process

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.