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Axioms for the norm residue isomorphism 433Corollary 3.3. The structural map H 1; 1 .M / ! H 1; 1 .k/ D k is injective.Proof. By (0.4) and Lemma 3.2, it suffices to show that Hom.Z;D.bC 1/Œ2b C 1/ D 0.By (0.5) this results from the vanishing of both Hom.Z;M.b C 1/Œ2b C 1/ andHom.Z; X ˝ L bCdC1 / which follows from Lemma 3.1.Lemma 3.4. H 0;1 .X/ D H 2;1 .X/ D 0 and H 1;1 .XI Z/ Š H 1;1 .Spec kI Z/ Š k .Proof. The spectral sequence E p;q1D H q .X pC1 I Z.1// ) H pCq;1 .XI Z/ degenerates,all rows vanishing except for q D 1 and q D 2, because Z.1/ Š O Œ 1; see[2, 4.2]. We compare this with the spectral sequence converging to H pCq .XI G m /;Hzar q .Y; O / ! H q ét . ; G m/ is an isomorphism for q D 0; 1 (and an injection forq D 2). Hence for q 2 we have H q;1 .X/ D H q;1ét.X/ D H q;1ét.k/ D H q 1ét.k; G m /.Remark. The proof of 3.4 also shows that H 3;1 .X/ injects into H 2 ét .k; G m/ D Br.k/.Lemma 3.5. H 2dC1;dC1 .DI Z/ is the kernel of H 1; 1 .M /y! H 1; 1 .k/ D k .Proof. From (0.5) we get an exact sequence with coefficients Z:H 2d;dC1 .X˝L d / ! H 2dC1;dC1 .D/ ! H 2dC1;dC1 .M / Dy ! H 2dC1;dC1 .X˝L d /:The first group is H 0;1 .X/, which is zero by 3.4, so it suffices to show that the map Dyidentifies with the structural map y. This follows from Axiom 0.3 (b), because for anyu in H 1; 1 .M / D Hom.Z; M.1/Œ1/, X tensored with the compositeX ˝ L d Dy ! M ˝ L d u ! Z.1/Œ1 ˝ L d D Z.d C 1/Œ2d C 1is the X-dual of X. 1/Œ 1u! M y ! X.Corollary 3.6. H 2dC1;dC1 .D/ D H 2dC1;dC1 .DI Z/ ˝ Z .`/ D 0.4 Between motivic and étale cohomologyDefinition 4.1 ([10, p. 90]). Let L.n/ denote the truncation nC1 Z ét .n/ of the complexin DM eff representing étale motivic cohomology; i. e., H p . ; L.n// Š H p;nét.`/. /for p n C 1. Let K.n/ denote the mapping cone of the canonical map Z .`/ .n/ !L.n/, and consider the triangle Z .`/ .n/ ! L.n/ ! K.n/ ! Z .`/ .n/Œ1.Lemma 4.2. The map H nC1 .X; K.n//y! H nC1 .M; K.n// is an injection.Proof. By triangle (0.4) we have an exact sequence:H n .D ˝ L b ; K.n// ! H nC1 .X; K.n//y! H nC1 .M; K.n//:
434 C. WeibelWe need to see that the left term vanishes. By (0.5), it is sufficient to show thatH n .M ˝ L b ; K.n// and H n 1 .X ˝ L b`; K.n// vanish. This follows from [10, 6.12],which says that H .Y.1/; K.n// D 0 for every smooth Y , the assumption that Mis a summand of X, and the consequent collapsing of the spectral sequence E p;q1DH q .X p ; K.n// ) H pCq .X; K.n//.Corollary 4.3. The map H nC1 .X; K.n// ! H nC1 .k.X/; K.n// is an injection.Proof. The map H nC1 .X; K.n// ! H nC1 .k.X/; K.n// is an injection by [2, 11.1,13.8, 13.10], or by [10, 7.4]. The corollary follows from Lemma 4.2, since H .X; / !H .M; / is a summand of H .X; / ! H .X; /.Proposition 4.4. There is an exact sequenceH nC1;n .X/ ! H nC1;nét.k/ ! H nC1;n .k.X//:Proof. By 1.5, X ! Spec.k/ is an isomorphism on étale motivic cohomology, so wehave H nC1;nét.k/ Š H nC1 .X; L.n//.The map Spec k.X/ ! X induces a commutative diagram with exact rows:étH nC1;n .X/H nC1 .X; L.n//H nC1 .X; K.n//4:3 into0=H nC1;n .k.X// H nC1 .k.X/; L.n// H nC1 .k.X/; K.n//:By 4.3, the right vertical map is an injection. The proposition now follows by a diagramchase.Acknowledgements. This paper is an attempt to clarify Voevodsky’s preprint [8]. It isdirectly inspired by the work of Markus Rost, especially [4], who explicitly suggestedthe line of attack, based upon Axiom 0.3 (c), which is presented in this paper. I amgreatly indebted to both of them.References[1] C. Haesemeyer and C. Weibel, Norm Varieties and the Chain Lemma (after Markus Rost),Preprint 2008, available at http://www.math.uiuc.edu/0900.[2] C. Mazza,V.Voevodsky and C. Weibel, Lecture notes on motivic cohomology, Clay Monogr.Math. 2, Amer. Math. Soc. , Providence, R.I., 2006.[3] M. Rost, Chain lemma for splitting fields of symbols, Preprint 1998, available athttp://www.math.uni-bielefeld.de/~rost/chain-lemma.html[4] M. Rost, On the Basic Correspondence of a Splitting Variety, Preprint 2006, available athttp://www.math.uni-bielefeld.de/~rost[5] N. Steenrod and D. Epstein, Cohomology Operations, Ann. of Math. Stud. 50, PrincetonUniversity Press, Princeton, N.J., 1962.
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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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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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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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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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Page 452 and 453: List of contributorsArthur Bartels,
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