally trivial, both J.L/ and L˝J M are locally isomorphic to J M .Therefore the sheaf Isom 0 .L ˝ J M ; J.L// is locally non-empty, hence a torsor.Corollary 7.3. The torsor Isom 0 .L ˝ J M ; J.L// is trivial, that is, Isom 0 .L ˝ J M ;J.L// WD .MI Isom 0 .L ˝ J M ; J.L/// ¤ ¿.Proof. Since the sheaf of groups J M;0 is soft we have H 1 .M; J M;0 / D 0 (see [8],Lemme 22, cf. also [6], Proposition 4.1.7). Therefore every J M;0 -torsor is trivial.Corollary 7.4. The set Isom 0 .L ˝ J M ; J.L// is an affine space with the underlyingvector space .MI J M;0 /.Let L 1 and L 2 be two line bundles, and f W L 1 ! L 2 an isomorphism. Thenf induces a map Isom 0 .L 2 ˝ J M ; J.L 2 // ! Isom 0 .L 1 ˝ J M ; J.L 1 // which wedenote by Ad f :Ad f./D .j 1 .f // 1 ı ı .f ˝ Id/:Let L be a line bundle on M and f W N ! M is a smooth map. Then there is apull-back map f W Isom 0 .L ˝ J M ; J.L// ! Isom 0 .f L ˝ J N ; J.f L//.If L 1 , L 2 are two line bundles, and i 2 Isom 0 .L i ˝ J M ; J.L i //, i D 1; 2. Thenwe denote by 1 ˝ 2 the induced element of Isom 0 ..L 1 ˝ L 2 / ˝ J M ; J.L 1 ˝ L 2 //.For a line bundle L let L be its dual. Then given 2 Isom 0 .L ˝ J M ; J.L//there exists a unique a unique 2 Isom 0 .L ˝ J M ; J.L // such that ˝ D Id.For any bundle E J.E/ has a canonical flat connection which we denote by r can .A choice of 2 Isom 0 .L ˝ J M ; J.L// induces the flat connection 1 ır canLı onL ˝ J M .Let r be a connection on L with the curvature . It gives rise to the connectionr˝Id C Id ˝r can on L ˝ J M .

Deformations of gerbes on smooth manifolds 381Lemma 7.5. 1. Choose , r as above. Then the differenceF.;r/ D 1 ır canL ı .r ˝Id C Id ˝rcan / (7.1)is an element of 2 1 ˝ End JM.L ˝ J M / Š 1 ˝ J M .2. Moreover, F satisfiesr can F.;r/ C D 0: (7.2)Proof. We leave the verification of the first claim to the reader. The flatness of 1 ıı implies the second claim.r canLThe following properties of our construction are immediateLemma 7.6. 1. Let L 1 and L 2 be two line bundles, and f W L 1 ! L 2 an isomorphism.Let r be a connection on L 2 and 2 Isom 0 .L 2 ˝ J M ; J.L 2 //. ThenF.;r/ D F.Ad f./;Ad f.r//:2. Let L be a line bundle on M , r a connection on L and 2 Isom 0 .L ˝ J M ;J.L//. Let f W N ! M be a smooth map. Thenf F.;r/ D F.f ; f r/:3. Let L be a line bundle on M , r a connection on L and 2 Isom 0 .L ˝ J M ;J.L//. Let 2 .MI J M;0 /. ThenF. ; r/ D F.;r/ Cr can :4. Let L 1 , L 2 be two line bundles with connections r 1 and r 2 respectively, andlet i 2 Isom 0 .L i ˝ J M ; J.L i //, i D 1; 2. ThenF. 1 ˝ 2 ; r 1 ˝ Id C Id ˝r 2 / D F. 1 ; r 1 / C F. 2 ; r 2 /:7.3 DGLAs of infinite jets. Suppose that .U; A/ is a descent datum representing atwisted form of O X . Thus, we have the matrix algebra Mat.A/ and the cosimplicialDGLA G.A/ of local C-linear Hochschild cochains.The descent datum .U; A/ gives rise to the descent datum .U; J.A//, J.A/ D.J.A/; J.A 01 /; j 1 .A 012 //, representing a twisted form of J X , hence to the matrixalgebra Mat.J.A// and the corresponding cosimplicial DGLA G.J.A// of local O-linear continuous Hochschild cochains.The canonical flat connection r can on J.A/ induces the flat connection, still denotedr can on Mat.J.A// p for each p; the product on Mat.J.A// p is horizontal withrespect to r can . The flat connection r can induces the flat connection, still denotedr can on C .Mat.J.A// p / loc Œ1 which acts by derivations of the Gerstenhaber bracketand commutes with the Hochschild differential ı. Therefore we have the sheaf of

382 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganDGLA DR.C .Mat.J.A// p / loc Œ1/ with the underlying sheaf of graded Lie algebras N p U ˝ C .Mat.J.A// p / loc Œ1 and the differential r can C ı.For W Œn ! letG DR .J.A// D .N .n/ UI .0n/ DR.C .Mat.J.A// .0/ / loc Œ1//be the DGLA of global sections. The “inclusion of horizontal sections” map inducesthe morphism of DGLAj 1 W G.A/ ! G DR .J.A// :For W Œm ! Œn in , D ı there is a morphism of DGLA W G DR .J.A// ! G DR .J.A// making the diagramG.A/ j 1 G DR .J.A// G.A/ j 1 G DR .J.A// commutative.Let G DR .J.A// n D Q GŒn! DR .J.A// . The cosimplicial DGLA G DR .J.A//is defined by the assignment 3 Œn 7! G DR .J.A// n , 7! .Proposition 7.7. The map j 1 W G.A/ ! G.J.A// extends to a quasiisomorphismof cosimplicial DGLA.j 1 W G.A/ ! G DR .J.A//:The goal of this section is to construct a quasiisomorphism of the latter DGLA withthe simpler DGLA.The canonical flat connection r can on J X induces a flat connection on xC .J X /Œ1,the complex of O-linear continuous normalized Hochschild cochains, still denotedr can which acts by derivations of the Gerstenhaber bracket and commutes with theHochschild differential ı. Therefore we have the sheaf of DGLA DR. xC .J X /Œ1/ withthe underlying graded Lie algebra X ˝ xC .J X /Œ1 and the differential ı Cr can .Recall that the Hochschild differential ı is zero on C 0 .J X / due to commutativityof J X . It follows that the action of the sheaf of abelian Lie algebras J X D C 0 .J X /on xC .J X /Œ1 via the restriction of the adjoint action (by derivations of degree 1)commutes with the Hochschild differential ı. Since the cochains we consider areO X -linear, the subsheaf O X D O X j 1 .1/ J X acts trivially. Hence the action ofJ X descends to an action of the quotient J X =O X . This action induces the action ofthe abelian graded Lie algebra X ˝ J X=O X by derivations on the graded Lie algebra X ˝ xC .J X /Œ1. Further, the subsheaf . X ˝J X=O X / cl WD ker.r can / acts by derivationwhich commute with the differential ı Cr can .For! 2 .XI .X 2 ˝ J X=O X / cl /

Deformations of gerbes on smooth manifolds 383we denote by DR. xC .J X /Œ1/ ! the sheaf of DGLA with the underlying graded Liealgebra X ˝ xC .J X /Œ1 and the differential ı Cr can C ! . Letg DR .J X / ! D .XI DR. xC .J X /Œ1/ ! /;be the corresponding DGLA of global sections.Suppose now that U isacoverofX; let W N U ! X denote the canonical map.For W Œn ! letG DR .J/ ! D .N .n/UI DR. xC .J X /Œ1/ ! /:For W Œm ! and a morphism W Œm ! Œn in such that D ı the map.m/ ! .n/ induces the map W G DR .J/ ! ! G DR.J/ ! :For n D 0;1;:::let G DR .J/ n ! D Q Œn !G DR .J/ ! . The assignment Œn 7! G DR.J/ n !extends to a cosimplicial DGLA G DR .J/ ! . If we need to explicitly indicate the coverwe will also denote this DGLA by G DR .J/ ! .U/.Lemma 7.8. The cosimplicial DGLA G DR .J/ ! is acyclic, i.e. satisfies the condition(3.8).Proof. Consider the cosimplicial vector space V withV n D .N n UI DR. xC .J X /Œ1/ ! /and the cosimplicial structure induced by the simplicial structure of N U. The cohomologyof the complex .V ;@/is the Čech cohomology of U with the coefficients inthe soft sheaf of vector spaces ˝ xC .J X /Œ1 and, therefore, vanishes in the positivedegrees. G DR .J/ ! as a cosimplicial vector space can be identified with yV in thenotations of Lemma 2.1. Hence the result follows from Lemma 2.1.We leave the proof of the following lemma to the reader.Lemma 7.9. The map W g DR .J X / ! ! G DR .J/ 0 ! induces an isomorphism of DGLAg DR .J X / ! Š ker.G DR .J/ 0 ! G DR.J/ 1 ! /where the two maps on the right are .@ 0 0 / and .@ 0 1 / .Two previous lemmas together with Corollary 3.13 imply the following:Proposition 7.10. Let ! 2 .XI . 2 X ˝ J X=O X / cl / and let m be a commutativenilpotent ring. Then the map induces equivalence of groupoids:MC 2 .g DR .J X / ! ˝ m/ Š Stack.G DR .J/ ! ˝ m/:

384 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganFor ˇ 2 .XI 1 ˝ J X =O X / there is a canonical isomorphism of cosimplicialDGLA exp.ˇ /W G DR .J/ !Cr canˇ ! G DR .J/ ! . Therefore, G DR .J/ ! depends only onthe class of ! in H 2 DR .J X=O X /.The rest of this section is devoted to the proof of the following theorem.Theorem 7.11. Suppose that .U; A/ is a descent datum representing a twisted formS of O X . There exists a quasi-isomorphism of cosimplicial DGLA G DR .J/ ŒS !G DR .J.A//.7.4 Quasiisomorphism. Suppose that .U; A/ is a descent datum for a twisted formof O X . Thus, A is identified with O N0 U and A 01 is a line bundle on N 1 U.7.4.1 Multiplicative connections. Let C .A 01 / denote the set of connections r onA 01 which satisfy1. Ad A 012 ..pr 2 02 / r/ D .pr 2 01 / r˝Id C Id ˝ .pr 2 12 / r,2. .pr 0 00 / r is the canonical flat connection on O N0 U.Let Isom 0 .A 01˝J N1 U; J.A 01 // denote the subset of Isom 0 .A 01˝J N1 U; J.A 01 //which consists of which satisfy1. Ad A 012 ..pr 2 02 / / D .pr 2 01 / ˝ .pr 2 12 / ,2. .pr 0 00 / D IdNote that the vector space xZ 1 .UI 1 / of cocycles in the normalized Čech complexof the cover U with coefficients in the sheaf of 1-forms 1 acts on the set C .A 01 /,with the action given by˛ r DrC˛: (7.3)Here r2C .A 01 /, ˛ 2 xZ 1 .UI 1 / 1 .N 1 U/.Similarly, the vector space xZ 1 .UI J 0 / acts on the set Isom 0 .A 01˝J N1 U; J.A 01 //,with the action given as in Corollary 7.4.Note that since the sheaves involved are soft, cocycles coincide with coboundaries:xZ 1 .UI 1 / D xB 1 .UI 1 /, xZ 1 .UI J 0 / D xB 1 .UI J 0 /.Proposition 7.12. The set C .A 01 / (respectively, Isom 0 .A 01 ˝ J N1 U; J.A 01 //)is an affine space with the underlying vector space being xZ 1 .UI 1 / (respectively,xZ 1 .UI J 0 /).Proof. Proofs of the both statements are completely analogous. Therefore we explainthe proof of the statement concerning Isom 0 .A 01 ˝ J N1 U; J.A 01 // only.We show first that Isom 0 .A 01 ˝ J N1 U; J.A 01 // is nonempty. Choose an arbitrary 2 Isom 0 .A 01 ˝ J N1 U; J.A ij // such that .pr 0 00 / D Id. Then, by Corollary7.4, there exists c 2 .N 2 UI J 0 / such that c.Ad A 012 ..d 1 / 02 // D .d 0 / 01˝.d 2 / 12 . It is easy to see that c 2 xZ 2 .UI exp J 0 /. Since the sheaf exp J 0 is soft,

Deformations of gerbes on smooth manifolds 385corresponding Čech cohomology is trivial. Therefore, there exists 2 xC 1 .UI J 0 /such that c D L @. Then, 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //.Suppose that ; 0 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //. By Corollary 7.4 D 0for some uniquely defined 2 .N 2 UI J 0 /. It is easy to see that 2 xZ 1 .UI J 0 /.We assume from now on that we have chosen 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //,r2C .A 01 /. Such a choice defines p ij 2 Isom 0.A ij ˝ J Np U; J.A ij // for every pand 0 i;j p by p ij D .prp ij / . This collection of p ijinduces for every p algebraisomorphism p W Mat.A/ p ˝ J Np U ! Mat.J.A// p . The following compatibilityholds for these isomorphisms. Let f W Œp ! Œq be a morphism in . Then thefollowing diagram commutes:f .Mat.A/ p ˝ J Np U/f f ] Mat.A ˝ J Nq U/ q(7.4)f . p /f ] . p /f Mat.J.A// p f f ] Mat.J.A// q .Similarly define the connections r p ijD .pr p ij / r. For p D 0;1;::: set r p D˚pi;jD0 r ij ; the connections r p on Mat.A/ p satisfyf r p D .Ad f /.f ] r q /: (7.5)Note that F.;r/ 2 .N 1 UI 1 N 1 U ˝ J N 1 U/ is a cocycle of degree 1 in LC .UI 1 X ˝J X /. Vanishing of the corresponding Čech cohomology implies that there exists F 0 2.N 0 UI 1 N 0 U ˝ J N 0 U/ such that.d 1 / F 0 .d 0 / F 0 D F.;r/: (7.6)For p D 0;1;:::, 0 i p, let F piiD .pr p i / F 0 ; put F pij D 0 for i ¤ j .Let F p 2 .N p UI 1 N p U ˝ Mat.A/p / ˝ J Np U denote the diagonal matrix withcomponents F pij.Forf W Œp ! Œq we havef F p D f ] F q : (7.7)Then, we obtain the following equality of connections on Mat.A/ p ˝ J Np U:. p / 1 ır can ı p Dr p ˝ Id C Id ˝r can C ad F p : (7.8)The matrices F p also have the following property. Let r can F p be the diagonalmatrix with the entries .r can F p / ii Dr can F pii . Denote by rcan F p the image of r can F punder the natural map .N p UI 2 N p U ˝ Mat.A/p ˝ J Np U/ ! .N p UI 2 N p U ˝Mat.A/ p ˝ .J Np U=O Np U//. Recall the canonical map p W N p U ! X. Then, wehave the following:

386 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganLemma 7.13. There exists a unique ! 2 .XI . 2 X ˝ J X=O X / cl / such thatr can F p D p ! ˝ Id p;where Id p denotes the .p C 1/ .p C 1/ identity matrix.Proof. Using the definition of F 0 and formula (7.2) we obtain: .d 1 / r can F 0.d 0 / r can F 0 Dr can F.;r/ 2 2 .N 1 U/. Hence .d 1 / r can F 0 .d 0 / r can F 0 D 0,and there exists a unique ! 2 .XI X 2 ˝ .J X=O X // such that 0 ! D rcan F 0 .Since .r can / 2 D 0 it follows that r can ! D 0. For any p we have: .r can F p / ii Dpr i rcan F 0 D p !, and the assertion of the lemma follows.Lemma 7.14. The class of ! in H 2 ..XI X ˝ J X=O X /; r can / does not depend onthe choices made in its construction.Proof. The construction of ! is dependent on the choice of F 0 , 2 Isom 0 .A 01 ˝J N1 UJ.A 01 //, and r 2 C .A 01 / satisfying the equation (7.6). Assume that wemake different choices: 0 D .@/ L , r 0 D .@˛/ L rand .F 0 / 0 satisfying @.F L 0 / 0 DF. 0 ; r 0 /. Here, 2 xC 0 .UI J 0 / and ˛ 2 xC 0 .UI 1 /. We have: F. 0 ; r 0 / DF.;r/ @˛ L C @r L can . It follows that @..F L 0 / 0 F 0 r can C ˛/ D 0. Therefore.F 0 / 0 F 0 r can C ˛ D 0ˇ for some ˇ 2 .XI 1 X ˝ J X/. Hence if ! 0 isconstructed using 0 , r 0 , .F 0 / 0 then ! 0 ! Dr can Nˇ where Nˇ is the image of ˇ underthe natural projection .XI X 1 ˝ J X/ ! .XI X 1 ˝ .J X=O X //.Let W V ! U be a refinement of the cover U, and .V; A / the correspondingdescent datum. Choice of , r, F 0 on U induces the corresponding choice .N/ ,.N/ r, .N/ .F 0 / on V. Let ! denotes the form constructed as in Lemma 7.13using .N/ , .N/ r, .N/ F 0 . Then,! D .N/ !:The following result now follows easily and we leave the details to the reader.Proposition 7.15. The class of ! in H 2 ..XI X ˝ J X=O X /; r can / coincides withthe image ŒS of the class of the gerbe.7.5 Construction of the quasiisomorphism. For W Œn ! letH WD .N .n/ UI N .n/ U ˝ .0n/ C .Mat.A/ .0/ ˝ J N.0/ U/ loc Œ1/;considered as a graded Lie algebra. For W Œm ! Œn, D ı there is a morphismof graded Lie algebras W H ! H .Forn D 0;1;:::let H n WD Q H . TheŒn! assignment 3 Œn 7! H n , 7! defines a cosimplicial graded Lie algebra H.For each W Œn ! the map WD Id ˝ .0n/ . .0/ /W H ! G DR .J.A//

