Artin-Tate motivic sheaves with finite coefficients over an algebraic variety, by Leonid Positselski

[ The original preprint, dated Jan 17, 2011, has been updated by the author. ]

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 of varieties over K quasi-finite over X. Relative homological motives of quasi-finite morphisms of smooth varieties are defined as certain objects of the derived category D^b(F_X^m).

A natural embedding of the category F_X^m into a hypothetical triangulated category of motivic sheves over X is constructed in the conventional assumptions about the latter. Assuming X is smooth, an isomorphism of the Z/m-modules Ext between the Tate motives Z/m(j) in F_X^m with the motivic cohomology modules predicted by the Beilinson-Lichtenbaum etale descent conjecture (proven by Voevodsky, Rost, et al.) holds if and only if a 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 shown in our previous paper 0966.

Leonid Positselski <>