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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):
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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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310 U. Bunke, T. Schick, M. Spitzwe
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Parshin’s conjecture revisitedTho
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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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