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Duality for topological abelian group stacks and T -duality 2573.3.12 Note that a site S which has finite products satisfies the assumption of Proposition3.12. In fact, for A 2 S we have by Lemma 3.6 thatA jY Š A Y ! Y :Therefore the restriction functor .:::/ jY preserves representables.Corollary 3.14. Let S be as in Proposition 3.12. For sheaves F;G 2 Sh Ab S and Y 2 Swe haveExt i Sh Ab S .F; G/ jY Š Ext i ShAb S=Y .F jY ;G jY /:Proof. Since .:::/ jY is exact (Lemma 3.8) we haveExt i Sh Ab S .F; G/ jY Š .H i RHom ShAb S .F; G// jYŠ H i .RHom ShAb S .F; G/ jY /Š H i .RHom ShAb S=Y .F jY ;G jY // (by Proposition 3.12)Š Ext i ShAbS=Y .F jY ;G jY /:3.3.13 Let us assume that S has finite fibre products. Fix U 2 S. Then we can definea morphism of sites U W S ! S=U which maps A 2 S to .U A ! U/2 S=U. Oneeasily checks the conditions given in [Tam94, 1.2.2]. This morphism of sites inducesan adjoint pair of functorsU W ShS , ShS=U W U :Let f W S=U ! S be the restriction defined in 3.3.2 (with Y replaced by U ) and letf W ShS=U , ShS W f be the corresponding pair of adjoint functors.Lemma 3.15. We assume that S has finite products. Then we have a canonical isomorphismf Š U .Proof. We first define this isomorphism on representable sheaves. Since every sheafcan be written as a colimit of representable sheaves, U commutes with colimits asa left-adjoint, and f commutes with colimits by Lemma 3.8, the isomorphism thenextends to all sheaves. Let W 2 S and F 2 ShS=U/. Then we haveHom Sh S=U . U W ;F/Š Hom Sh S .W ; U .F //Š U .F /.W /Š F.W U ! U/Š Hom Sh S=U ..W U ! U/;F/ (by Lemma 3.6)Š Hom Sh S=U .f W ;F/This isomorphism is represented by a canonical isomorphismU .W / Š f .W /in ShS=U.
258 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.3.14 Let f W S ! S 0 be a morphism of sites. It induces an adjoint pairf W ShS , ShS 0 W f :Lemma 3.16. For F 2 ShS the functor f has the explicit descriptionProof. In fact, for G 2 ShS 0 we havef .F / WD colim .U !F/2Sh.S=F / f.U/:Hom Sh S 0.colim .U !F/2Sh.S/=F f.U/;G/Š lim .U !F/2Sh.S/=F Hom Sh S 0.f .U /;G/Š lim .U !F/2Sh.S/=F G.f.U//Š lim .U !F/2Sh.S/=F f G.U /Š lim .U !F/2Sh.S/=F Hom Sh S 0.U ;f G/Š Hom Sh S .colim .U !F/2Sh.S/=F U ;f G/Š Hom Sh S .F; f G/:Lemma 3.17. For A 2 S we havef .A/ D f .A/:Proof. Since .id A W A ! A/ 2 S=A is final we havef .A/ Š colim .U !A/2Sh.S/=A f.U/Š colim .U !A/2S=A f.U/Š f .A/:Lemma 3.18. If f W S ! S 0 is fully faithful, then we have for A 2 S that f f A Š A.Proof. We calculate for U 2 S thatf f A.U / Š f f .A/.U / (by Lemma 3.17)Š f .A/.f .U //Š Hom S 0.f .U /; f .A//Š Hom S .U; A/Š A.U /3.3.15 For U 2 S we have an induced morphism of sites U f W S=U ! S 0 =f .U /which maps .A ! U/to .f .A/ ! f.U//.If f is fully faithful, then U f is fully faithful for all U 2 S.Lemma 3.19. For G 2 ShS 0 we have in S=U the identityU f .G jf.U/ / Š .f G/ jU :If we know in addition that S, S 0 have finite products which are preserved by f W S ! S 0 ,then for F 2 S we have in S 0 =f .U / the identityU f F jU Š .f F/ jf.U/ :
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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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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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