Deformations of gerbes on smooth manifolds 387is an isomorphism of graded Lie algebras. It follows from (7.4) that the maps yieldan isomorphism of cosimplicial graded Lie algebras W H ! G DR .J.A//:Moreover, the equation (7.8) shows that if we equip H with the differential given on.N .n/ UI N .n/ U ˝ .0n/ C .Mat.A/ .0/ ˝ J N.0/ U/ loc Œ1/ byı C .0n/ .r .0/ / ˝ Id C Id ˝r can C ad .0n/ .F .0/ /D ı C .0n/ ] .r .n/ / ˝ Id C Id ˝r can C ad .0n/ ] .F .n/ // (7.9)then becomes an isomorphism of DGLA. Consider now an automorphism exp F ofthe cosimplicial graded Lie algebra H given on H by exp .0n/ F p. Note the fact thatthis morphism preserves the cosimplicial structure follows from the relation (7.7).The following result is proved by the direct calculation; see [4], Lemma 16.Lemma 7.16.exp. Fp / ı .ı Cr p ˝ Id C Id ˝r can C ad F p / ı exp. Fp /D ı Cr p ˝ Id C Id ˝r can rF p:(7.10)Therefore the morphismexp F W H ! H (7.11)conjugates the differential given by the formula (7.9) into the differential which on H is given byConsider the mapdefined as follows:ı C .0n/ .r .0/ / ˝ Id C Id ˝r can .0n/ .r can F .0/ / : (7.12)cotr W xC .J Np U/Œ1 ! C .Mat.A/ p ˝ J Np U/Œ1 (7.13)cotr.D/.a 1 ˝ j 1 ;:::;a n ˝ j n / D a 0 :::a n D.j 1 ;:::;j n /: (7.14)The map cotr is a quasiisomorphism of DGLAs (cf. [15], section 1.5.6; see also [4]Proposition 14).Lemma 7.17. For every p the mapId ˝ cotr W N p U ˝ xC .J Np U/Œ1 ! N p U ˝ C .Mat.A/ p ˝ J Np U/Œ1 (7.15)is a quasiisomorphism of DGLA, where the source and the target are equipped with thedifferentials ı Cr can C ! and ı Cr p ˝ Id C Id ˝r can rF p respectively.

388 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganProof. It is easy to see that Id˝cotr is a morphism of graded Lie algebras, which satisfies.r p˝IdCId˝rcan /ı.Id˝cotr/ D .Id˝cotr/ır can and ıı.Id˝cotr/ D .Id˝cotr/ıı.Since the domain of .Id ˝ cotr/ is the normalized complex, in view of Lemma 7.13 wealso have rF p ı .Id ˝ cotr/ D .Id ˝ cotr/ ı !. This implies that .Id ˝ cotr/ is amorphism of DGLA.To see that this map is a quasiisomorphism, introduce filtration on N p U byF i N p U D iN p U and consider the complexes xC .J Np U/Œ1 and C .Mat.A// p ˝J Np U/Œ1 equipped with the trivial filtration. The map (7.15) is a morphism of filteredcomplexes with respect to the induced filtrations on the source and the target. Thedifferentials induced on the associated graded complexes are ı (or, more precisely,Id ˝ ı) and the induced map of the associated graded objects is Id ˝ cotr which is aquasi-isomorphism. Therefore, the map (7.15) is a quasiisomorphism as claimed.The map (7.15) therefore induces a morphism Id ˝cotr W G DR .J/ ! ! H for everyW Œn ! . These morphisms are clearly compatible with the cosimplicial structureand hence induce a quasiisomorphism of cosimplicial DGLAsId ˝ cotr W G DR .J/ ! ! Hwhere the differential in the right hand side is given by (7.12).We summarize our consideration in the following:Theorem 7.18. For a any choice of 2 Isom 0 .A 01 ˝J N1 U; J.A 01 //, r2C .A 01 /and F 0 as in (7.6) the composition ˆ;r;F WD ı exp. F / ı .Id ˝ cotr/is a quasiisomorphism of cosimplicial DGLAs.ˆ;r;F W G DR .J/ ! ! G DR .J.A// (7.16)Let W V ! U be a refinement of the cover U and let .V; A / be the induced descentdatum. We will denote the corresponding cosimplicial DGLAs by G DR .J/ ! .U/ andG DR .J/ ! .V/ respectively. Then the map NW N V ! N U induces a morphism ofcosimplicial DGLAs.N/ W G DR .J.A// ! G DR .J.A // (7.17)and.N/ W G DR .J/ ! .U/ ! G DR .J/ ! .V/: (7.18)Notice also that the choice that the choice of data , r, F 0 on N U induces thecorresponding data .N/ , .N/ r, .N/ F 0 on N V. This data allows one toconstruct using the equation (7.16) the mapˆ.N/ ;.N/ r;.N/ F W G DR .J/ ! ! G DR .J.A //: (7.19)The following proposition is an easy consequence of the description of the map ˆ,and we leave the proof to the reader.

Deformations of gerbes on smooth manifolds 389Proposition 7.19. The following diagram commutes:G DR .J/ ! .U/ .N/ G DR .J/ ! .V/ˆ;r;FG DR .J.A// .N/ G DR .J.A //.ˆ.N/ ;.N/ r; .N/ F(7.20)8 Proof of the main theoremIn this section we prove the main result of this paper. Recall the statement of thetheorem from the introduction:Theorem 1. Suppose that X is a C 1 manifold and S is an algebroid stack on X whichis a twisted form of O X . Then, there is an equivalence of 2-groupoid valued functorsof commutative Artin C-algebrasDef X .S/ Š MC 2 .g DR .J X / ŒS /:Proof. Suppose U isacoverofX such that 0 S.N 0U/ is nonempty. There is adescent datum .U; A/ 2 Desc C .U/ whose image under the functor Desc C .U/ !AlgStack C .X/ is equivalent to S.The proof proceeds as follows.Recall the 2-groupoids Def 0 .U; A/.R/ and Def.U; A/.R/ of deformations of andstar-products on the descent datum .U; A/ defined in Section 6.2. Note that the compositionDesc R .U/ ! Triv R .X/ ! AlgStack R .X/ induces functors Def 0 .U; A/.R/ !Def.S/.R/ and Def.U; A/.R/ ! Def.S/.R/, the second one being the compositionof the first one with the equivalence Def.U; A/.R/ ! Def 0 .U; A/.R/. We are goingto show that for a commutative Artin C-algebra R there are equivalences1. Def.U; A/.R/ Š MC 2 .g DR .J X / ŒS /.R/ and2. the functor Def.U; A/.R/ Š Def.S/.R/ induced by the functor Def.U; A/.R/ !Def.S/.R/ above.Let J.A/ D .J.A 01 /; j 1 .A 012 //. Then, .U; J.A// is a descent datum for atwisted form of J X . Let R be a commutative Artin C-algebra with maximal ideal m R .Then the first statement follows from the equivalencesDef.U; A/.R/ Š Stack.G.A/ ˝ m R / (8.1)Š Stack.G DR .J.A/// ˝ m R / (8.2)Š Stack.G DR .J X / ŒS ˝ m R / (8.3)Š MC 2 .g DR .J X / ŒS ˝ m R /: (8.4)Here the equivalence (8.1) is the subject of Proposition 6.4. The inclusion of horizontalsections is a quasi-isomorphism and the induced map in (8.2) is an equivalenceby Theorem 3.6. In Theorem 7.18 we have constructed a quasi-isomorphism

390 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganG.J X / ŒS ! G.J.A//; the induced map (8.3) is an equivalence by another applicationof Theorem 3.6. Finally, the equivalence (8.4) is shown in Proposition 7.10We now prove the second statement. We begin by considering the behavior ofDef 0 .U; A/.R/ under the refinement. Consider a refinement W V ! U . Recallthat by Proposition 7.10 the map induces equivalences MC 2 .g DR .J X / ! ˝ m/ !Stack.G DR .J/ ! .U/ ˝ m/ and MC 2 .g DR .J X / ! ˝ m/ ! Stack.G DR .J/ ! .V/ ˝ m/.It is clear that the diagramMC 2 .g DR .J X / ! ˝ m/ .N/Stack.G DR .J/ ! .U/ ˝ m/ Stack.G DR .J/ ! .V/ ˝ m/commutes, and therefore .N/ W Stack.G DR .J/ ! .U/˝m/ ! Stack.G DR .J/ ! .V/˝m/ is an equivalence. Then Proposition 7.19 together with Theorem 3.6 implies that.N/ W Stack.G DR .J.A/// ˝ m/ ! Stack.G DR .J.A /// ˝ m/ is an equivalence.It follows that the functor W Def.U; A/.R/ ! Def.V; A /.R/ is an equivalence.Note also that the diagramDef.U; A/.R/ Def.V; A /.R/Def 0 .U; A/.R/ Def 0 .V; A /.R/is commutative with the top horizontal and both vertical maps being equivalences.Hence it follows that the bottom horizontal map is an equivalence.Recall now that the embedding Def.U; A/.R/ ! Def 0 .U; A/.R/ is an equivalenceby Proposition 6.3. Therefore it is sufficient to show that the functor Def 0 .U; A/.R/ !Def.S/.R/ is an equivalence. Suppose that C is an R-deformation of S. It follows fromLemma 6.2 that C is an algebroid stack. Therefore, there exists a cover V and an R-descent datum .V; B/ whose image under the functor Desc R .V/ ! AlgStack R .X/ isequivalent to C. Replacing V by a common refinement of U and V if necessary we mayassume that there is a morphism of covers W V ! U. Clearly, .V; B/ is a deformationof .V; A /. Since the functor W Def 0 .U; A/.R/ ! Def 0 .V; A / is an equivalencethere exists a deformation .U; B 0 / such that .U; B 0 / is isomorphic to .V; B/. LetC 0 denote the image of .U; B 0 / under the functor Desc R .V/ ! AlgStack R .X/. Sincethe images of .U; B 0 / and .U; B 0 / in AlgStack R .X/ are equivalent it follows thatC 0 is equivalent to C. This shows that the functor Def 0 .U; A/.R/ ! Def.C/.R/ isessentially surjective.Suppose now that .U; B .i/ /, i D 1; 2, are R-deformations of .U; A/. Let C .i/denote the image of .U; B .i/ / in Def.S/.R/. Suppose that F W C .1/ ! C .2/ is a1-morphism.

Deformations of gerbes on smooth manifolds 391Let L denote the image of the canonical trivialization under the composition 0 .F /W A B .1/C Š 0 C .1/ F ! 0 C .1/ ŠA B .2/C :Thus, L is a B .1/ ˝R B .2/op -module such that the line bundle L ˝R C is trivial.Therefore, L admits a non-vanishing global section. Moreover, there is an isomorphismf W B .1/01 ˝.B .1/ / 1 .pr 1 1 / L ! .pr 1 0 / 1L˝.B .2/ / 1 B .2/01 of .B.1/ / 1 0 ˝R ..B .2/ / 1 1 /op -0modules.A choice of a non-vanishing global section of L gives rise to isomorphisms B .1/B .1/01 ˝.B .1/ / 1 1The composition.pr 1 1 / L and B .2/01 Š .pr1 0 / L ˝.B .2/ / 1 0B .2/01 .B .1/01 Š B.1/ 01 ˝.B .1/ / 1 .pr 1 1 / L ! f .pr 1 0 / L1˝.B .2/ / 1 B .2/01 Š B.2/ 01001 Šdefines a 1-morphism of deformations of .U; A/ such that the induced 1-morphismC .1/ ! C .2/ is isomorphic to F . This shows that the functor Def 0 .U; A/.R/ !Def.C/ induces essentially surjective functors on groupoids of morphisms. By similararguments left to the reader one shows that these are fully faithful.This completes the proof of Theorem 1.References[1] Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris), 3.Société Mathématique de France, Paris, 2003, Séminaire de géométrie algébrique du BoisMarie 1960–61, directed by A. Grothendieck, with two papers by M. Raynaud, updatedand annotated reprint of the 1971 original.[2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theoryand quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978),61–110.[3] L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque 225 (1994).[4] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan, Deformations of Azumaya algebras,in Actas del XVI Coloquio Latinoamericano de Algebra (Colonia del Sacramento, Uruguay,Agosto 2005), Biblioteca de la Revista Matemática Iberoamericana, 2008, 131–152.[5] P. Bressler, A. Gorokhovsky, R.Nest, and B. Tsygan, Deformation quantization of gerbes,Adv. Math. 214 (2007), 230–266.[6] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progr.Math. 107, Birkhäuser, Boston, MA, 1993.[7] A. D’Agnolo and P. Polesello, Stacks of twisted modules and integral transforms, in Geometricaspects of Dwork theory, Vol. I, Walter de Gruyter, Berlin 2004, 463–507.[8] J. Dixmier and A. Douady, Champs continus d’espaces hilbertiens et de C -algèbres, Bull.Soc. Math. France 91 (1963), 227–284.[9] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2), 79 (1964),59–103.

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Torsion classes of finite type and spectraGrigory Garkusha and Mike Prest1 IntroductionNon-commutative geometry comes in various flavours. One is based on abelian and triangulatedcategories, the latter being replacements of classical schemes. This is basedon classical results of Gabriel and later extensions, in particular byThomason. Precisely,Gabriel [6] proved that any noetherian scheme X can be reconstructed uniquely up toisomorphism from the abelian category, Qcoh X, of quasi-coherent sheaves over X.This reconstruction result has been generalized to quasi-compact schemes by Rosenbergin [15]. Based on Thomason’s classification theorem, Balmer [3] reconstructs anoetherian scheme X from the triangulated category of perfect complexes D per .X/.This result has been generalized to quasi-compact, quasi-separated schemes by Buan–Krause–Solberg [5].In this paper we reconstruct affine and projective schemes from appropriate abeliancategories. Our approach, similar to that used in [8], [9], is different from Rosenberg’s[15] and less abstract. Moreover, some results of the paper are of independentinterest.Let Mod R (respectively QGr A) denote the category of R-modules (respectivelygraded A-modules modulo torsion modules) with R (respectively A D L n0 A n)acommutative ring (respectively a commutative graded ring). We first demonstrate thefollowing result (cf. [8], [9]).Theorem (Classification). Let R (respectively A) be a commutative ring (respectivelycommutative graded ring which is finitely generated as an A 0 -algebra). Then the mapsV 7! S DfM 2 Mod R j supp R .M / V g;S 7! V D [ M 2Ssupp R .M /andV 7! S DfM 2 QGr A j supp A .M / V g;S 7! V D [ M 2Ssupp A .M /induce bijections between1. the set of all subsets V Spec R (respectively V Proj A) of the form V DSi2 Y i with Spec R n Y i (respectively Proj A n Y i ) quasi-compact and open forall i 2 , This paper was written during the visit of the first author to the University of Manchester supported bythe MODNET Research Training Network in Model Theory. He would like to thank the University for thekind hospitality.

