### 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 <posic@mccme.ru>