A note on Tannakian categories and mixed motives

TL;DR

This paper explores the properties of tensor functors on the triangulated category of mixed motives, showing that non-trivial functors have zero kernel and classifying certain motives over a field under motivic conjectures.

Contribution

It demonstrates that all non-zero motives generate the entire category under the tensor triangulated structure assuming motivic conjectures, and classifies triangulated étale motives over a field.

Findings

Non-trivial tensor functors have zero kernel under motivic conjectures.

Complete classification of triangulated étale motives over a field.

Every non-zero motive generates the whole category.

Abstract

We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F 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 \'etale motives over F with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of F.