394 G. Garkusha and M. Prest2. the set of all torsion classes of finite type in Mod R (respectively tensor torsionclasses of finite type in QGr A).This theorem says that Spec R and Proj A contain all the information about finitelocalizations in Mod R and QGr A respectively. The next result says that there isa 1-1 correspondence between the finite localizations in Mod R and the triangulatedlocalizations in D per .R/ (cf. [11], [8]).Theorem. Let R be a commutative ring. The mapinduces a bijection betweenS 7! T DfX 2 D per .R/ j H n .X/ 2 S for all n 2 Zg1. the set of all torsion classes of finite type in Mod R,2. the set of all thick subcategories of D per .R/.Following Buan–Krause–Solberg [5] we consider the lattices L tor .Mod R/ andL tor .QGr A/ of (tensor) torsion classes of finite type in Mod R and QGr A, as well astheir prime ideal spectra Spec.Mod R/ and Spec.QGr A/. These spaces come naturallyequipped with sheaves of rings O Mod R and O QGr A . The following result says that theschemes .Spec R;O R / and .Proj A; O Proj A / are isomorphic to .Spec.Mod R/; O Mod R /and .Spec.QGr A/; O QGr A / respectively.Theorem (Reconstruction). Let R (respectively A) be a commutative ring (respectivelycommutative graded ring which is finitely generated as an A 0 -algebra). Then there arenatural isomorphisms of ringed spacesand.Spec R;O R /.Proj A; O Proj A /! .Spec.Mod R/; OMod R /! .Spec.QGr A/; OQGr A /:2 Torsion classes of finite typeWe refer the reader to the appendix for necessary facts about localization and torsionclasses in Grothendieck categories.Proposition 2.1. Assume that B is a set of finitely generated ideals of a commutativering R. The set of those ideals which contain a finite products of ideals belonging toB is a Gabriel filter of finite type.Proof. See [17, VI.6.10].Given a module M , we denote by supp R .M / DfP 2 Spec R j M P ¤ 0g. HereM P denotes the localization of M at P , that is, the module of fractions M Œ.R n P/ 1 .

Torsion classes of finite type and spectra 395Note that V.I/ DfP 2 Spec R j I P g is equal to supp R .R=I / for every ideal Iandsupp R .M / D [V.ann R .x//; M 2 Mod R:x2MRecall from [10] that a topological space is spectral if it is T 0 , quasi-compact, ifthe quasi-compact open subsets are closed under finite intersections and form an openbasis, and if every non-empty irreducible closed subset has a generic point. Given aspectral topological space, X, Hochster [10] endows the underlying set with a new,“dual", topology, denoted X , by taking as open sets those of the form Y D S i2 Y iwhere Y i has quasi-compact open complement X nY i for all i 2 . Then X is spectraland .X / D X (see [10, Proposition 8]). The spaces, X, which we shall consider arenot in general spectral; nevertheless we make the same definition and denote the spaceso obtained by X .Given a commutative ring R, every closed subset of Spec R with quasi-compactcomplement has the form V.I/ for some finitely generated ideal, I ,ofR (see [2,Chapter 1, Ex. 17 (vii)]). Therefore a subset of Spec R is open if and only if it is of theform S V.I / with each I finitely generated. Notice that V.I/with I a non-finitelygenerated ideal is not open in Spec R in general. For instance (see [18, 3.16.2]), letR D CŒx 1 ;x 2 ;:::and m D .x 1 ;x 2 ;:::/. It is clear that V .m/ Dfmg is not open inSpec CŒx 1 ;x 2 ;:::.For definitions of terms used in the next result see the Appendix to this paper.Theorem 2.2 (Classification). Let R be a commutative ring.betweenThere are bijections1. the set of all open subsets V Spec R,2. the set of all Gabriel filters F of finite type,3. the set of all torsion classes S of finite type in Mod R.These bijections are defined as follows:´FV D¹I R j V.I/ V º;V 7!S V D¹M 2 Mod R j supp R .M / V ºI8ˆ< V F D [ V.I/;F 7!I 2Fˆ:S F D¹M 2 Mod R j ann R .x/ 2 F for every x 2 M ºI8ˆ< F S D¹I R j R=I 2 Sº;S 7! V ˆ: S D [ supp R .M /:M 2S

396 G. Garkusha and M. PrestProof. The bijection between Gabriel filters of finite type and torsion classes of finitetype is a consequence of a theorem of Gabriel (see, e.g., [7, 5.8]).Let F be a Gabriel filter of finite type. Then the set ƒ F of finitely generated idealsI belonging to F is a filter basis for F. Therefore V F D S I 2ƒ FV.I/ is open inSpec R.Now let V be an open subset of Spec R. Let ƒ denote the set of finitely generatedideals I such that V.I/ V . By definition of the topology V D S I 2ƒV.I/ andI 1 I n 2 ƒ for any I 1 ;:::;I n 2 ƒ. We denote by F 0 Vthe set of ideals I Rsuch that I J for some J 2 ƒ. By Proposition 2.1 F 0 Vis a Gabriel filter of finitetype. Clearly, F 0 V F V DfI R j V.I/ V g. Suppose I 2 F V n F 0 V; by [17,VI.6.13–15] (cf. the proof of Theorem 6.4) there exists a prime ideal P 2 V.I/ suchthat P 62 F 0 V . But V.I/ V and therefore P J for some J 2 ƒ, soP 2 F0 V ,acontradiction. Thus F 0 V D F V .Clearly, V D V FV for every open subset V Spec R. Let F be a Gabriel filterof finite type and I 2 F. Clearly F F VF and, as above, there is no ideal belongingto F VF n F. We have shown the bijection between the sets of all Gabriel filters of finitetype and all open subsets in Spec R. The description of the bijection between the setof torsion classes of finite type and the set of open subsets in Spec R is now easilychecked.3 The fg-topologyLet Inj R denote the set of isomorphism classes of indecomposable injective modules.Given a finitely generated ideal I of R, we denote by S I the torsion class of finitetype corresponding to the Gabriel filter of finite type having fI n g n1 as a basis (seeProposition 2.1 and Theorem 2.2). Note that a module M has S I -torsion if and only ifevery element x 2 M is annihilated by some power I n.x/ of the ideal I . Let us setD fg .I / WD fE 2 Inj R j E is S I -torsion freeg; V fg .I / WD Inj R n D fg .I /(“fg” referring to this topology being defined using only finitely generated ideals).Let E be any indecomposable injective R-module. Set P D P.E/to be the sum ofannihilator ideals of non-zero elements, equivalently non-zero submodules, of E. SinceE is uniform the set of annihilator ideals of non-zero elements of E is closed underfinite sum. It is easy to check ([14, 9.2]) that P.E/ is a prime ideal and P.E P / D P .Here E P stands for the injective hull of R=P. There is an embedding˛ W Spec R ! Inj R; P 7! E P ;which need not be surjective. We shall identify Spec R with its image in Inj R.If P is a prime ideal of a commutative ring R its complement in R is a multiplicativelyclosed set S. Given a module M we denote the module of fractions MŒS 1 by M P . There is a corresponding Gabriel filterF P DfI j P … V.I/g:

Torsion classes of finite type and spectra 397Clearly, F P is of finite type. The F P -torsion modules are characterized by the propertythat M P D 0 (see [17, p. 151]).More generally, let P be a subset of Spec R. ToP we associate a Gabriel filterF P D \F P DfI j P \ V.I/ D;g:P 2PThe corresponding torsion class consists of all modules M with M P D 0 for all P 2 P .Given a family of injective R-modules E, denote by F E the Gabriel filter determinedby E. By definition, this corresponds to the localizing subcategory S E DfM 2 Mod R j Hom R .M; E/ D 0 for all E 2 Eg.Proposition 3.1. A Gabriel filter F is of finite type if and only if it is of the form F Pwith P a closed set in Spec R. Moreover, F P is determined by E P DfE P j P 2 P gvia F P DfI j Hom R .R=I; E P / D 0g.Proof. This is a consequence of Theorem 2.2.Proposition 3.2. Let P be the closure of P in Spec R. Then P DfQ 2 Spec R jQ P g. Also F P D F P .Proof. This is direct from the definition of the topology.Recall that for any ideal I of a ring, R, and r 2 R we have an isomorphismR=.I W r/ Š .rR C I /=I , where .I W r/ Dfs 2 R j rs 2 I g, induced by sending1 C .I W r/ to r C I .Proposition 3.3. Let E be an indecomposable injective module and let P.E/ be theprime ideal defined before. Let I be a finitely generated ideal of R. Then E 2 V fg .I /if and only if E P.E/ 2 V fg .I /.Proof. Let I be such that E D E.R=I/. For each r 2 R n I we have, by the remarkjust above, that the annihilator of r CI 2 E is .I W r/and so, by definition of P.E/;wehave .I W r/ P.E/. The natural projection .rR C I /=I Š R=.I W r/ ! R=P.E/extends to a morphism from E to E P.E/ which is non-zero on r C I . Forming theproduct of these morphisms as r varies over R n I , we obtain a morphism from E toa product of copies of E P.E/ which is monic on R=I and hence is monic. ThereforeE is a direct summand of a product of copies of E P.E/ and so E 2 V fg .J / impliesE P.E/ 2 V fg .J /, where J is a finitely generated ideal.Now, E P.E/ 2 V fg .I /, where I is a finitely generated ideal, means that there isa non-zero morphism f W R=I n ! E P.E/ for some n. Since R=P.E/ is essential inE P.E/ the image of f has non-zero intersection with R=P.E/ so there is an ideal J ,without loss of generality finitely generated, with I n

398 G. Garkusha and M. Prestranges over the annihilators of non-zero elements of E. Since R=I m is finitely presented,0 ¤ f 0 g factorises through one of the maps R=I ! R=P.E/. In particular,there is a non-zero morphism R=I m ! E showing that E 2 V fg .I /, as required.Given a module M ,wesetŒM WD fE 2 Inj R j Hom R .M; E/ D 0g; .M/ WD Inj R n ŒM :Remark 3.4. For any finitely generated ideal I we have: D fg .I / \ Spec R D D.I /and V fg .I / \ Spec R D V.I/. Moreover, D fg .I / D ŒR=I .If I;J are finitely generated ideals, then D.IJ / D D.I / \ D.J /. It follows fromProposition 3.3 and Remark 3.4 that D fg .I / \ D fg .J / D D fg .IJ /. Thus the setsD fg .I / with I running over finitely generated ideals form a basis for a topology onInj R which we call the fg-ideals topology. This topological space will be denoted byInj fg R. Observe that if R is coherent then the fg-topology equals the Zariski topologyon Inj R (see [14], [8]). The latter topological space is defined by taking the ŒM withM finitely presented as a basis of open sets.Theorem 3.5. (cf. Prest [14, 9.6]) Let R be a commutative ring, let E be an indecomposableinjective module and let P.E/ be the prime ideal defined before. Then E andE P.E/ are topologically indistinguishable in Inj fg R.Proof. This follows from Proposition 3.3 and Remark 3.4.Theorem 3.6. (cf. Garkusha–Prest [8, Theorem A]) Let R be a commutative ring.The space Spec R is dense and a retract in Inj fg R. A left inverse to the embeddingSpec R,! Inj fg R takes an indecomposable injective module E to the prime idealP.E/. Moreover, Inj fg R is quasi-compact, the basic open subsets D fg .I /, with I finitelygenerated, are quasi-compact, the intersection of two quasi-compact open subsets isquasi-compact, and every non-empty irreducible closed subset has a generic point.Proof. For any finitely generated ideal I we haveD fg .I / \ Spec R D D.I /(see Remark 3.4). From this relation and Theorem 3.5 it follows that Spec R is densein Inj fg R and that ˛ W Spec R ! Inj fg R is a continuous map.One may check (see [14, 9.2]) thatˇ W Inj fg R ! Spec R;E 7! P.E/;is left inverse to ˛. Remark 3.4 implies that ˇ is continuous. Thus Spec R is a retractof Inj fg R.Let us show that each basic open set D fg .I / is quasi-compact (in particular Inj fg R DD fg .R/ is quasi-compact). Let D fg .I / D S i2 Dfg .I i / with each I i finitely generated.It follows from Remark 3.4 that D.I / D S i2 D.I i/. Since I is finitely generated,

Torsion classes of finite type and spectra 399D.I / is quasi-compact in Spec R by [2, Chapter 1, Ex. 17 (vii)]. We see that D.I / DSi2 0D.I i / for some finite subset 0 .Assume E 2 D fg .I / n S i2 0D fg .I i /. It follows from Theorem 3.5 that E P.E/ 2D fg .I /n S i2 0D fg .I i /. But E P.E/ 2 D fg .I /\Spec R D D.I / D S i2 0D.I i /, andhence it is in D.I i0 / D D fg .I i0 /\Spec R for some i 0 2 0 , a contradiction. So D fg .I /is quasi-compact. It also follows that the intersection D fg .I / \ D fg .J / D D fg .IJ / oftwo quasi-compact open subsets is quasi-compact. Furthermore, every quasi-compactopen subset in Inj fg R must therefore have the form D fg .I / with I finitely generated.Finally, it follows from Remark 3.4 and Theorem 3.5 that a subset V of Inj fg R isZariski-closed and irreducible if and only if there is a prime ideal Q of R such thatV DfE j P.E/ Qg. This obviously implies that the point E Q 2 V is generic.Notice that Inj fg R is not a spectral space in general, for it is not necessarily T 0 .Lemma 3.7. Let the ring R be commutative. Then the mapsandSpec R V7! Q V DfE 2 Inj R j P.E/ 2 V g.Inj fg R/ Q 7! V Q DfP.E/ 2 Spec R j E 2 Qg DQ \ Spec Rinduce a 1-1 correspondence between the lattices of open sets of Spec R and those of.Inj fg R/ .Proof. First note that E P 2 Q V for any P 2 V (see [14, 9.2]). Let us check thatQ V is an open set in .Inj fg R/ . Every closed subset of Spec R with quasi-compactcomplement has the form V.I/ for some finitely generated ideal, I ,ofR (see [2,Chapter 1, Exercise 17 (vii)]), so there are finitely generated ideals I R such thatV D S V.I /. Since the points E and E P.E/ are, by Theorem 3.5, indistinguishablein .Inj fg R/ we see that Q V D S V fg .I /, hence this set is open in .Inj fg R/ .The same arguments imply that V Q is open in Spec R. It is now easy to see thatV QV D V and Q VQ D Q.4 Torsion classes and thick subcategoriesWe shall write L.Spec R/, L..Inj fg R/ /, L thick .D per .R//, L tor .Mod R/ to denote:• the lattice of all open subsets of Spec R,• the lattice of all open subsets of .Inj fg R/ ,• the lattice of all thick subcategories of D per .R/,• the lattice of all torsion classes of finite type in Mod R, ordered by inclusion.

