Mixed Motives

TL;DR

This paper introduces the concept of a universal mixed Weil cohomology within Voevodsky's motives, establishing its existence and implications for Nori motives and mixed motives in various characteristics.

Contribution

It proves the existence of a universal mixed Weil cohomology and relates Nori motives to Betti cohomology through a new localization approach.

Findings

Universal mixed Weil cohomology exists.

Nori motives are a universal enrichment of Betti cohomology.

Implications for mixed motives in arbitrary characteristic.

Abstract

A mixed Weil cohomology with values in an abelian rigid tensor category is a cohomological functor on Voevodsky's category of motives which is satisfying K\"unneth formula and such that its restriction to Chow motives is a Weil cohomology. We show that the universal mixed Weil cohomology exists. Nori motives can be recovered as a universal enrichment of Betti cohomology via a localisation. This new picture is drawing some consequences with respect to the theory of mixed motives in arbitrary characteristic.