Coniveau filtration and mixed motives
The main purpose of this paper is to extend Quillen's results on the Gersten conjecture for higher \(K\)-theory to the theory of Voevodsky's motives. In [\textit{M. Rost}, Doc. Math., J. DMV 1, 319--393 (1996; Zbl 0864.14002)] a theory of cycle premodules has been defined. For a cycle premodule \(M\) over an algebraic scheme \(k\)-scheme \(X\), with \(k\) a field, one has a canonical morphism NEWLINE\[NEWLINEd^m_{X,M}: \bigoplus_{x\in X^{(n)}} M(k(x))\to \prod_{y\in X^{(n+1)}} M(k(x))NEWLINE\]NEWLINE which can be viewed as a generalization of the divisor class map for Milnor \(K\)-theory NEWLINE\[NEWLINEd^0_X: K^M_* (k(X))\to \bigoplus_{x\in X^{(1)}} K^M_*(k(X)).NEWLINE\]NEWLINE Let \(DM_{gm}(k)\) be the category of geometrical motives of Voevodsky over a field \(k\) and let \({\mathcal A}\) be an abelian category. Let \(H: DM_{gm}(k)^{op}\to{\mathcal A}\) be a cohomological functor sending distinguished triangles into long exact sequences. By previous results of the same author, for any couple \((q,n)\in\mathbb{Z}^2\), there exists a canonical cycle premodule \(\widetilde H^{q,n}\) with coefficients in \({\mathcal A}\). Given a smooth \(k\)-scheme \(X\) and an integer \(n\) there is a coniveau spectral sequence NEWLINE\[NEWLINEE^{p,q}_1(X, n)=\bigoplus_{x\in X^{(p)}}\widetilde H^{q,n}_* (k(X))\Rightarrow H^{p+q, n}_*(X)\tag{1}NEWLINE\]NEWLINE with coefficients in the category of graded objects in \({\mathcal A}\). It converges to the coniveau filtration on the cohomology \(H\). -- The main result of this paper is the followingNEWLINENEWLINE Theorem 1. Let \(H: DM_{gm}(k)^{op}\to{\mathcal A}\) be a cohomological functor and denote by \(d^{p,q}_1\) the differentials in the spectral sequence (1). Then for any couple of integers \((p,q)\in\mathbb{Z}^2\), \(d^{p,q}_1= d^p_{\widetilde H^{q,n}}\).NEWLINENEWLINE This gives back the computation of Quillen, but replacing \(K\)-theory with motivic cohomology. Also, as a corollary of these computations, one gets back the results of Bloch-Ogus for the cohomology \(H\). More preciselyNEWLINENEWLINE Corollary 1. Consider as above a cohomology functor \(H\). For any smooth scheme \(X\) let \({\mathcal H}^{q,n}_*(X)\) be the kernel of the divisor map \(d^0_{X,\widetilde H^{q,n}_*}\). Then \({\mathcal H}^{q,n}_*(X)\) is a homotopy invariant Nisnevich sheaf with transfers in the sense of Voevodsky. It coincides with the Zariski sheaf associated with \(H^{q,n}_*\). The coniveau spectral sequence in (1) can be written from \(E_2\) as NEWLINE\[NEWLINEE^{p,q}_2(X, n)= H^p_{Zar}(X,{\mathcal H}^{q,n}_*)\simeq H^p_{Nis}(X,{\mathcal H}^{q,n}_*)\Rightarrow H^{p+q,n}_*(X).NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].
scientific article; zbMATH DE number 6077543