400 G. Garkusha and M. Prest(A thick subcategory is a triangulated subcategory closed under direct summands).Given a perfect complex X 2 D per .R/ denote by supp.X/ DfP 2 Spec R jX ˝LR R P ¤ 0g. It is easy to see thatsupp.X/ D [ n2Zsupp R .H n .X//;where H n .X/ is the nth homology group of X.Theorem 4.1 (Thomason [18]). Let R be a commutative ring. The assignmentsT 2 L thick .D per .R// 7! [supp.X/andX2TV 2 L.Spec R/ 7! fX 2 D per .R/ j supp.X/ V gare mutually inverse lattice isomorphisms.Given a subcategory X in Mod R, we may consider the smallest torsion class offinite type in Mod R containing X. This torsion class we denote byp \X D fS Mod R j S X is a torsion class of finite typeg:Theorem 4.2. (cf. Garkusha–Prest [8, Theorem C]) Let R be a commutative ring.There are bijections between• the set of all open subsets Y .Inj fg R/ ,• the set of all torsion classes of finite type in Mod R,• the set of all thick subcategories of D per .R/.These bijections are defined as follows:´S D¹M j .M / Y º;Y 7!T D¹X 2 D per .R/ j .H n .X// Y for all n 2 ZºI8ˆ< Y D [ /;S 7! M 2S.Mˆ:T D¹X 2 D per .R/ j H n .X/ 2 S for all n 2 ZºI8ˆ< Y D [.H n .X//;T 7! X2T ;n2Zˆ:S D p ¹H n .X/ j X 2 T ;n2 Zº:

Torsion classes of finite type and spectra 401Proof. That S Y DfM j .M / Y g is a torsion class follows because it is defined asthe class of modules having no non-zero morphism to a family of injective modules,E WD Inj R n Y . By Lemma 3.7, E \ Spec R D U is a closed set in Spec R, that isP.E/ 2 U for all E 2 E. S Y is also determined by the family of injective modulesfE P g P 2U . Indeed, any E 2 E is a direct summand of some power of E P.E/ by the proofof Proposition 3.3. Therefore Hom R .M; E P.E/ / D 0 implies Hom R .M; E/ D 0. ByProposition 3.1 S Y is of finite type. Conversely, given a torsion class of finite type S, theset Y S D S M 2S .M / is plainly open in .Inj fg R/ . Moreover, S YS D S and Y D Y SY .Consider the following diagram:L.Spec R/ L..Inj fg R/ / ıL thick .D per .R//L tor .Mod R/;where ; are as in Lemma 3.7, ; are as in Theorem 4.1 and the remaining maps arethe corresponding maps indicated in the formulation of the theorem. We have D 1by Theorem 4.1, D 1 by Lemma 3.7, and D ı 1 by the above.By construction,.V/ D ˚X j S n2Z supp R.H n .X// V DfX j supp.X/ V gfor all V 2 L.Spec R/. Thus D . Since , , are bijections so is .On the other hand,ı.T / D [supp R .H n .X// D [supp.X/X2T ;n2ZX2Tfor any T 2 L thick .D per .R//. We have used here the relation[supp R .M / D [supp R .H n .X//:M 2.T /X2T ;n2ZOne sees that ı D . Since ı; ; are bijections so is . Obviously, D 1and the diagram above yields the desired bijective correspondences. The theorem isproved.To conclude this section, we should mention the relation between torsion classesof finite type in Mod R and the Ziegler subspace topology on Inj R (we denote thisspace by Inj zg R). The latter topology arises from Ziegler’s work on the model theoryof modules [20]. The points of the Ziegler spectrum of R are the isomorphism classesof indecomposable pure-injective R-modules and the closed subsets correspond tocomplete theories of modules. It is well known (see [14, 9.12]) that for every coherentring R there is a 1-1 correspondence between the open (equivalently closed) subsets

402 G. Garkusha and M. Prestof Inj zg R and torsion classes of finite type in Mod R. However, this is not the case forgeneral commutative rings.The topology on Inj zg R can be defined as follows. Let M be the set of those modulesM which are kernels of homomorphisms between finitely presented modules; that isM D Ker.Kf ! L/ with K; L finitely presented. The sets .M / with M 2 M forma basis of open sets for Inj zg R. We claim that there is a ring R and a module M 2 Msuch that the intersection .M / \ Spec R is not open in Spec R, and hence such thatthe open subset .M / cannot correspond to any torsion class of finite type on Mod R.Such a ring has been pointed out by G. Puninski.LLet V be a commutative valuation domain with value group isomorphic to Dn2ZZ,aZ-indexed direct sum of copies of Z. The order on is defined as follows..a n / n2Z >.b n / n2Z if a i >b i for some i and a k D b k for every k.an 0 / n for some .an 0 / n with an 0 D 0 for all n N for some fixed N . (Recall thatthe valuation v on R satisfies v.r C s/ minfv.r/; v.s/g and v.rs/ D v.r/ C v.s/.)It follows that there is a prime ideal properly between I and J.R/. This gives acontradiction, as required.5 Graded rings and modulesIn this section we recall some basic facts about graded rings and modules.Definition. A(positively) graded ring is a ring A together with a direct sum decompositionA D A 0 ˚ A 1 ˚ A 2 ˚ as abelian groups, such that A i A j A iCj fori;j 0. Ahomogeneous element of A is simply an element of one of the groups A j ,and a homogeneous ideal of A is an ideal that is generated by homogeneous elements.A graded A-module is an A-module M together with a direct sum decompositionM D L j 2Z M j as abelian groups, such that A i M j M iCj for i 0; j 2 Z. Onecalls M j the j th homogeneous component of M . The elements x 2 M j are said to behomogeneous (of degree j ).

Torsion classes of finite type and spectra 403Note that A 0 is a commutative ring with 1 2 A 0 , that all summands M j are A 0 -modules, and that M D L j 2Z M j is a direct sum decomposition of M as an A 0 -module.Let A be a graded ring. The category of graded A-modules, denoted by Gr A, hasas objects the graded A-modules. A morphism of graded A-modules f W M ! N isan A-module homomorphism satisfying f.M j / N j for all j 2 Z. An A-modulehomomorphism which is a morphism in Gr A will be called homogeneous.Let M be a graded A-module and let N be a submodule of M . Say that N is agraded submodule if it is a graded module such that the inclusion map is a morphismin Gr A. The graded submodules of A are called graded ideals. Ifd is an integer thetail M d is the graded submodule of M having the same homogeneous components.M d / j as M in degrees j d and zero for j < d. We also denote the ideal A 1by A C .For n 2 Z, GrA comes equipped with a shift functor M 7! M.n/ where M.n/ isdefined by M.n/ j D M nCj . Then Gr A is a Grothendieck category with generatingfamily fA.n/g n2Z . The tensor product for the category of all A-modules induces atensor product on Gr A: given two graded A-modules M; N and homogeneous elementsx 2 M i ;y 2 N j , set deg.x ˝ y/ WD i C j . We define the homomorphism A-moduleHom A .M; N / to be the graded A-module which is, in dimension n 2 Z, the groupHom A .M; N / n of graded A-module homomorphisms of degree n, i.e.,Hom A .M; N / n D Gr A.M; N.n//:We say that a graded A-module M is finitely generated if it is a quotient of a freegraded module of finite rank L nsD1 A.d s/ where d 1 ;:::;d s 2 Z. Say that M is finitelypresented if there is an exact sequencemMA.c t / !tD1nMA.d s / ! M ! 0:sD1The full subcategory of finitely presented graded modules will be denoted by gr A. Notethat any graded A-module is a direct limit of finitely presented graded A-modules, andtherefore Gr A is a locally finitely presented Grothendieck category.Let E be any indecomposable injective graded A-module (we remind the reader thatthe corresponding ungraded module, L n E n, need not be injective in the category ofungraded A-modules). Set P D P.E/to be the sum of the annihilator ideals ann A .x/of non-zero homogeneous elements x 2 E. Observe that each ideal ann A .x/ is homogeneous.Since E is uniform the set of annihilator ideals of non-zero homogeneouselements of E is upwards closed so the only issue is whether the sum, P.E/, of themall is itself one of these annihilator ideals.Given a prime homogeneous ideal P , we use the notation E P to denote the injectivehull, E.A=P/, ofA=P . Notice that E P is indecomposable. We also denote the set ofisomorphism classes of indecomposable injective graded A-modules by Inj A.

404 G. Garkusha and M. PrestLemma 5.1. If E 2 Inj A then P.E/is a homogeneous prime ideal. If the module E hasthe form E P .n/ for some prime homogeneous ideal P and integer n, then P D P.E/.Proof. The proof is similar to that of [14, 9.2].It follows from the preceding lemma that the mapP A 7! E P 2 Inj Afrom the set of homogeneous prime ideals to Inj A is injective.A tensor torsion class in Gr A is a torsion class S Gr A such that for any X 2 Sand any Y 2 Gr A the tensor product X ˝ Y is in S.Lemma 5.2. Let A be a graded ring. Then a torsion class S is a tensor torsion classof Gr A if and only if it is closed under shifts of objects, i.e. X 2 S implies X.n/ 2 Sfor any n 2 Z.Proof. Suppose that S is a tensor torsion class of Gr A. Then it is closed under shiftsof objects, because X.n/ Š X ˝ A.n/.Assume the converse. Let X 2 S and Y 2 Gr A. Then there is a surjectionLi2I A.i/f Y . It follows that 1X ˝ f W L i2IX.i/ ! X ˝ Y is a surjection.Since each X.i/ belongs to S then so does X ˝ Y .Lemma 5.3. The mapS 7! F.S/ Dfa A j A=a 2 Sgestablishes a bijection between the tensor torsion classes in Gr A and the sets F ofhomogeneous ideals satisfying the following axioms:T1: A 2 F;T2: if a 2 F and if a is a homogeneous element of A then it holds that .a W a/ Dfx 2 A j xa 2 ag 2F;T3: if a and b are homogeneous ideals of A such that a 2 F and .b W a/ 2 F forevery homogeneous element a 2 a then b 2 F.We shall refer to such filters as t-filters. Moreover, S is of finite type if and only if F.S/has a basis of finitely generated ideals, that is every ideal in F.S/ contains a finitelygenerated ideal belonging to F.S/. In this case F.S/ will be referred to as a t-filterof finite type.Proof. It is enough to observe that there is a bijection between the Gabriel filters onthe family fA.n/g n2Z of generators closed under the shift functor (i.e., if a belongs tothe Gabriel filter then so does a.n/ for all n 2 Z) and the t-filters.Proposition 5.4. The following statements are true:1. Let F be a t-filter. If I;J belong to F, then IJ 2 F.

Torsion classes of finite type and spectra 4052. Assume that B is a set of homogeneous finitely generated ideals. The set B 0 offinite products of ideals belonging to B is a basis for a t-filter of finite type.Proof. 1. For any homogeneous element a 2 I we have .IJ W a/ J ,soIJ 2 F byT3and the fact that every homogeneous ideal containing an ideal from F must belongto F.2. We follow [17, VI.6.10]. We must check that the set F of homogeneous idealscontaining ideals in B 0 is a t-filter of finite type. T1 is plainly satisfied. Let a be ahomogeneous element in A and I 2 F. There is an ideal I 0 2 B 0 contained in I . Then.I W a/ I 0 and therefore .I W a/ 2 F, hence T2is satisfied as well.Next we verify that F satisfies T3. Suppose that I is a homogeneous ideal and thereexists J 2 F such that .I W a/ 2 F for every homogeneous element a 2 J . We mayassume that J 2 B 0 . Let a 1 ;:::;a n be generators of J . Then .I W a i / 2 F, i n,and .I W a i / J i for some J i 2 B 0 . It follows that a i J i I for each i, and henceJJ 1 :::J n J.J 1 \\J n / I ,soI 2 F.6 Torsion modules and the category QGr ALet A be a commutative graded ring. In this section we introduce the category QGr A,which is analogous to the category of quasi-coherent sheaves on a projective variety.The non-commutative analog of the category QGr A plays a prominent role in “noncommutativeprojective geometry" (see, e.g., [1], [16], [19]).Recall that the projective scheme Proj A is a topological space whose points are thehomogeneous prime ideals not containing A C . The topology of Proj A is defined bytaking the closed sets to be the sets of the form V.I/ DfP 2 Proj A j P I g for I ahomogeneous ideal of A. We set D.I / WD Proj A n V.I/.In the remainder of this section the homogeneous ideal A C A is assumed tobe finitely generated. This is equivalent to assuming that A is a finitely generatedA 0 -algebra. The space Proj A is spectral and the quasi-compact open sets are thoseof the form D.I / with I finitely generated (see, e.g., [9, 5.1]). Let Tors A denote thetensor torsion class of finite type corresponding to the family of homogeneous finitelygenerated ideals fA n C g n1 (see Proposition 5.4). We refer to the objects of Tors A astorsion graded modules.Let QGr A D Gr A= Tors A. Let Q denote the quotient functor Gr A ! QGr A.We shall identify QGr A with the full subcategory of Tors-closed modules. The shiftfunctor M 7! M.n/ defines a shift functor on QGr A for which we shall use the samenotation. Observe that Q commutes with the shift functor. Finally we shall writeO D Q.A/. Note that QGr A is a locally finitely generated Grothendieck categorywith the family, fQ.M /g M 2gr A , of finitely generated generators (see [7, 5.8]).The tensor product in Gr A induces a tensor product in QGr A, denoted by . Moreprecisely, one setsX Y WD Q.X ˝ Y/for any X; Y 2 QGr A.

406 G. Garkusha and M. PrestLemma 6.1. Given X; Y 2 Gr A there is a natural isomorphism in QGr A: Q.X/ Q.Y / Š Q.X ˝ Y/. Moreover, the functor Y W QGr A ! QGr A is right exactand preserves direct limits.Proof. See [9, 4.2].As a consequence of this lemma we get an isomorphism X.d/ Š O.d/ X forany X 2 QGr A and d 2 Z.The notion of a tensor torsion class of QGr A (with respect to the tensor product) is defined analogously to that in Gr A. The proof of the next lemma is like that ofLemma 5.2 (also use Lemma 6.1).Lemma 6.2. A torsion class S is a tensor torsion class of QGr A if and only if it isclosed under shifts of objects, i.e. X 2 S implies X.n/ 2 S for any n 2 Z.Given a prime ideal P 2 Proj A and a graded module M , denote by M P thehomogeneous localization of M at P .Iff is a homogeneous element of A,byM f wedenote the localization of M at the multiplicative set S f Dff n g n0 .Lemma 6.3. If T is a torsion module then T P D 0 and T f D 0 for any P 2 Proj A andf 2 A C . As a consequence, M P Š Q.M / P and M f Š Q.M / f for any M 2 Gr A.Proof. See [9, 5.5].Denote by L tor .Gr A; Tors A/ (respectively L tor .QGr A/) the lattice of the tensortorsion classes of finite type in Gr A with torsion classes containing Tors A (respectivelythe tensor torsion classes of finite type in QGr A) ordered by inclusion. The map`W L tor .Gr A; Tors A/ ! L tor .QGr A/;S 7! S= Tors Ais a lattice isomorphism, where S= Tors A DfQ.M / j M 2 Sg (see, e.g., [7, 1.7]).We shall consider the map ` as an identification.Theorem 6.4 (Classification). Let A be a commutative graded ring which is finitelygenerated as an A 0 -algebra. Then the mapsV 7! S DfM 2 QGr A j supp A .M / V g and S 7! V D [ supp A .M /M 2Sinduce bijections between1. the set of all open subsets V Proj A,2. the set of all tensor torsion classes of finite type in QGr A.Proof. By Lemma 5.3 it is enough to show that the mapsV 7! F V DfI 2 A j V.I/ V g and F 7! V F D [ V.I/I 2F

Torsion classes of finite type and spectra 407induce bijections between the set of all open subsets V Proj A and the set of allt-filters of finite type containing fA n C g n1.Let F be such a t-filter. Then the set ƒ F of finitely generated graded ideals Ibelonging to F is a basis for F. Clearly V F D S I 2ƒ FV.I/,soV F is open in Proj A.Now let V be an open subset of Proj A. Let ƒ be the set of finitely generatedhomogeneous ideals I such that V.I/ V . Then V D S I 2ƒ V.I/and I 1 I n 2 ƒfor any I 1 ;:::;I n 2 ƒ. We denote by F 0 Vthe set of homogeneous ideals I A suchthat I J for some J 2 ƒ. By Proposition 5.4(2) F 0 Vis a t-filter of finite type.Clearly, F 0 V F V . Suppose I 2 F V n F 0 V .We can use Zorn’s lemma to find an ideal J I which is maximal with respect toJ … F 0 V (we use the fact that F0 Vhas a basis of finitely generated ideals). We claim thatJ is prime. Indeed, suppose a; b 2 A are two homogeneous elements not belonging toJ . Then J CaA and J CbA must be members of F 0 V , and also .J CaA/.J CbA/ 2 F0 Vby Proposition 5.4(1). But .J C aA/.J C bA/ J C abA, and therefore ab … J .We see that J 2 V.I/ V , and hence J 2 V.I 0 / for some I 0 2 ƒ. But this impliesJ 2 F 0 V , a contradiction. Thus F0 V D F V . Clearly, V D V FV for every open subsetV Proj A. Let F be a t-filter of finite type and I 2 F. Then I J for someJ 2 ƒ F , and hence V.I/ V.J/ V F . It follows that F F VF . As above, there isno ideal belonging to F VF n F. We have shown the desired bijection between the setsof all t-filters of finite type and all open subsets in Proj A.7 The prime spectrum of an ideal latticeInspired by recent work of Balmer [4], Buan, Krause, and Solberg [5] introduce thenotion of an ideal lattice and study its prime ideal spectrum. Applications arise fromabelian or triangulated tensor categories.Definition (Buan, Krause, Solberg [5]). An ideal lattice is by definition a partiallyordered set L D .L; /, together with an associative multiplication L L ! L, suchthat the following holds.(L1) The poset L is a complete lattice, that is,sup A D _ a and inf A D ^aa2Aa2Aexist in L for every subset A L.(L2) The lattice L is compactly generated, that is, every element in L is the supremumof a set of compact elements. (An element a 2 L is compact, if for all A Lwith a sup A there exists some finite A 0 A with a sup A 0 .)(L3) We have for all a; b; c 2 La.b _ c/ D ab _ ac and .a _ b/c D ac _ bc:

408 G. Garkusha and M. Prest(L4) The element 1 D sup L is compact, and 1a D a D a1 for all a 2 L.(L5) The product of two compact elements is again compact.A morphism W L ! L 0 of ideal lattices is a map satisfying _a2Aa D _ .a/ for A L;a2A.1/ D 1 and .ab/ D .a/.b/ for a; b 2 L:Let L be an ideal lattice. Following [5] we define the spectrum of prime elementsin L. An element p ¤ 1 in L is prime if ab p implies a p or b p for alla; b 2 L. We denote by Spec L the set of prime elements in L and define for eacha 2 LV.a/ Dfp 2 Spec L j a pg and D.a/ Dfp 2 Spec L j a 6 pg:The subsets of Spec L of the form V.a/are closed under forming arbitrary intersectionsand finite unions. More precisely, _ V a i D \ V.a i / and V.ab/ D V.a/[ V.b/:i2i2Thus we obtain the Zariski topology on Spec L by declaring a subset of Spec L to beclosed if it is of the form V.a/ for some a 2 L. The set Spec L endowed with thistopology is called the prime spectrum of L. Note that the sets of the form D.a/ withcompact a 2 L form a basis of open sets. The prime spectrum Spec L of an ideallattice L is spectral [5, 2.5].There is a close relation between spectral spaces and ideal lattices. Given a topologicalspace X, we denote by L open .X/ the lattice of open subsets of X and considerthe multiplication mapL open .X/ L open .X/ ! L open .X/; .U; V / 7! UV D U \ V:The lattice L open .X/ is complete.The following result, which appears in [5], is part of the Stone Duality Theorem(see, for instance, [12]).Proposition 7.1. Let X be a spectral space. Then L open .X/ is an ideal lattice. Moreover,the mapX ! Spec L open .X/; x 7! X n fxg;is a homeomorphism.We deduce from the classification of torsion classes of finite type (Theorems 2.2and 6.4) the following.

