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

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 étale 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 c c 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 ⟶ X act on the exact categories F X m . Assuming the existence of triangulated categories of motivic sheaves D M ( X , Z / m ) over algebraic varieties X over K and a weak version of the ‘six operations’ in these categories, we identify F X m with the exact subcategory in D M ( X , Z / m ) consisting of all the iterated extensions of the Tate twists M c c m ( Y / X ) ( j ) of the motives M c c 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 étale descent conjecture (recently proved 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 shown in our previous paper [Positselski, ‘Mixed Artin–Tate motives with finite coefficients’, Mosc. Math. J. 11 (2011) 317–402].