Mixed Artin-Tate motives over number rings

TL;DR

This paper develops the theory of Artin-Tate motives over number rings, establishing a motivic t-structure, stability properties, and a weight filtration, thereby advancing the understanding of their categorical and cohomological structure.

Contribution

It introduces a functorial weight filtration for mixed Artin-Tate motives and refines the category definition, enhancing the structural understanding of these motives over number rings.

Findings

The cohomological dimension of mixed Artin-Tate motives is two.

An equivalence between the triangulated category and the derived category of motives is established.

A motivic t-structure with exactness properties similar to perverse sheaves is constructed.

Abstract

This paper studies Artin-Tate motives over number rings. As a subcategory of geometric motives, the triangulated category of Artin-Tate motives DATM(S) is generated by motives of schemes that are finite over the base S. After establishing stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives is two, and there is an equivalence of the triangulated category of Artin-Tate motives with the derived category of mixed Artin-Tate motives. Update in second version: a functorial and strict weight filtration for mixed Artin-Tate motives is established. Moreover, two minor corrections have been performed: first, the category of Artin-Tate motives is now defined to be the triangulated category generated by direct factors of $f_* \mathbf 1(n)$---as opposed to the thick category generated by these generators. Secondly, the exactness of $f^*$ for a finite map is only stated for etale maps $f$.