process

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: