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Duality for topological abelian group stacks and T -duality 313An inspection of the formulas 4.2.7 for the differential of the complex U shows thatd 1 W E q;01! E qC1;01vanishes for even q, and is the identity for odd q. In other wordsthis complex is isomorphic to0 ! Z ! 0 Z ! id Z ! 0 Z ! id Z ! 0 :This implies that E q;02D 0 for q 1 and therefore the assertion of the lemma.We now turn to 2. We calculateH 1 .G q I Z/ Š Œ.ƒ yG/˝Zq 1 D yG ˚˚ yG :„ ƒ‚ …q summandsBy (42) we getE q;11Š yG q :We see that .E ;11 ;d 1/ is the sheafification of the complex of discrete groupsyG W 0 ! yG ! yG 2 ! yG 3 ! ; (43)where the differential is the dual of the differential of the complex0 G G 2 G 3 induced by the maps given in 4.2.7. Let us describe the differential more explicitly. If 2 yG, and W G G ! G is the multiplication map, then we have D .; / 22G G Š yG yG. We can write @W yG q ! yG qC1 asqC1X@ D . 1/ i @ i :Using the formulas of 4.2.7 we getfor i D 1;:::;q. FurthermoreiD0@ i . 1 ;:::; q / D . 1 ;:::; i ; i ;:::; q /@ 0 . 1 ;:::; q / D .0; 1 ;:::; q /and@ qC1 . 1 ;:::; q / D . 1 ;:::; q ; 0/:Note that 0 D @W yG 0 ! yG 1 . We see that we can write the higher ( 1) degree part ofthe complex (43) in the formK ˝ZyG;where K q D Z q for q 1 and @W Z q ! Z qC1 is given by the same formulas as above.One now shows 12 that´H q Z; q D 1;.K / D0; q 2:12 We leave this as an exercise in combinatorics to the interested reader.
314 U. Bunke, T. Schick, M. Spitzweck, and A. ThomSince yG is torsion-free and hence a flat Z-module we haveH q . yG / Š H q .K ˝ZyG/ Š H q .K / ˝ZyG:Therefore H q . yG / D 0 for q 2. This implies the assertion of the lemma in thecase 2.Lemma 4.77. Let F 0 F 1 F k 1 F k D F be a filtered sheaf of Z mult -modules such that Gr l .F / has weight l, and such that F 0 D F 1 D 0. IfV F is atorsion-free sheaf of weight 1, then V D 0.Proof. Assume that V 6D 0. We show by induction (downwards) that F l \ V 6D 0 forall l 1. The case l D 1 gives the contradiction. Assume that l>1. We consider theexact sequence0 ! F l 1 \ V ! F l \ V ! Gr l .F /:First of all, by induction assumption, the sheaf V \ F l is non-trivial, and a as asubsheaf of V it is torsion-free of weight 1. Since Gr l .F / has weight l 6D 1, the mapV \ F l ! Gr l .F / can not be injective. Otherwise its image would be a torsion-freesheaf of two different weights l and 1, and this is impossible by Lemma 4.74. HenceF l 1 \ V 6D 0.4.6.14 We now show that Lemma 4.64 extends to connected compact groups.Lemma 4.78. Let G be a compact connected abelian group. Then the followingassertions are equivalent.1. Ext i Sh Ab S lc.G; Z/ is torsion-free for i D 2; 3.2. Ext i Sh Ab S lc.G; Z/ Š 0 for i D 2; 3.Proof. The non-trivial direction is that 1. implies 2. Therefore let us assume thatExt i Sh Ab S lc.G; Z/ is torsion-free for i D 2; 3. We now look at the spectral sequence.F r ;d r /. The left lower corner of its second page was calculated in 4.60. We seethat F 1;22D Ext 1 Sh Ab S lc..ƒ 2 Z G/] ; Z/ has weight 2, while Ext 3 Sh Ab S lc.G; Z/ has weight 1.Since Ext 3 Sh Ab S lc.G; Z/ is torsion-free by assumption, it follows from Lemma 4.74 thatthe differential d 1;22W Ext 1 Sh Ab S lc..ƒ 2 Z G/] ; Z/ ! Ext 3 Sh Ab S lc.G; Z/ is trivial.We conclude that Ext 2 Sh Ab S lc.G; Z/ and Ext 3 Sh Ab S lc.G; Z/ survive to the limit of thespectral sequence. We see that Ext i Sh Ab S lc.G; Z/ are torsion-free sub-sheaves of weight1 of H iC1 for i D 2; 3. Using the structure of H iC1 given in 4.75 in conjunction withLemmas 4.76 and 4.77 we get Ext i Sh Ab S lc.G; Z/ D 0 for i D 2; 3.4.6.15 The combination of Lemma 4.62 (resp. Lemma 4.62) and Lemma 4.78 givesthe following result.Theorem 4.79. 1. A compact connected abelian group G which satisfies the two-threecondition is admissible on the site S lc-acyc .2. If G is in addition locally topologically divisible, then it is admissible on S lc .
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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 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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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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