Artin-Tate motivic sheaves with finite coefficients over an algebraic variety

TL;DR

This paper constructs a tensor exact category of Artin-Tate motivic sheaves with finite coefficients over algebraic varieties, linking it to motivic cohomology and the Beilinson-Lichtenbaum conjecture, under certain assumptions.

Contribution

It introduces a new exact category of motivic sheaves with finite coefficients and relates it to existing motivic categories and conjectures, providing a framework for understanding Ext groups and motivic cohomology.

Findings

Identification of F_X^m with a subcategory of DM(X,Z/m)

Verification of Ext isomorphisms under the Beilinson-Lichtenbaum conjecture

Equivalence conditions involving primitive m-root of unity and Koszulity hypothesis

Abstract

We propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X. Among the objects of F_X^m, in addition to the Tate motives Z/m(j), there are the cohomological relative motives with compact support M_cc^m(Y/X) of varieties Y quasi-finite over X. Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties Y\to X act on the exact categories F_X^m. Assuming the existence of triangulated categories of motivic sheaves DM(X,Z/m) over algebraic varities X over K and a weak version of the "six operations" in these categories, we identify F_X^m with the exact subcategory in DM(X,Z/m) consisting of all the iterated extensions of the Tate twists M_cc^m(Y/X)(j) of the motives M_cc^m(Y/X). An isomorphism of the Z/m-modules Ext between the Tate motives Z/m(j) in the exact category F_X^m with the motivic cohomology modules predicted by the Beilinson-Lichtenbaum etale descent conjecture (recently proven by Voevodsky, Rost, et al.) holds for smooth varieties X over K if and only if the similar isomorphism holds for Artin-Tate motives over fields containing K. When K contains a primitive m-root of unity, the latter condition is equivalent to a certain Koszulity hypothesis, as it was shown in our previous paper arXiv:1006.4343