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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.
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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 97s
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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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