Torsion classes of finite type and spectra 409Proposition 7.2. Let R (respectively A) be a commutative ring (respectively gradedcommutative ring which is finitely generated as an A 0 -algebra). Then L tor .Mod R/and L tor .QGr A/ are ideal lattices.Proof. The spaces Spec R and Proj A are spectral. Thus Spec R and Proj A arespectral, also L open .Spec R/ and L open .Proj A/ are ideal lattices by Proposition 7.1.By Theorems 2.2 and 6.4 we have isomorphisms L open .Spec R/ Š L tor .Mod R/ andL open .Proj A/ Š L tor .QGr A/. Therefore L tor .Mod R/ and L tor .QGr A/ are ideallattices.Corollary 7.3. Let R (respectively A) be a commutative ring (respectively the gradedcommutative ring which is finitely generated as an A 0 -algebra). The points ofSpec L tor .Mod R/ (respectively Spec L tor .QGr A/) are the \-irreducible torsionclasses of finite type in Mod R (respectively tensor torsion classes of finite type inQGr A) and the mapsf W Spec R ! Spec L tor .Mod R/; P 7! S P DfM 2 Mod R j M P D 0g;f W Proj A ! Spec L tor .QGr A/; P 7! S P DfM 2 QGr A j M P D 0gare homeomorphisms of spaces.Proof. This is a consequence of Theorems 2.2, 6.4 and Propositions 7.1, 7.2.8 Reconstructing affine and projective schemesLet R (respectively A) be a commutative ring (respectively graded commutative ringwhich is finitely generated as an A 0 -algebra). We shall write Spec.Mod R/ WDSpec L tor .Mod R/ (resp. Spec.QGr A/ WD Spec L tor .QGr A/) and supp.M / WDfP 2 Spec.Mod R/ j M 62 P g (resp. supp.M / WD fP 2 Spec.QGr A/ j M 62 P g)for M 2 Mod R (resp. M 2 QGr A). It follows from Corollary 7.3 thatsupp R .M / D f 1 .supp.M // (resp. supp A .M / D f 1 .supp.M //):Following [4], [5], we define a structure sheaf on Spec.Mod R/ (Spec.QGr A/)asfollows. For an open subset U Spec.Mod R/ (U Spec.QGr A/), letS U DfM 2 Mod R.QGr A/ j supp.M / \ U D;gand observe that S U DfM j M P D 0 for all P 2 f1 .U /g is a (tensor) torsion class.We obtain a presheaf of rings on Spec.Mod R/ (Spec.QGr A/) byU 7! End Mod R=SU .R/.End QGr A=SU .O//:If V U are open subsets, then the restriction mapEnd Mod R=SU .R/ ! End Mod R=SV .R/.End QGr A=SU .O/ ! End QGr A=SV .O//

410 G. Garkusha and M. Prestis induced by the quotient functorMod R=S U ! Mod R=S V .QGr A=S U ! QGr A=S V /:The sheafification is called the structure sheaf of Mod R (QGr A) and is denoted byO Mod R (O QGr A ). This is a sheaf of commutative rings by [13, XI.2.4]. Next letP 2 Spec.Mod R/ and P WD f1 .P /. WehaveO Mod R;P D lim !P 2UEnd Mod R=SU .R/ D lim !f …PEnd Mod R=SD.f /.R/ Š lim !f …PR f D O R;P :Similarly, for P 2 Spec.QGr A/ and P WD fO QGr A;P Š O Proj A;P :1 .P / we haveThe next theorem says that the abelian category Mod R (QGr A) contains all thenecessary information to reconstruct the affine (projective) scheme .Spec R;O R / (respectively.Proj A; O Proj A /).Theorem 8.1 (Reconstruction). Let R (respectively A) be a ring (respectively gradedring which is finitely generated as an A 0 -algebra). The maps of Corollary 7.3 induceisomorphisms of ringed spacesandf W .Spec R;O R /f W .Proj A; O Proj A /! .Spec.Mod R/; OMod R /! .Spec.QGr A/; OQGr A /:Proof. The proof is similar to that of [5, 9.4]. Fix an open subset U Spec.Mod R/and consider the composition of the functorsF W Mod R . /! Qcoh Spec R. /j f 1 .U /! Qcoh f 1 .U /:Here, for any R-module M , we denote by M its associated sheaf. By definition,the stalk of M at a prime P equals the localized module M P . We claim that Fannihilates S U . In fact, M 2 S U implies f 1 .supp.M //\f 1 .U / D;and thereforesupp R .M /\f 1 .U / D;. Thus M P D 0 for all P 2 f 1 .U / and therefore F.M/ D0. It follows that F factors through Mod R=S U and induces a map End Mod R=SU .R/ !O R .f 1 .U // which extends to a map O Mod R .U / ! O R .f 1 .U //. This yields themorphism of sheaves f ] W O Mod R ! f O R .By the above f ] induces an isomorphism f ]P W O Mod R;f.P/ ! O R;P at each pointP 2 Spec R. We conclude that f ]Pis an isomorphism. It follows that f is an isomorphismof ringed spaces if the map f W Spec R ! Spec.Mod R/ is a homeomorphism.This last condition is a consequence of Propositions 7.1 and 7.2. The same argumentsapply to show thatf W .Proj A; O Proj A /! .Spec.QGr A/; OQGr A /is an isomorphism of ringed spaces.

Torsion classes of finite type and spectra 4119 AppendixA subcategory S of a Grothendieck category C is said to be Serre if for any short exactsequence0 ! X 0 ! X ! X 00 ! 0X 0 ;X 00 2 S if and only if X 2 S. A Serre subcategory S of C is said to be atorsion class if S is closed under taking coproducts. An object C of C is said to betorsionfree if .S;C/D 0. The pair consisting of a torsion class and the correspondingclass of torsionfree objects is referred to as a torsion theory. Given a torsion class Sin C the quotient category C=S is the full subcategory with objects those torsionfreeobjects C 2 C satisfying Ext 1 .T; C / D 0 for every T 2 S. The inclusion functori W S ! C admits the right adjoint t W C ! S which takes every object X 2 C to themaximal subobject t.X/of X belonging to S. The functor t we call the torsion functor.Moreover, the inclusion functor i W C=S ! C has a left adjoint, the localization functor. / S W C ! C=S, which is also exact. Then,Hom C .X; Y / Š Hom C=S .X S ;Y/for all X 2 C and Y 2 C=S. A torsion class S is of finite type if the functor i W C=S !C preserves directed sums. If C is a locally coherent Grothendieck category then S isof finite type if and only if i W C=S ! C preserves direct limits (see, e.g., [7]).Let C be a Grothendieck category having a family of finitely generated projectivegenerators A DfP i g i2I . Let F D S i2I Fi be a family of subobjects, where each F iis a family of subobjects of P i . We refer to F as a Gabriel filter if the following axiomsare satisfied:T1: P i 2 F i for every i 2 I ;T2: if a 2 F i and W P j ! P i then fa W g D 1 .a/ 2 F j ;T3: if a and b are subobjects of P i such that a 2 F i and fb W g 2F j for anyW P j ! P i with Im a then b 2 F i .In particular each F i is a filter of subobjects of P i . A Gabriel filter is of finite type ifeach of these filters has a cofinal set of finitely generated objects (that is, if for each iand each a 2 F i there is a finitely generated b 2 F i with a b).Note that if A DfAg is a ring and a is a right ideal of A, then for every endomorphismW A ! A 1 .a/ Dfa W .1/g Dfa 2 A j .1/a 2 ag:On the other hand, if x 2 A, then fa W xg D 1 .a/, where 2 End A is such that.1/ D x.It is well-known (see, e.g., [7]) that the mapS 7 ! F.S/ Dfa P i j i 2 I; P i =a 2 Sgestablishes a bijection between the Gabriel filters (respectively Gabriel filters of finitetype) and the torsion classes on C (respectively torsion classes of finite type on C).

412 G. Garkusha and M. PrestReferences[1] M. Artin, J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109(2) (1994),228–287.[2] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-WesleyPublishing Co., Reading, Mass, 1969.[3] P. Balmer, Presheaves of triangulated categories and reconstruction of schemes, Math. Ann.324(3) (2002), 557–580.[4] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew.Math. 588 (2005), 149–168.[5] A. B. Buan, H. Krause, Ø. Solberg, Support varieties – an ideal approach, Homology,Homotopy Appl. 9 (2007), 45–74.[6] P. Gabriel, Des catégories abeliénnes, Bull. Soc. Math. France 90 (1962), 323–448.[7] G. Garkusha, Grothendieck categories, Algebra i Analiz 13(2) (2001), 1–68; English transl.St. Petersburg Math. J. 13 (2002), 149–200.[8] G. Garkusha, M. Prest, Classifying Serre subcategories of finitely presented modules, Proc.Amer. Math. Soc. 136 (2008), 761–770.[9] G. Garkusha, M. Prest, Reconstructing projective schemes from Serre subcategories, J. Algebra319 (2008), 1132–1153.[10] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142(1969), 43–60.[11] M. Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353 (2001),3181—3191.[12] P. T. Johnstone, Stone Spaces, Cambridge Stud. in Adv. Math. 3, Cambridge UniversityPress, Cambridge 1982.[13] Ch. Kassel, Quantum groups, Grad. Texts in Math. 155, Springer-Verlag, New York 1995.[14] M. Prest, The Zariski spectrum of the category of finitely presented modules, preprintavailable at maths.man.ac.uk/mprest.[15] A. L. Rosenberg, The spectrum of abelian categories and reconstruction of schemes, inRings, Hopf algebras, and Brauer groups, Lect. Notes PureAppl. Math. 197, Marcel Dekker,New York 1998, 257–274.[16] J. T. Stafford, Noncommutative projective geometry, in Proceedings of the InternationalCongress of Mathematicians (Beijing, 2002), Vol. II, Higher Ed. Press, Beijing 2002,93–103.[17] B. Stenström, Rings of quotients, Grundlehren Math. Wiss. 217, Springer-Verlag, Berlin,Heidelberg, New York 1975.[18] R. W. Thomason, The classification of triangulated subcategories, Compos. Math. 105(1997), 1–27.[19] A. B. Verevkin, On a noncommutative analogue of the category of coherent sheaves on aprojective scheme, in Algebra and analysis (Tomsk, 1989), Amer. Math. Soc. Transl. (2)151, Amer. Math. Soc., Providence, RI, 1992.[20] M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149–213.

Parshin’s conjecture revisitedThomas Geisser 1 IntroductionParshin’s conjecture states that K i .X/ Q D 0 for i>0and X smooth and projectiveover a finite field F q . The purpose of this paper is to break up Parshin’s conjectureinto several independent statements, in the hope that each of them is easier to attackindividually. If CH n .X; i/ is Bloch’s higher Chow group of cycles of relative dimensionn, then in view of K i .X/ Q Š L n CH n.X; i/ Q , Parshin’s conjecture is equivalentto Conjecture P.n/ for all n, stating that CH n .X; i/ Q D 0 for i>0, and all smooth andprojective X. We show, assuming resolution of singularities, that Conjecture P.n/ isequivalent to the conjunction of three conjectures A.n/, B.n/ and C.n/, and give severalequivalent versions of these conjectures. This is most conveniently formulated in termsof weight homology. We define HW .X; Q.n// to be the homology of the complexCH n .W.X// Q , where W.X/is the weight complex defined by Gillet–Soulé [8] shiftedby 2n. Then, in a nutshell, Conjecture A.n/ states that for all schemes X over F q ,the niveau spectral sequence of CH n .X; / Q degenerates to one line, Conjecture C.n/states that for all schemes X over F q the niveau spectral sequence of HW .X; Q.n//degenerates to one line, and Conjecture B.n/ states that, for all X over F q , the twolines are isomorphic. The conjunction of A.n/, B.n/, and C.n/ clearly implies P.n/,because HiW .X; Q.n// D 0 for i 6D 2n and X smooth and projective over F q , and weshow the converse. Note that in the above formulation, Parshin’s conjecture impliesstatements on higher Chow groups not only for smooth and projective schemes, butgives a way to calculate CH n .X; i/ Q for arbitrary X.A more concrete version of A.n/ is that, for every smooth and projective scheme Xof dimension d over F q , CH n .X; i/ Q D 0 for i>d n. A reformulation of B.n/ isthat for every smooth and projective scheme X of dimension d>nC 1, the followingsequence is exact0 ! K M d n .k.X// Q ! MK M d n 1 .k.x// Q ! MK M d n 2 .k.x// Qx2X .1/ x2X .2/and that this sequence is exact at K M 1 .k.X// Q if d D dim X D n C 1.In the second half of the paper, we focus on the case n D 0, because of its applicationsin [5]. A different version of weight homology has been studied by Jannsen [11],and he proved that Conjecture C.0/ holds under resolution of singularities. We use thisto give two more versions of Conjecture P.0/. The first is that there is an isomorphism Supported in part by NSF grant No.0556263

414 T. Geisserfrom higher Chow groups with Q l -coefficients to l-adic cohomology, for all X and i,CH 0 .X; i/ Ql ˚ CH 0 .X; i C 1/ Ql ! H i .X et ; yQ l /:Conjecture P.0/ can also be recovered from, and implies a structure theorem for higherChow groups of smooth affine schemes: For all smooth and affine schemes U ofdimension d over F q , the groups CH 0 .U; i/ are torsion for i 6D d, and the canonicalmap CH 0 .U; d/ Ql ! H d . xU et ; yQ l / Gal.Fq/ is an isomorphism.Finally, we reproduce an argument of Levine showing that if F is the absoluteFrobenius, the push-forward F acts like q n on CH n .X; i/, and the pull-back F actson motivic cohomology HM i .X; Z.n// like qn for all n. As a corollary, ConjectureP.0/ follows from finite dimensionality of smooth and projective schemes over finitefields in the sense of Kimura [13].Acknowledgements. This paper was inspired by the work of, and discussions with,U. Jannsen and S. Saito. We are indebted to the referee, whose careful reading helpedto improve the exposition.2 Parshin’s conjectureWe fix a perfect field k of characteristic p, and consider the category of separatedschemes of finite type over k. We recall some facts on Bloch’s higher Chow groups[1], see [3] for a survey. Let z n .X; i/ be the free abelian group generated by cycles ofdimension n C i on X k i which meet all faces properly, and let z n .X; / be thecomplex of abelian groups obtained by taking the alternating sum of intersection withface maps as differential. We define CH n .X; i/ as the ith homology of this complexand motivic Borel–Moore homology to beHi c .X; Z.n// D CH n.X; i 2n/:For a proper map f W X ! Y we have a push-forward z n .X; / ! z n .Y; /, for a flat,quasi-finite map f W X ! Y we have a pull-back z n .X; / ! z n .Y; /, and a closedembedding i W Z ! X with open complement j W U ! X induces a localizationsequence!Hi c .Z; Z.n// i ! Hi c .X; Z.n// j ! Hi c .U; Z.n// ! :If X is smooth of pure dimension d, then Hi c.X;Z.n// Š H 2d i .X; Z.d n//, wherethe right-hand side is Voevodsky’s motivic cohomology [15]. For a finitely generatedfield F over k, we define Hi c.F;Z.n// D colim U Hi c .U; Z.n//, where the colimitruns through U of finite type over k with field of functions F . For the reader who ismore familiar with motivic cohomology, we mention that Voevodsky’s theorem impliesthat for a field F of transcendence degree d over k, wehaveH ci .F; Z.n// Š H 2d i .F; Z.d n// Š(0; i < d C n;K M .F /;d ni D d C n:

Parshin’s conjecture revisited 415The latter isomorphism is due to Nesterenko–Suslin and Totaro. It follows formallyfrom localization that there are spectral sequencesEs;t 1 .X/ D MHsCt c .k.x/; Z.n// ) H sCt c .X; Z.n//: (1)x2X .s/Here X .s/ denotes points of X of dimension s. Since Hi c .F; Z.n// D 0 for i 0. If Tate’s conjecture holds and rational equivalenceand homological equivalence agree up to torsion for all X, then Parshin’s conjectureholds by [2]. Since K i .X/ Q D L n CH n.X; i/ Q , it follows that Parshin’s conjectureis equivalent to the following conjecture for all n.Conjecture P.n/. For all smooth and projective schemes X over the finite field F q ,the groups Hi c .X; Q.n// vanish for i 6D 2n.We will refer to the following equivalent statements as Conjecture A.n/:Proposition 2.1. For a fixed integer n, the following statements are equivalent:a) For all schemes X=F q and all i, ˛ induces an isomorphism Hi c .X; Q.n// ŠzHi c .X; Q.n//.b) For all finitely generated fields k=F q with d WD trdeg k=F q , and all i 6D d C n,we have Hi c .k; Q.n// D 0.c) For all smooth and projective X over F q and all i > dim X C n, we haveHi c .X; Q.n// D 0.d) For all smooth and affine schemes U over F q and all i>dim U C n, we have.U; Q.n// D 0.H ciProof. a) ) c), d): The complex (2) is concentrated in degrees Œ2n; d C n.c) ) b): This is proved by induction on the transcendence degree. By de Jong’stheorem, we find a smooth and proper model X of a finite extension of k. Lookingat the niveau spectral sequence (1), we see that the induction hypothesis impliesH ci.X; Q.n// D 0 for i>dC n (see [2] for details).

