process

Parshin’s conjecture revisited 417Given an additive covariant functor F from C to an abelian category, we defineweight (Borel–Moore) homology HiW .X; F / as the ith homology of the complexF.W.X//. Weight homology has the functorialities inherited from b) and c), andsatisfies a localization sequence deduced from d). If K is a finitely generated fieldover k, then we define HiW .K; F / to be colim HiW .U; F /, where the (filtered) limitruns through integral varieties having K as their function field. Similarly, a contravariantfunctor G from C to an abelian category gives rise to weight cohomology (with compactsupport) HW i .X; G/.As a special case, we define the weight homology group HiW .X; Z.n// as thei 2nth homology of the homological complex of abelian groups CH n .W.X//.Lemma 3.1. We have HiW .X; Z.n// D 0 for i>dim X C n. In particular we haveHi W .K; Z.n// D 0 for every finitely generated field K=k and every i>trdeg k K C n.Proof. This follows from the first property of weight complexes together withCH n .T / D 0 for n>dim T .It follows from Lemma 3.1 that the niveau spectral sequenceEs;t 1 .X/ D MHsCt W .k.x/; Z.n// ) H sCt W .X; Z.n// (3)x2X .s/is concentrated on and below the line t D n. LetzH iW .X; Z.n// D EiCn;n 2 .X/be the ith homology of the complexM0x2X .n/H W 2n .k.x/; Z.n// Mx2X .s/H W sCn.k.x/; Z.n// ; (4)where L x2X .s/HsCn W .k.x/; Z.n// is placed in degree sCn. Then we obtain a canonicaland natural map W zH Wi.X; Z.n// ! HiW .X; Z.n//:Consider the canonical map of covariant functors 0 W z n . ; / ! CH n . / on thecategory of smooth projective schemes over k, sending the cycle complex to its highestcohomology. Then by [11, Theorem 5.13, Remark 5.15], the set of associated homologyfunctors extends to a homology theory on the category of all varieties over k. Theargument of [11, Proposition 5.16] show that the extension of the associated homologyfunctors for z n . ; / are higher Chow groups CH n . ;i/. The extension CH n . / areby definition the functors HiW . ; Z.n//. We obtain a functorial map W H ciW.X; Z.n// ! Hi.X; Z.n//:Lemma 3.2. For i D 2n, and for all schemes X, the map is an isomorphismH2n c .X; Z.n// Š H 2n W .X; Z.n//. In particular, H W .K; Z.d// Š Z for all fields K of2dtranscendence degree d over k.