Galois descent for motivic theories

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.