416 T. Geisserd) ) b): follows by writing k as a colimit of smooth affine scheme schemes ofdimension d.b) ) a): The niveau spectral sequence collapses to the complex (2).Using the Gersten resolution, the statement in the proposition implies that on asmooth X of dimension d, the motivic complex Q.d n/ is concentrated in degreed n, and if C n WD H d n .Q.d n// D CH n . ;d n/ Q , then CH n .X; i/ DH d n i .X; C n /.Statement 2.1 d) is part of the following affine analog of P.n/:Conjecture L.n/. For all smooth and affine schemes U of dimension d over the finitefield F q , the group Hi c .U; Q.n// vanishes unless d i d C n.Since Hi c.U;Q.n// Š H 2d i .U; Q.d n//, Conjecture L.n/ can be thought ofas an analog of the affine Lefschetz theorem.3 Weight homologyThis section is inspired by Jannsen [11]. Throughout this section we assume resolutionof singularities over the field k. Let C be the category with objects smooth projectivevarieties over a field k of characteristic 0, and Hom C .X; Y / D L i CHdim Y i.X Y i /,where Y i runs through the connected components of Y . Let H be the homotopycategory of bounded complexes over C . Gillet and Soulé define in [8], for everyseparated scheme of finite type, a weight complex W.X/ 2 H satisfying the followingproperties [8, Theorem 2] (our notation differs from loc. cit. in variance):a) W.X/ is represented by a bounded complexM.X 0 / M.X 1 / M.X k /with dim X i dim X i, where M.X i / placed in degree i.b) W. / is covariant functorial for proper maps.c) W. / is contravariant functorial for open embeddings.d) If T ! X is a closed embedding with open complement U , then there is adistinguished triangleW.T/ i ! W.X/ j ! W.U/:e) If D is a divisor with normal crossings in a scheme X, with irreducible componentsY i , and if Y .r/ D `#I Dr \ i2I Y i , then W.X D/ is represented byM.X/ M.Y .1/ / M.Y .dim X/ /:The argument in loc. cit. only uses that resolution of singularities exists over k, andwe assume from now on that k is such a field.

Parshin’s conjecture revisited 417Given an additive covariant functor F from C to an abelian category, we defineweight (Borel–Moore) homology HiW .X; F / as the ith homology of the complexF.W.X//. Weight homology has the functorialities inherited from b) and c), andsatisfies a localization sequence deduced from d). If K is a finitely generated fieldover k, then we define HiW .K; F / to be colim HiW .U; F /, where the (filtered) limitruns through integral varieties having K as their function field. Similarly, a contravariantfunctor G from C to an abelian category gives rise to weight cohomology (with compactsupport) HW i .X; G/.As a special case, we define the weight homology group HiW .X; Z.n// as thei 2nth homology of the homological complex of abelian groups CH n .W.X//.Lemma 3.1. We have HiW .X; Z.n// D 0 for i>dim X C n. In particular we haveHi W .K; Z.n// D 0 for every finitely generated field K=k and every i>trdeg k K C n.Proof. This follows from the first property of weight complexes together withCH n .T / D 0 for n>dim T .It follows from Lemma 3.1 that the niveau spectral sequenceEs;t 1 .X/ D MHsCt W .k.x/; Z.n// ) H sCt W .X; Z.n// (3)x2X .s/is concentrated on and below the line t D n. LetzH iW .X; Z.n// D EiCn;n 2 .X/be the ith homology of the complexM0x2X .n/H W 2n .k.x/; Z.n// Mx2X .s/H W sCn.k.x/; Z.n// ; (4)where L x2X .s/HsCn W .k.x/; Z.n// is placed in degree sCn. Then we obtain a canonicaland natural map W zH Wi.X; Z.n// ! HiW .X; Z.n//:Consider the canonical map of covariant functors 0 W z n . ; / ! CH n . / on thecategory of smooth projective schemes over k, sending the cycle complex to its highestcohomology. Then by [11, Theorem 5.13, Remark 5.15], the set of associated homologyfunctors extends to a homology theory on the category of all varieties over k. Theargument of [11, Proposition 5.16] show that the extension of the associated homologyfunctors for z n . ; / are higher Chow groups CH n . ;i/. The extension CH n . / areby definition the functors HiW . ; Z.n//. We obtain a functorial map W H ciW.X; Z.n// ! Hi.X; Z.n//:Lemma 3.2. For i D 2n, and for all schemes X, the map is an isomorphismH2n c .X; Z.n// Š H 2n W .X; Z.n//. In particular, H W .K; Z.d// Š Z for all fields K of2dtranscendence degree d over k.

418 T. GeisserProof. The statement is clear for X smooth and projective. We proceed by induction onthe dimension of X. Using the localization sequence for both theories, we can assumethat X is proper. Let f W X 0 ! X be a resolution of singularities of X, Z be the closedsubscheme (of lower dimension) where f is not an isomorphism, and Z 0 D Z X X 0 .Then we conclude by comparing localization sequencesH2n c .Z0 ; Z.n// H2n c .Z; Z.n// ˚ H 2n c .X 0 ; Z.n// H2n c .X; Z.n// 0H2n W .Z0 ; Z.n// HW2n.Z; Z.n// ˚ H2n W .X 0 ; Z.n// HW2n.X; Z.n// 0.The map for fields induces a map ˇ W zHi c.X;Q.n// ! zHiW .X; Q.n//, which fitsinto the (non-commutative) diagramH czH cii .X; Z.n// ˛ˇ.X; Z.n//HiW.X; Z.n// zHiW .X; Z.n//.We now return to the situation k D F q , and compare weight homology to higherChow groups using their niveau spectral sequences. We saw that the niveau spectralsequence for higher Chow groups is concentrated above the line t D n, and that theniveau spectral sequence for weight homology is concentrated below the line t D n.Our aim is to show that Parshin’s conjecture is equivalent to both being rationallyconcentrated on this line, and that the resulting complexes are isomorphic.The following statements will be referred to as Conjecture B.n/:Proposition 3.3. For a fixed integer n, the following statements are equivalent:a) The map ˇ induces an isomorphism zH i .X; Q.n// Š zHiW .X; Q.n// for allschemes X and all i.b) The map induces an isomorphism H c dCn .k; Q.n// Š H W .k; Q.n// for alldCnfinitely generated fields k=F q , where d D trdeg k=F q .c) For every smooth and projective X over F q the following holds: if d D dim X>n C 1, then zH c dCn .X; Q.n// D zH c .X; Q.n// D 0, and if dim X D n C 1,dCn 1then zH2nC1 c .X; Q.n// D 0.Note that assuming the conjecture A.n/, c) is equivalent to H c .X; Q.n// DdCnH c dCn 1 .X; Q.n// D 0 and H 2nC1 c .X; Q.n// D 0 for all smooth and projective X ofdimension d>nC 1 and d D n C 1, respectively, hence are part of Conjecture P.n/.

Parshin’s conjecture revisited 419Proof. b) ) a) is trivial, and a) ) b) follows by a colimit argument because(colim zH i c .X; Q.n// Š H c .k.X/; Q.n//;dCn i D d C nIU X0; otherwise.a) ) c) follows because for X smooth and projective of dimension d, the cohomologyof the complex (2) tensored with Q equals zHi c.X;Q.n// D zHiW .X; Q.n// fori D d C n and i D d C n 1 (or i D 2n C 1 in case d D n C 1). An inspection ofthe niveau spectral sequence (3) shows that this is a subgroup of Hi W .X; Q.n// D 0.c) ) b): For n > d, both sides vanish, whereas for d D n, both sides arecanonically isomorphic to Q. For nnC 1. The upper sequence is exact by hypothesis, and an inspection of (3) showsthat the lower sequence is exact because HiW .X; Q.n// D 0 for i>2n.We refer to the following statements as Conjecture C.n/:Proposition 3.4. For a fixed integer n, the following statements are equivalent:a) For all schemes X over F q and for all i, the map induces an isomorphismzHiW .X; Q.n/ Š HiW .X; Q.n//.b) For all finitely generated fields k=F q and for all i 6D trdeg k=F q C n, we haveHi W .k; Q.n// D 0.c) For all smooth and projective X, the map induces an isomorphismzH iW .X; Q.n// D(0; i > 0ICH n .X/ Q ; i D 2n:Proof. b) ) a) is trivial and a) ) b) follows by a colimit argument.a) ) c) is trivial for i>2n, and Lemma 3.2 for i D 2n.c) ) b): This is proved like Proposition 2.1, by induction on the transcendencedegree of k. Let X be a smooth and projective model of k. The induction hypothesisimplies that the niveau spectral sequence (3) collapses to the horizontal line t D nand the vertical line s D d. Since it converges to HiW .X; Q.n//, which is zero fori>2n, we obtain isomorphisms H W dCniC1 .k.X/; Q.n// d i! zH W .X; Q.n// fordCn i

420 T. Geisser1 0, henceConjecture P.n/ follows. Conversely, Conjecture P.n/ implies Proposition 2.1 c),then 3.3 c), and finally 3.4 c) by using 2.1 a) and 3.3 a).Remark. Propositions 2.1, 3.4, 3.3 as well as Theorem 3.5 remain true if we restrictourselves to schemes of dimension at most N , and to fields of transcendence degree atmost N , for a fixed integer N .Remark. Gillet announced that one can obtain a rational version of weight complexesby using de Jong’s theorem on alterations instead of resolution of singularities. Thesame argument should then give the generalization [11, Theorem 5.13]. In this case,all arguments of this section hold true rationally, except the proof of c) ) b) in Propositions3.3 and 3.4, and the proof of P.n/ ) A.n/; B.n/; C.n/ in Theorem 3.5, whichrequire that every finitely generated field over F q has a smooth and projective model.4 The case n D 0Since H ci .X; Q.0// D CH 0.X; i/ Q , we use higher Chow groups in this section.Proposition 4.1. We have CH 0 .X; i/ Q Š zHi c .X; Q.0// for i 2, and the mapCH 0 .X; 3/ Q ! zH3 c .X; Q.0// is surjective for all X. In particular, A.0/ holds indimensions at most 2.Proof. Since H i .k.x/; Q.1// D 0 for i 6D 1 and H i .k.x/; Q.0// D 0 for i 6D 0, thisfollows from an inspection of the niveau spectral sequence.Jannsen [11] defines a variant of weight homology with coefficients A, HiW .X; A/,as the homology of the complex Hom.CH 0 .W.X//; A/. Note that HiW .X; Q/ DHi W .X; Q.0// because Hom.CH 0 .X/; Q/ Š CH 0 .X/ Q for smooth and projective X,in a functorial way. Indeed, a map of connected, smooth and projective varieties inducesthe identity pull-back on CH 0 and the identity push-forward on CH 0 .

Parshin’s conjecture revisited 421Theorem 4.2 (Jannsen). Under resolution of singularities, HaW .k; A/ D 0 for a 6Dtrdeg k=F q , hence HiW .X; A/ is the homology of the complexM0 H0 W .k.x/; A/ MHs W .k.x/; A/ (5)x2X .0/ x2X .s/for all schemes X. In particular, Conjecture C.0/ holds.This is proved in [11, Proposition 5.4, Theorem 5.10]. The proof only works forn D 0, because it uses the bijectivity of CH 0 .Y / ! CH 0 .X/ for a map of connectedsmooth and projective schemes X ! Y . The second statement follows using the niveauspectral sequence, which exists because HiW .X; A/ satisfies the localization propertyby property (4) of weight complexes.Let Z c .0/ be the complex of etale sheaves z 0 . ; /. For any prime l, considerl-adic cohomologyH i .X et ; yQ l / WD Q ˝Z lim H i .X et ; Z c =l r .0//:In [4], we showed that for every positive integer m, and every scheme f W X ! k overa perfect field, there is a quasi-isomorphism Z c =m.0/ Š Rf Š Z=m. In particular, theabove definition agrees with the usual definition of l-adic homology if l 6D p D char F q .If xX D X FqxF q and yG D Gal.xF q =F q /, then there is a short exact sequence0 ! H iC1 . xX et ; yQ l / O G ! H i.X et ; yQ l / ! H i . xX et ; yQ l / OG ! 0and for U affine and smooth, H i .U et ; yQ l / vanishes for i 6D d;d 1 and l 6D p bythe affine Lefschetz theorem and a weight argument [12, Theorem 3a)]. The map fromZariski-hypercohomology of Z c =m.0/ to etale-hypercohomology of Z c =m.0/ inducesa functorial maphence in the limit a mapCH 0 .X; i/=m ! CH 0 .X; i; Z=m/ ! H i .X et ; Z c =m.0//;! W CH 0 .X; i/ Ql ! H i .X et ; yQ l /:Similarly, the map x! W CH 0 . xX;i/ Ql! H i . xX et ; yQ l / induces a map W CH 0 .X; i C 1/ Ql.CH0 . xX;i C 1/ Ql / O G ! H iC1. xX et ; yQ l / O G ! H i.X et ; yQ l /(the left map is an isomorphism by a trace argument). The sum' i X W CH 0.X; i/ Ql ˚ CH 0 .X; i C 1/ Ql ! H i .X et ; yQ l / (6)is compatible with localization sequences, because all maps involved in the definitionare. The following proposition shows that Parshin’s conjecture can be recovered fromand implies a structure theorem for higher Chow groups of smooth affine schemes;compare to Jannsen [11, Conjecture 12.4b)].

