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 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.