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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/:
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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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42 A. Bartels, S. Echterhoff, and W
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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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Equivariant cyclic homology for qua
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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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228 U. Bunke, T. Schick, M. Spitzwe
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Deformations of gerbes on smooth ma
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Torsion classes of finite type and
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Parshin’s conjecture revisitedTho
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Parshin’s conjecture revisited 41
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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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