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Axioms for the norm residue isomorphism 429Theorems 3.8 and 4.4 of [8] upon Lemmas 2.2 and 2.3 of [8], whose proofs are currentlyunavailable. Instead, [8, 5.11] proves that the canonical map X ! X factors throughthe map y W M ! X. Since this paper was written, the gap in the proof was patchedin [12].In Rost’s approach, defines an equivalence class of idempotent endomorphismse of X and Rost defines M to be the Chow motive .X; e/. By construction, his Msatisfies Axioms 0.3 (a)–(b). Hopefully it also satisfies Axiom 0.3 (c), the existence oftriangles (0.4) and (0.5).Notation. The integer n and the prime `>2will be fixed. We will work over a fixedfield k in which ` is invertible. The integer d will always be `n 1 1 and b will alwaysbe d=.` 1/ D 1 CC`n 2 .We fix the sequence of units a D .a 1 ;:::;a n /, and X will always denote ad-dimensional Rost variety relative to a, satisfying Axioms 0.3.We will work in the triangulated category of motives DM eff described in [2]. TheLefschetz motive is L D Z.1/Œ2. Unless explicitly stated otherwise, motivic cohomologywill always be taken with coefficients Z .`/ . The notation H p;qét. / refers tothe étale motivic cohomology H p ét . ; Z .`/.q// defined in [2, 10.1].Finally, I would like to thank the Institute for Advanced Study for allowing me topresent these results in a seminar during the Fall of 2006, and Pierre Deligne for hiscontinued interest and encouragement.1 Proof of Theorem 0.1Recall [6, 1.20] that a n 1 -variety over a field k is a smooth projective variety Xof dimension d D `n 1 1, with deg s d .X/ 6 0.mod `2/. Here s d .X/ is thePcharacteristic class of the tangent bundle T X corresponding to the symmetric polynomialtdjin the Chern roots t j of T X ; see [9, 14.3]. We also recall that H 1; 1 .Y / is thegroup Hom.Z; Y.1/Œ1/, and is covariant in Y ; see [2, 14.17].Definition 1.1. A Rost variety for a sequence a D .a 1 ;:::;a n / of units in k is a n 1 -variety such that: fa 1 ;:::;a n g vanishes in Kn M .k.X//=`; for each i < n there is a i -variety mapping to X; and the motivic homology sequenceH 1; 1 .X 2 / 0 1! H 1; 1 .X/ ! H 1; 1 .k/ .D k / (1.2)is exact. As mentioned in the Introduction, Rost varieties exist for every a by [1], [6].Example 1.2.1. If n D 1 and L D k. `pa1 /, then Spec.L/ is a Rost variety for a 1 ,with the convention that s 0 .X/ D ŒL W k D `. When n D 2, the Severi–Brauer varietycorresponding to the degree ` algebra with symbol a is a Rost variety.Proof of Theorem 0.1. By [10, 6.10], it suffices to prove the assertionH 90.n; `/:H nC1;nét.k/ D 0:
430 C. WeibelAs pointed out in the proof of [10, 7.4], it suffices to show that for every symbola 2 Kn M .k/ there is a field extension k F such that a vanishes in KM n .F /=` andH nC1;nét.k/ ! H nC1;nét.F / is an injection. When F D k.X/, this is the bottom rightarrow in the flowchart (1.3), which is implicit in [8, 6.11].H 2C2b`;1Cb`.X/ ontoH 2dC1;dC1 .D/ D 01:43.6into 2:4(1.3)H nC1;n .X/4:4exactH nC1;nét.k/ sequence HnC1;nét.k.X//:The other decorations in (1.3) refer to the properties we need and where they areestablished. Proposition 4.4 states that the bottom row of the flowchart (1.3) is exact.By Lemma 2.4 the group H nC1;n .X/ injects (by a cohomology operation) intoH 2C2b`;1Cb`.X/, which is in turn a quotient of H 2dC1;dC1 .D/ by 1.4. In Corollary3.6, we prove that H 2dC1;dC1 .D/ D 0. All of this implies that H nC1;nét.k/ !H nC1;nét.k.X// is an injection, finishing the proof of Theorem 0.1.One part of the flowchart (1.3) is easy to establish.Lemma 1.4. The map s W X ! D ˝ L b Œ1 in (0.4) induces a surjection:H 2dC1;dC1 .D/ Š H 2b`C2;b`C1 .D ˝ L b Œ1/s! H 2b`C2;b`C1 .X/:Proof. In the cohomology exact sequence arising from (0.4), the next term isH 2b`C2;b`C1 .M /, which is a summand of H 2b`C2;b`C1 .X/. Because b` D d C b>d,this group is zero by the Vanishing Theorem [2, 3.6] – which says that H n;i .X/ D 0whenever n>iC dim.X/.The lower left map in flowchart (1.3) is just the motivic-to-étale map by the followingbasic result, which we quote from [10, 7.3].Lemma 1.5. The structure map X ! Spec.k/ induces isomorphismsH ;ét .k/ Š H ;ét .X/ and H ;ét .kI Z=`/ Š H ;ét .XI Z=`/:2 Motivic cohomology operationsIn the course of the proof, we will need some facts about the motivic cohomologyoperations constructed in [9]. When `>2, there are operations P i on H ; . I Z=`/of bidegree .2i.` 1/; i.` 1//, for each i 0 (with P 0 D 1), as well as the Bocksteinoperation ˇ of bidegree .1; 0/. These satisfy the Adem relations given in [5] for theusual topological cohomology operations. In fact, the subring of all (stable) motivic
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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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