422 T. GeisserProposition 4.3. The following statements are equivalent:a) Conjecture P.0/.b) For all schemes X, and all i, the map 'X i is an isomorphism.c) For all smooth and affine schemes U of dimension d, the groups CH 0 .U; i/ aretorsion for i 6D d, and the composition ! W CH 0 .U; d/ Ql ! H d .U et ; yQ l / !H d . xU et ; yQ l / OG is an isomorphism.Proof. a) ) b): First consider the case that X is smooth and proper. Then ConjectureP(0) is equivalent to the vanishing of the left-hand side of (6) for i 6D 0; 1, whereas theright-hand side of (6) vanishes by the Weil-conjectures. On the other hand, 'X 0 inducesan isomorphism CH 0 .X/ ˝ Q l Š H 2d .X et ; yQ l .d// and 'X1 induces an isomorphismCH 0 .X/ ˝ Q l Š .CH 0 . xX/ ˝ Q l /G O Š H 2d . xX; yQ l .d//G O . Indeed, both sides areisomorphic to Q l if X is connected. Using localization and alterations, the statementfor smooth and proper X implies the statement for all X.b) ) a): The right-hand side of (6) is zero for i 6D 0; 1 by weight reasons forsmooth and projective X, hence so is the left side.b) ) c): This follows because H i .U et ; yQ l / D 0 unless i D d;d 1 for smoothand affine U .c) ) b): We first assume that X is smooth and affine. By hypothesis and the affineLefschetz theorem, both sides of (6) vanish for i 6D d;d 1, and are isomorphic fori D d. Fori D d 1, the vertical maps in the following diagram are isomorphismsby semi-simplicity,CH 0 .X; d/ Ql Š CH 0 . xX;d/ O GQl Hd . xX; yQ l / O G.CH 0 . xX;d/ Ql / O G Hd . xX; yQ l / O G Hd 1 .X; yQ l /.Hence the lower map 'Xd 1 is an isomorphism because the upper map is. Using localization,the statement for smooth and affine X implies the statement for all X.Proposition 4.4. Under resolution of singularities, the following are equivalent:a) Conjecture P.0/.b) For every smooth affine U of dimension d over F q , we have CH 0 .U; i/ Q ŠHi W .U; Q/ for all i, and these group vanish for i 6D d.c) For every smooth affine U of dimension d over F q , the groups CH 0 .U; i/ Q vanishfor i>d, and CH 0 .U; d/ Q Š H W .U; Q/.dProof. a) ) b): It follows from the previous proposition that CH 0 .U; i/ Q D 0 fori 6D d. On the other hand, P.0/ for all X implies that CH 0 .X; i/ Š HiW .X; Q/ forall i and X.c) ) a): The statement implies Conjecture A.0/, and then Conjecture B.0/, versionb), for all X. By Theorems 3.5 and 4.2, P.0/ follows.

Parshin’s conjecture revisited 423Proposition 4.5. Conjecture P.0/ for all smooth and projective X implies the followingstatements:a) (Affine Gersten) For every smooth affine U of dimension d, the following sequenceis exact:CH 0 .U; d/ Q ,! MH d .k.x/; Q.d//! MH d 1 .k.x/; Q.d 1//! :x2U .0/ x2U .1/b) Let X D X d X d 1 X 1 X 0 be a filtration such that U i D X i X i 1is smooth and affine of dimension i. Then CH 0 .X; i/ Q is isomorphic to the ithhomology of the complex0 ! CH 0 .U d ;d/ Q ! CH 0 .U d 1 ;d 1/ Q !!CH 0 .U 0 ;0/ Q ! 0:The maps CH 0 .U i ;i/ Q ! CH 0 .X i 1 ;i 1/ Q ! CH 0 .U i 1 ;i 1/ Q arisefrom the localization sequence.Proof. a) follows because the spectral sequence (1) collapses, and b) by a diagramchase.Remark. If we fix a smooth scheme X of dimension d, and use cohomological notation,then by Proposition 2.1 and the Gersten resolution, we get that the rational motiviccomplex Q.d/ is conjecturally concentrated in degree d, say C d D H d .Q.d// DCH 0 . ;d/ Q D H W d. ; Q/. Then Conjecture L.0/ says that H i .U; C d / D 0 forU X affine and i>0. This is analog to the mod p situation, where the motiviccomplex agrees with the logarithmic de Rham–Witt sheaf Z=p.n/ Š d Œ d, andH i .U et ; d / D 0 for U X affine and i>0. The latter can be proved by writing d as the kernel of a map of coherent sheaves and using the vanishing of cohomologyof coherent sheaves on affine schemes. This suggest that one might try to do the samefor C d .4.1 Frobenius action. Let F W X ! X be the Frobenius morphism induced by theqth power map on the structure sheaf.Theorem 4.6. The push-forward F acts like q n on CH n .X; i/, and the pull-back F acts on H i .X; Z.n// as q n for all n.The theorem is well known, but we could not find a proof in the literature. Theproof of Soulé [14, Proposition 2] for Chow groups does not carry over to higherChow groups, because the Frobenius does not act on the simplices n , hence a cycleZ n X is not send to a multiple of itself by the Frobenius. We give an argumentdue to M. Levine.Proof. Let DM be Voevodsky’s derived category of bounded above complexes ofNisnevich sheaves with transfers with homotopy invariant cohomology sheaves. Then

424 T. Geisserwe have the isomorphismsCH n .X; i/ Š Hom DMH i .X; Z.n// Š Hom DM.Z.n/Œ2n C i;M c .X//;.M.X/; Z.n/Œi/:The action of the Frobenius is given by composition with F W M c .X/ ! M c .X/ andF W M.X/ ! M.X/, respectively.The Frobenius acts on the category DM , i.e. for every ˛ 2 Hom DM .X; Y / wehave F Y ı ˛ D ˛ ı F X . This follows by considering composition of correspondences.Hence it suffices to calculate the action of the Frobenius on Z.n/, i.e. show thatF D q n 2 Hom DM .Z.n/; Z.n// Š Z. But Hom DM .Z.n/; Z.n// is a directfactor of Hom DM .Z.n/; P n Œ 2n/ D CH n .P n /. The latter is the free abelian groupgenerated by the generic point, and the Frobenius acts by q n on it.Remark. 1) It would be interesting to write down an explicit chain homotopy betweenF and q n on z n .X; /.2) The proposition implies that the groups CH n .F q ;i/are killed by q n 1, and thatCH n .X; i/ is q-divisible for n

Parshin’s conjecture revisited 425[6] T. Geisser, M. Levine, The p-part of K-theory of fields in characteristic p, Invent. Math.139 (2000), 459–494.[7] H. Gillet, Homological descent for the K-theory of coherent sheaves, in Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), Lecture Notes in Math.1046, Springer-Verlag, Berlin 1984, 80–103.[8] H. Gillet, C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996),127–176.[9] U. Jannsen, Mixed motives and algebraic K-theory, Lecture Notes in Math. 1400, Springer-Verlag, Berlin 1990.[10] U. Jannsen, On finite-dimensional motives and Murre’s conjecture, in Algebraic cycles andmotives, Vol. 2, London Math. Soc. Lecture Note Ser. 344, Cambridge University Press,Cambridge 2007, 112–142. finite dimensional motives and Murre’s conjecture, 2006.[11] U. Jannsen, Hasse principles for higher-dimensional fields, Preprint Universität Regensburg18/2004.[12] B. Kahn, Some finiteness results for etale cohomology, J. Number Theory 99 (2003), 57–73.[13] S. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005),173–201.[14] C. Soulé, Groupes de Chow et K-théorie de variétés sur un corps fini, Math. Ann. 268(1984), 317–345.[15] V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in anycharacteristic, Internat. Math. Res. Not. 2002 (2002), 351–355.

Axioms for the norm residue isomorphismCharles Weibel IntroductionThe purpose of this paper is to give an axiomatic framework for proving (by inductionon n) that the norm residue map Kn M .k/=` ! H ét n .k; ˝n/ is an isomorphism, for any`prime `>2and for any field k such that ` is invertible in k.Fix ` and n. ByaRost variety for a sequence a D .a 1 ;:::;a n / of units in k, weshall mean a smooth projective variety X of dimension d D `n 1 1 satisfying theconditions of [8, 6.3] or [6, (0.1)]; the exact definition is given in Definition 1.1 below.One key requirement is that fa 1 ;:::;a n g vanishes in Kn M .k.X//=`. Rost constructedsuch a variety in his 1998 preprint [3]; the proof that it satisfies these properties waspublished in [1] and [6].Theorem 0.1. Let n and ` be such that the norm residues maps are isomorphisms forall i

428 C. WeibelGiven (0.4) and axiom 0.3 (b), triangle (0.5) is equivalent to the duality assertionthat D ˝ L d b Š D; (0.5) is isomorphic to L d times the dual of triangle (0.4).Remark 0.6. In the language of Chow motives, any direct summand M D .X; e/ ofX has a dual motive M D .X; e t ;d/and a transpose summand M 0 D .X; e t / of X.Thus the summand M 0 D M ˝ L d of X is a priori different from M . Axiom 0.3 (b)says that M 0 ! X ! M is an isomorphism. (The differences between these languagesis partly due to the fact that the category of Chow motives embeds contravariantly intoDM eff .)Although we only use X-duals fleetingly in Lemma 3.5 below, they fit into a largercontext, which we shall now quickly describe. For this, we need to assume that kadmits resolution of singularities, so that A exists by [2, 20.3].DM X and X-duals 0.7. Let DM X denote the full subcategory of DM consisting ofobjects M isomorphic to A ˝ X for a geometric motive A over k. For this, we assumethat k admits resolution of singularities, so that A exists by [2, 20.3]. As pointed outin [11, 6.11], Hom.M; B/ Š Hom.M; B ˝ X/ for all B in DM eff . ThusHom.U ˝X;A ˝X/ D Hom.U ˝X;A / D Hom.U ˝X˝A; Z/ D Hom.U ˝M; X/:That is, Hom X .M; X/ D A ˝ X is an internal Hom object from M to X in the tensorcategory DM X . As such, Hom X .M; X/ is unique up to canonical isomorphism (cf. [11,8.1]). Any map M 1 ! M 2 induces a map Hom X .M 2 ; X/ ! Hom X .M 1 ; X/ viaHom.A 1 ˝ X;A 2 ˝ X/ Š Hom.A 1 ˝ X ˝ A 2 ; X/ Š Hom.A 2 ˝ X;A 1 ˝ X/:It is easy to check that Hom X . ; X/ is a contravariant functor, and that it is a dualityin the sense that Hom X .Hom X .M; X/; X/ Š M .In this paper we consider the X-dual DM. / D Hom X . ; X/ ˝ L d , which alsosatisfies D 2 .M / Š M . Our choice of the twist is such that D.X/ Š X, D.M / is thetranspose M 0 of Remark 0.6, and Dy is given by (0.2).This paper is an attempt to clarify the ending of the Voevodsky–Rost programto prove that the norm residue map is an isomorphism, and specifically the proof aspresented in Voevodsky’s 2003 preprint [8]. One important feature of our Theorem 0.1is that only published results (and [3]) are used.When ` D 2, triangle (0.4) was constructed in [10, 4.4] using D D X, and (0.5) isits X-dual. Axioms 0.3 (a)–(b) hold by a result of Rost (cited as [10, 4.3]).When `>2, there are two different programs for constructing M . Both start byusing the norm residue symbol of a to construct a nonzero element of H 2bC1;b .X/.The construction of is given in [10, p. 95] and [8, (5.2)] (see 2.5 below).In Voevodsky’s approach, the triangles (0.4) and (0.5) are constructed in (5.5) and(5.6) of [8] with M D M` 1 and D D M` 2 , starting with 2 H 2bC1;b .X/. Theduality axiom 0.3 (b) for M and D is given by [8, 5.7].Unfortunately, there is a gap in the proof of Lemma [8, 5.15], which asserts thatAxiom 0.3 (a) holds, i. e., that M` 1 is a summand of X. That proof depends via

Axioms for the norm residue isomorphism 429Theorems 3.8 and 4.4 of [8] upon Lemmas 2.2 and 2.3 of [8], whose proofs are currentlyunavailable. Instead, [8, 5.11] proves that the canonical map X ! X factors throughthe map y W M ! X. Since this paper was written, the gap in the proof was patchedin [12].In Rost’s approach, defines an equivalence class of idempotent endomorphismse of X and Rost defines M to be the Chow motive .X; e/. By construction, his Msatisfies Axioms 0.3 (a)–(b). Hopefully it also satisfies Axiom 0.3 (c), the existence oftriangles (0.4) and (0.5).Notation. The integer n and the prime `>2will be fixed. We will work over a fixedfield k in which ` is invertible. The integer d will always be `n 1 1 and b will alwaysbe d=.` 1/ D 1 CC`n 2 .We fix the sequence of units a D .a 1 ;:::;a n /, and X will always denote ad-dimensional Rost variety relative to a, satisfying Axioms 0.3.We will work in the triangulated category of motives DM eff described in [2]. TheLefschetz motive is L D Z.1/Œ2. Unless explicitly stated otherwise, motivic cohomologywill always be taken with coefficients Z .`/ . The notation H p;qét. / refers tothe étale motivic cohomology H p ét . ; Z .`/.q// defined in [2, 10.1].Finally, I would like to thank the Institute for Advanced Study for allowing me topresent these results in a seminar during the Fall of 2006, and Pierre Deligne for hiscontinued interest and encouragement.1 Proof of Theorem 0.1Recall [6, 1.20] that a n 1 -variety over a field k is a smooth projective variety Xof dimension d D `n 1 1, with deg s d .X/ 6 0.mod `2/. Here s d .X/ is thePcharacteristic class of the tangent bundle T X corresponding to the symmetric polynomialtdjin the Chern roots t j of T X ; see [9, 14.3]. We also recall that H 1; 1 .Y / is thegroup Hom.Z; Y.1/Œ1/, and is covariant in Y ; see [2, 14.17].Definition 1.1. A Rost variety for a sequence a D .a 1 ;:::;a n / of units in k is a n 1 -variety such that: fa 1 ;:::;a n g vanishes in Kn M .k.X//=`; for each i < n there is a i -variety mapping to X; and the motivic homology sequenceH 1; 1 .X 2 / 0 1! H 1; 1 .X/ ! H 1; 1 .k/ .D k / (1.2)is exact. As mentioned in the Introduction, Rost varieties exist for every a by [1], [6].Example 1.2.1. If n D 1 and L D k. `pa1 /, then Spec.L/ is a Rost variety for a 1 ,with the convention that s 0 .X/ D ŒL W k D `. When n D 2, the Severi–Brauer varietycorresponding to the degree ` algebra with symbol a is a Rost variety.Proof of Theorem 0.1. By [10, 6.10], it suffices to prove the assertionH 90.n; `/:H nC1;nét.k/ D 0:

430 C. WeibelAs pointed out in the proof of [10, 7.4], it suffices to show that for every symbola 2 Kn M .k/ there is a field extension k F such that a vanishes in KM n .F /=` andH nC1;nét.k/ ! H nC1;nét.F / is an injection. When F D k.X/, this is the bottom rightarrow in the flowchart (1.3), which is implicit in [8, 6.11].H 2C2b`;1Cb`.X/ ontoH 2dC1;dC1 .D/ D 01:43.6into 2:4(1.3)H nC1;n .X/4:4exactH nC1;nét.k/ sequence HnC1;nét.k.X//:The other decorations in (1.3) refer to the properties we need and where they areestablished. Proposition 4.4 states that the bottom row of the flowchart (1.3) is exact.By Lemma 2.4 the group H nC1;n .X/ injects (by a cohomology operation) intoH 2C2b`;1Cb`.X/, which is in turn a quotient of H 2dC1;dC1 .D/ by 1.4. In Corollary3.6, we prove that H 2dC1;dC1 .D/ D 0. All of this implies that H nC1;nét.k/ !H nC1;nét.k.X// is an injection, finishing the proof of Theorem 0.1.One part of the flowchart (1.3) is easy to establish.Lemma 1.4. The map s W X ! D ˝ L b Œ1 in (0.4) induces a surjection:H 2dC1;dC1 .D/ Š H 2b`C2;b`C1 .D ˝ L b Œ1/s! H 2b`C2;b`C1 .X/:Proof. In the cohomology exact sequence arising from (0.4), the next term isH 2b`C2;b`C1 .M /, which is a summand of H 2b`C2;b`C1 .X/. Because b` D d C b>d,this group is zero by the Vanishing Theorem [2, 3.6] – which says that H n;i .X/ D 0whenever n>iC dim.X/.The lower left map in flowchart (1.3) is just the motivic-to-étale map by the followingbasic result, which we quote from [10, 7.3].Lemma 1.5. The structure map X ! Spec.k/ induces isomorphismsH ;ét .k/ Š H ;ét .X/ and H ;ét .kI Z=`/ Š H ;ét .XI Z=`/:2 Motivic cohomology operationsIn the course of the proof, we will need some facts about the motivic cohomologyoperations constructed in [9]. When `>2, there are operations P i on H ; . I Z=`/of bidegree .2i.` 1/; i.` 1//, for each i 0 (with P 0 D 1), as well as the Bocksteinoperation ˇ of bidegree .1; 0/. These satisfy the Adem relations given in [5] for theusual topological cohomology operations. In fact, the subring of all (stable) motivic

