A note on Tannakian categories and mixed motives

Abstract

We explain why every non‐trivial exact tensor functor on the triangulated category of mixed motives over a field double-struckF has zero kernel, if one assumes ‘all’ motivic conjectures. In other words, every non‐zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated étale motives over double-struckF with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of double-struckF.

Supports for constructible systems

“…Let us then denote by DM c (X) 'the' category of constructible motivic sheaves on X with coefficients in a characteristic zero field, for example Beilinson motives [19] or Ayoub's étale motives [4]. If X = Spec(k) is the spectrum of a field, we had already established in [31,Theorem 3.8] the analogue of Theorem 1.1 assuming 'all conjectures on motives over k'. Here we show that this special case implies the result for arbitrary X and obtain the following result in motivic tensortriangular geometry.…”

“…The theories mentioned in Theorems 1.1 and 1.3 are shown to be examples in Section 3, which involves establishing that C(X) is a simple tensor triangulated category generically. This uses crucially (a step in) Beilinson's argument on the derived category of perverse sheaves [11], as well as [31] which shows that the derived category of a Tannakian category in characteristic zero is simple. In Section 4 (resp.…”

“…The following criterion for generic simplicity leverages an observation from [31] together with Deligne's internal characterization of Tannakian categories. For both of these the characteristic zero assumption is crucial.…”