Mixed Artin–Tate motives with finite coefficients

The Beilinson–Lichtenbaum conjectures about motivic cohomology with finite coefficients (proven by Voevodsky–Rost–$\dots$) suggest a possibility of describing the categories of motivic sheaves with finite coefficients in terms of the etale site. In this talk I will explain how to describe the triangulated category of Artin–Tate motives with finite coefficients over a field as the derived category of a certain exact category of filtered modules over the absolute Galois group. The description depends on the assumption of existence of silly filtrations, which in many cases can be interpreted as a Koszulity hypothesis. Similar Koszulity conjectures appear in some other branches of the theory of motives, in particular, for motives with rational coefficients over a field of finite characteristic.