process

432 C. WeibelConsider the cohomology operation Q D Q n 1 :::Q 2 Q 1 .Lemma 2.4. The operation Q W H nC1;n .XI Z=`/ ! H 2C2b`;1Cb`.XI Z=`/ is aninjection, and induces an injectionH nC1;n .X/ ,! H 2C2b`;1Cb`.X/:Proof (Voevodsky). By the above remarks, it suffices to show that the operation Qfrom H nC2;n .†XI Z=`/ to H 3C2b`;1Cb`.†XI Z=`/ is injective. As illustrated in Example2.2.2, it is easy to see from 2.2 and (2.1) that each Q i is injective on thegroup H ; .†XI Z=`/ containing Q i 1 :::Q 1 H nC2;n .†XI Z=`/, because the precedingterm in 2.2 is zero.Remark 2.5. The same argument, given in [8, 6.7], shows that Q 0 D Q n 2 :::Q 0 isan injection from H n;n 1 .XI Z=`/ to H 2bC1;b .X/ H 2bC1;b .XI Z=`/.If fa 1 ;:::;a n g¤0 in Kn M .k/=`, Voevodsky shows in [8, 6.5] that its norm residuesymbol in Hét n .k; ˝n/ lifts to a nonzero element ı 2 H n;n 1 .XI Z=`/. Using injectivityof Q 0 , we get a nonzero symbol D Q 0 .ı/ 2 H 2bC1;b .X/. This symbol is the`starting point of the construction of the motive M in both the program of Voevodsky[8] and that of Rost [4].3 Motivic homologyIn this section, we prove Corollary 3.6, which depends uponAxiom 0.3 (b) and exactnessof (1.2) via results about the motivic homology of X and M .We will make repeated use of the following basic lemma. In this section, the notationH p; q .Y / refers to the group Hom.Z; Y.q/Œp/; see [2, 14.17].Lemma 3.1. For every smooth (simplicial) Y and p>q, Hom.Z; Y.q/Œp/ D 0.Proof. We have Hom.Z; Y.q/Œp/ Š Hzar p q .k; C .Y Gm// q D H p q C .Y Gm/.k/;qsee [2, 14.16]. The chain complex C .Y Gm/ q is zero in positive cohomological degrees,so the H p q group vanishes.Lemma 3.2. The structural map H 1; 1 .X/ ! H 1; 1 .k/ D k is injective.Proof (Voevodsky). By Lemma 3.1, Hom.Z;X p .1/Œn/ D 0 for all n 2 and all p.Therefore the right half-plane homological spectral sequenceE 1 p;q D Hom.Z;XpC1 .1/Œ q/ ) Hom.Z; X.1/Œpq/has no nonzero rows below q D 1, and the row q D 1 yields the exact sequence0 H 1; 1 .X/ H 1; 1 .X/ H 1; 1 .X X/:Since (1.2) is exact, this implies the result.