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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/:
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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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