Axioms for the norm residue isomorphism 431cohomology operations generated by the P i and ˇ is isomorphic to the topologists’Steenrod algebra A , developed in [5].In addition, the motivic operations satisfy the usual Cartan formula P n .xy/ DP P i .x/P n i .y/, P i .x/ D x` if x 2 H 2i;i .Y I Z=`/, and P i D 0 on H p;q .Y I Z=`/when .p; q/ is in the region q i, p 2, the dual to the usual Steenrod algebra A is a gradedcommutativealgebra on generators i in (even) degrees .2`i 2; `i 1/ and i in(odd) degrees .2`i 1; `i 1/; see [9, 12.6]. The dual to i is the motivic cohomologyoperation Q i , which has bidegree .2`i 1; `i 1/. Because it is true in A , the Q iare derivations which form an exterior subalgebra of all (stable) motivic cohomologyoperations. They may be inductively defined by Q 0 D ˇ and Q iC1 D ŒP `i;Q i .We now quickly establish those portions of [8] that we need, concerning theseoperations on the cohomology of X and the unreduced simplicial suspension †X ofX (the cofiber of X ! cone.X/). The proofs we give are due to Voevodsky, and onlydepend upon [10] and [9]. In particular, they do not depend upon the missing lemmasin op. cit., or upon the Axioms 0.3.Fix q

432 C. WeibelConsider the cohomology operation Q D Q n 1 :::Q 2 Q 1 .Lemma 2.4. The operation Q W H nC1;n .XI Z=`/ ! H 2C2b`;1Cb`.XI Z=`/ is aninjection, and induces an injectionH nC1;n .X/ ,! H 2C2b`;1Cb`.X/:Proof (Voevodsky). By the above remarks, it suffices to show that the operation Qfrom H nC2;n .†XI Z=`/ to H 3C2b`;1Cb`.†XI Z=`/ is injective. As illustrated in Example2.2.2, it is easy to see from 2.2 and (2.1) that each Q i is injective on thegroup H ; .†XI Z=`/ containing Q i 1 :::Q 1 H nC2;n .†XI Z=`/, because the precedingterm in 2.2 is zero.Remark 2.5. The same argument, given in [8, 6.7], shows that Q 0 D Q n 2 :::Q 0 isan injection from H n;n 1 .XI Z=`/ to H 2bC1;b .X/ H 2bC1;b .XI Z=`/.If fa 1 ;:::;a n g¤0 in Kn M .k/=`, Voevodsky shows in [8, 6.5] that its norm residuesymbol in Hét n .k; ˝n/ lifts to a nonzero element ı 2 H n;n 1 .XI Z=`/. Using injectivityof Q 0 , we get a nonzero symbol D Q 0 .ı/ 2 H 2bC1;b .X/. This symbol is the`starting point of the construction of the motive M in both the program of Voevodsky[8] and that of Rost [4].3 Motivic homologyIn this section, we prove Corollary 3.6, which depends uponAxiom 0.3 (b) and exactnessof (1.2) via results about the motivic homology of X and M .We will make repeated use of the following basic lemma. In this section, the notationH p; q .Y / refers to the group Hom.Z; Y.q/Œp/; see [2, 14.17].Lemma 3.1. For every smooth (simplicial) Y and p>q, Hom.Z; Y.q/Œp/ D 0.Proof. We have Hom.Z; Y.q/Œp/ Š Hzar p q .k; C .Y Gm// q D H p q C .Y Gm/.k/;qsee [2, 14.16]. The chain complex C .Y Gm/ q is zero in positive cohomological degrees,so the H p q group vanishes.Lemma 3.2. The structural map H 1; 1 .X/ ! H 1; 1 .k/ D k is injective.Proof (Voevodsky). By Lemma 3.1, Hom.Z;X p .1/Œn/ D 0 for all n 2 and all p.Therefore the right half-plane homological spectral sequenceE 1 p;q D Hom.Z;XpC1 .1/Œ q/ ) Hom.Z; X.1/Œpq/has no nonzero rows below q D 1, and the row q D 1 yields the exact sequence0 H 1; 1 .X/ H 1; 1 .X/ H 1; 1 .X X/:Since (1.2) is exact, this implies the result.

Axioms for the norm residue isomorphism 433Corollary 3.3. The structural map H 1; 1 .M / ! H 1; 1 .k/ D k is injective.Proof. By (0.4) and Lemma 3.2, it suffices to show that Hom.Z;D.bC 1/Œ2b C 1/ D 0.By (0.5) this results from the vanishing of both Hom.Z;M.b C 1/Œ2b C 1/ andHom.Z; X ˝ L bCdC1 / which follows from Lemma 3.1.Lemma 3.4. H 0;1 .X/ D H 2;1 .X/ D 0 and H 1;1 .XI Z/ Š H 1;1 .Spec kI Z/ Š k .Proof. The spectral sequence E p;q1D H q .X pC1 I Z.1// ) H pCq;1 .XI Z/ degenerates,all rows vanishing except for q D 1 and q D 2, because Z.1/ Š O Œ 1; see[2, 4.2]. We compare this with the spectral sequence converging to H pCq .XI G m /;Hzar q .Y; O / ! H q ét . ; G m/ is an isomorphism for q D 0; 1 (and an injection forq D 2). Hence for q 2 we have H q;1 .X/ D H q;1ét.X/ D H q;1ét.k/ D H q 1ét.k; G m /.Remark. The proof of 3.4 also shows that H 3;1 .X/ injects into H 2 ét .k; G m/ D Br.k/.Lemma 3.5. H 2dC1;dC1 .DI Z/ is the kernel of H 1; 1 .M /y! H 1; 1 .k/ D k .Proof. From (0.5) we get an exact sequence with coefficients Z:H 2d;dC1 .X˝L d / ! H 2dC1;dC1 .D/ ! H 2dC1;dC1 .M / Dy ! H 2dC1;dC1 .X˝L d /:The first group is H 0;1 .X/, which is zero by 3.4, so it suffices to show that the map Dyidentifies with the structural map y. This follows from Axiom 0.3 (b), because for anyu in H 1; 1 .M / D Hom.Z; M.1/Œ1/, X tensored with the compositeX ˝ L d Dy ! M ˝ L d u ! Z.1/Œ1 ˝ L d D Z.d C 1/Œ2d C 1is the X-dual of X. 1/Œ 1u! M y ! X.Corollary 3.6. H 2dC1;dC1 .D/ D H 2dC1;dC1 .DI Z/ ˝ Z .`/ D 0.4 Between motivic and étale cohomologyDefinition 4.1 ([10, p. 90]). Let L.n/ denote the truncation nC1 Z ét .n/ of the complexin DM eff representing étale motivic cohomology; i. e., H p . ; L.n// Š H p;nét.`/. /for p n C 1. Let K.n/ denote the mapping cone of the canonical map Z .`/ .n/ !L.n/, and consider the triangle Z .`/ .n/ ! L.n/ ! K.n/ ! Z .`/ .n/Œ1.Lemma 4.2. The map H nC1 .X; K.n//y! H nC1 .M; K.n// is an injection.Proof. By triangle (0.4) we have an exact sequence:H n .D ˝ L b ; K.n// ! H nC1 .X; K.n//y! H nC1 .M; K.n//:

434 C. WeibelWe need to see that the left term vanishes. By (0.5), it is sufficient to show thatH n .M ˝ L b ; K.n// and H n 1 .X ˝ L b`; K.n// vanish. This follows from [10, 6.12],which says that H .Y.1/; K.n// D 0 for every smooth Y , the assumption that Mis a summand of X, and the consequent collapsing of the spectral sequence E p;q1DH q .X p ; K.n// ) H pCq .X; K.n//.Corollary 4.3. The map H nC1 .X; K.n// ! H nC1 .k.X/; K.n// is an injection.Proof. The map H nC1 .X; K.n// ! H nC1 .k.X/; K.n// is an injection by [2, 11.1,13.8, 13.10], or by [10, 7.4]. The corollary follows from Lemma 4.2, since H .X; / !H .M; / is a summand of H .X; / ! H .X; /.Proposition 4.4. There is an exact sequenceH nC1;n .X/ ! H nC1;nét.k/ ! H nC1;n .k.X//:Proof. By 1.5, X ! Spec.k/ is an isomorphism on étale motivic cohomology, so wehave H nC1;nét.k/ Š H nC1 .X; L.n//.The map Spec k.X/ ! X induces a commutative diagram with exact rows:étH nC1;n .X/H nC1 .X; L.n//H nC1 .X; K.n//4:3 into0=H nC1;n .k.X// H nC1 .k.X/; L.n// H nC1 .k.X/; K.n//:By 4.3, the right vertical map is an injection. The proposition now follows by a diagramchase.Acknowledgements. This paper is an attempt to clarify Voevodsky’s preprint [8]. It isdirectly inspired by the work of Markus Rost, especially [4], who explicitly suggestedthe line of attack, based upon Axiom 0.3 (c), which is presented in this paper. I amgreatly indebted to both of them.References[1] C. Haesemeyer and C. Weibel, Norm Varieties and the Chain Lemma (after Markus Rost),Preprint 2008, available at http://www.math.uiuc.edu/0900.[2] C. Mazza,V.Voevodsky and C. Weibel, Lecture notes on motivic cohomology, Clay Monogr.Math. 2, Amer. Math. Soc. , Providence, R.I., 2006.[3] M. Rost, Chain lemma for splitting fields of symbols, Preprint 1998, available athttp://www.math.uni-bielefeld.de/~rost/chain-lemma.html[4] M. Rost, On the Basic Correspondence of a Splitting Variety, Preprint 2006, available athttp://www.math.uni-bielefeld.de/~rost[5] N. Steenrod and D. Epstein, Cohomology Operations, Ann. of Math. Stud. 50, PrincetonUniversity Press, Princeton, N.J., 1962.

Axioms for the norm residue isomorphism 435[6] A. Suslin and S. Joukhovitski, Norm Varieties, J. Pure Appl. Alg. 206 (2006), 245–276.[7] A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finitecoefficients, in The arithmetic and geometry of algebraic cycles (Banff, 1998), NATO Sci.Ser. C Math. Phys. Sci. 548, Kluwer, Dordrecht 2000, 117–189 .[8] V. Voevodsky, On Motivic Cohomology with Z/l coefficients, Preprint 2003, available athttp://www.math.uiuc. edu/K-theory/0639/.[9] V. Voevodsky, Reduced Power operations in Motivic Cohomology, Inst. Hautes Études Sci.Publ. Math. 98 (2003), 1–57.[10] V. Voevodsky, Motivic Cohomology with Z=2 coefficients, Inst. Hautes Études Sci. Publ.Math. 98 (2003), 59–104.[11] V. Voevodsky, Motives over simplicial schemes, Preprint, available at http://www.math.uiuc. edu/K-theory/0638/, 2003.[12] C. Weibel, The Norm Residue Isomorphism Theorem, Preprint 2007, available athttp://www.math.uiuc.edu/0844.

List of contributorsArthur Bartels, Mathematisches Institut, Westfälische Wilhelms-Universität Münster,Einsteinstraße 62, 48149 Münster, Germanybartelsa@math.uni-muenster.dePaul Bressler, Institute forAdvanced Study, Einstein Drive, Princeton, NJ 08540, U.S.A.bressler@math.ias.eduUlrich Bunke, NWF I – Mathematik, Universität Regensburg, 93040 Regensburg,Germanyulrich.bunke@mathematik.uni-regensburg.dePaulo Carrillo Rouse, Projet d’algèbres d’opérateurs, Université de Paris 7, 175, rue deChevaleret, 75013 Paris, Francecarrillo@math.jussieu.frJoachim Cuntz, Mathematisches Institut, Westfälische Wilhelms-Universität Münster,Einsteinstraße 62, 48149 Münster, Germanycuntz@math.uni-muenster.deSiegfried Echterhoff, Mathematisches Institut,WestfälischeWilhelms-Universität Münster,Einsteinstraße 62, 48149 Münster, Germanyechters@math.uni-muenster.deGrigory Garkusha, Department of Mathematics, University of Wales Swansea, SingletonPark, Swansea SA2 8PP, UKG.Garkusha@swansea.ac.ukThomas Geisser, Department of Mathematics, University of Southern California, 3620S Vermont Av. KAP 108, Los Angeles, CA 90089, U.S.A.geisser@usc.eduAlexander Gorokhovsky, Department of Mathematics, University of Colorado, UCB395, Boulder, CO 80309-0395, U.S.A.Alexander.Gorokhovsky@colorado.eduMax Karoubi, Université Paris 7, 2 place Jussieu, 75251 Paris, Francemax.karoubi@gmail.comWolfgang Lück, Mathematisches Institut, Westfälische Wilhelms-Universität Münster,Einsteinstraße 62, 48149 Münster, Germanylueck@math.uni-muenster.deRalf Meyer, Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße3–5, 37073 Göttingen, Germanyrameyer@uni-math.gwdg.de

438 List of contributorsFernando Muro, Universitat de Barcelona, Departament d’Àlgebra i Geometria, GranVia de les Corts Catalanes 585, 08007 Barcelona, Spainfmuro@ub.eduRyszard Nest, Department of Mathematics, Copenhagen University, Universitetsparken5, 2100 Copenhagen, Denmarkrnest@math.ku.dkMike Prest, School of Mathematics, University of Manchester, Oxford Road, ManchesterM13 9PL, UKmprest@manchester.ac.ukThomas Schick, Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße3–5, 37073 Göttingen, Germanythomas.schick@uni-math.gwdg.deMarkus Spitzweck, Mathematisches Institut, Georg-August-Universität Göttingen,Bunsenstraße 3–5, 37073 Göttingen, Germanyspitz@uni-math.gwdg.deAndreas Thom, Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße3–5, 37073 Göttingen, Germanythom@uni-math.gwdg.deAndrew Tonks, London Metropolitan University, 166–220 Holloway Road, London N78DB, UKa.tonks@londonmet.ac.ukBoris Tsygan, Department of Mathematics, Northwestern University, Evanston, IL60208-2730, U.S.A.tsygan@math.northwestern.eduChristian Voigt, Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken5, 2100 Copenhagen, Denmark; Mathematisches Institut, WestfälischeWilhelms-Universität Münster, Einsteinstraße 62, 48149 Münster, Germanycvoigt@math.uni-muenster.deCharles Weibel, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540;Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway,NJ 08854, U.S.A.weibel@math.rutgers.eduWend Werner, Fachbereich Mathematik und Informatik, Westfälische Wilhelms-UniversitätMünster, Einsteinstraße 62, 48149 Münster, Germanywwerner@math.uni-muenster.de

List of participantsNatalia Abad Santos, Universidad Nacional del Sur, Bahía BlancaAlexander Alldridge, Universität PaderbornMarian Anton, University of KentuckyPaul Balmer, ETH, ZürichGrzegroz Banaszak, Posnan UniversityMiquel Brustenga, Universtitat Autónoma de BarcelonaUlrich Bunke, Universität GöttingenRubén Burga, Instituto de Matemática y Ciencias Afines, LimaLeandro Cagliero, Universidad Nacional de CórdobaPaulo Carrillo Rouse, Institut JussieuGuillermo Cortiñas, Universidad de Valladolid & Universidad de Buenos AiresJoachim Cuntz, Universität MünsterEugenia Ellis, Universidad de ValladolidMatthias Fuchs, Facultat des Sciences de LuminyImmaculada Gálvez, London Metropolitan UniversityGrigory Garkusha, University of ManchesterThomas Geisser, University of Southern CaliforniaStefan Gille, ETH, ZürichVictor Ginzburg, University of ChicagoAlexander Gorokhovsky, University of Colorado, BoulderJoachim Grünewald, Universität MünsterJuan José Guccione, Universidad de Buenos AiresDaniel Hernández Ruipérez, Universidad de SalamancaJohn Hunton, University of LeicesterTroels Johansen, Universität PaderbornBernhard Keller, Institut JussieuJoachim Kock, Universtitat Autónoma de BarcelonaMaxim Kontsevich, Institut de Hautes Études ScientifiquesMarc Levine, Northeastern University, ChicagoCatherine Llewellyn-Jones, University of LeicesterAna Cristina López Martín, Universidad de SalamancaWolfgang Lück, Universität MünsterRalf Meyer, Universität GöttingenFernando Muro, Max Planck Institut, Bonn

440 List of participantsAmnon Neeman, Canberra UniversityRyszard Nest, University of CopenhagenPere Pascual, Universitat Politécnica de CatalunyaLlorenç Rubió, Universitat Politécnica de CatalunyaSelene Sánchez Florez, Université de Montpellier 2Darío Sánchez Gómez, Universidad de SalamancaFernando Sancho de Salas, Universidad de SalamancaPaolo Stellari, Università degli Studi di MilanoPatrick Szuta, University of Illinois at Urbana-ChampaignGonçalo Tabuada, Institut JussieuAndreas Thom, Universität GöttingenAndrew Tonks, London Metropolitan UniversityBoris Tsygan, Northwestern UniversityChristian Voigt, Universität MünsterWend Werner, Universität MünsterMariusz Wodzicki, California State University, Berkeley