Galois descent for motivic theories

TL;DR

This paper establishes criteria for when categories of motives form stacks over profinite groups, proves the exactness of motivic Galois groups under Galois extensions, and clarifies the construction and Galois groups of Chow-Lefschetz motives.

Contribution

It provides necessary and sufficient conditions for motivic categories to be stacks, offers a simple proof of Galois group exactness, and clarifies the construction of Chow-Lefschetz motives.

Findings

Criteria for motivic categories to be stacks over profinite groups

Proof of exactness of motivic Galois groups under Galois extensions

Simplified construction and computation of Galois groups for Chow-Lefschetz motives

Abstract

We give necessary conditions for a category fibred in pseudo-abelian additive categories over the classifying topos of a profinite group to be a stack; these conditions are sufficient when the coefficients are $\mathbf{Q}$-linear. This applies to pure motives over a field in the sense of Grothendieck, Deligne-Milne and Andr\'e, to mixed motives in the sense of Nori and to several motivic categories considered in arXiv:1506.08386 [math.AG]. We also give a simple proof of the exactness of a sequence of motivic Galois groups under a Galois extension of the base field, which applies to all the above (Tannakian) situations. Finally, we clarify the construction of the categories of Chow-Lefschetz motives given in arXiv:2302.08327 [math.AG] and simplify the computation of their motivic Galois group in the numerical case.