Mixed Artin-Tate motives with finite coefficients

TL;DR

This paper provides an explicit description of the triangulated categories of Tate and Artin-Tate motives with finite coefficients over a field containing a primitive root of unity, linking it to Koszulity hypotheses in Galois cohomology.

Contribution

It offers a new explicit description of Tate and Artin-Tate motives with finite coefficients based on filtered modules and explores the conditions under which this description holds, connecting to Koszulity hypotheses.

Findings

Description is conditional on Koszulity hypotheses.

Connects motives with filtered modules over Galois groups.

Discusses implications for Tate motives with integral coefficients.

Abstract

The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin-Tate motives with finite coefficients Z/m over a field K containing a primitive m-root of unity as the derived categories of exact categories of filtered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equivalent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded ring to be Koszul. Exact categories, silly filtrations, and the K(\pi,1)-conjecture are discussed in the appendices. Tate motives with integral coefficients are considered in the "Conclusions" section.