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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.
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EMS Series of Congress ReportsEMS S
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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82 H. Emerson and R. MeyerAs a resu
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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On K 1 of a Waldhausen category 93t
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On K 1 of a Waldhausen category 95(
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On K 1 of a Waldhausen category 992
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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142 M. KaroubiWe would like to poin
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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148 M. Karoubi[6] M. F. Atiyah, R.
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174 C. Voigtand using ˇ.x; y/ D .S
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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A Schwartz type algebra for the tan
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202 J. CuntzWe denote by N the set
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204 J. CuntzConsider the action of
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206 J. Cuntzprime numbers p and q,
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208 J. Cuntz6 Representations as cr
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210 J. CuntzProof. The spectral pro
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212 J. CuntzThe operator s 1 plays
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214 J. CuntzProposition 7.4. The K-
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On a class of Hilbert C*-manifoldsW
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On a class of Hilbert C*-manifolds
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428 C. WeibelGiven (0.4) and axiom
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430 C. WeibelAs pointed out in the
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432 C. WeibelConsider the cohomolog
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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