Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Abstract

In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum$(k{)}_F$ of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum$(k{)}_F$ is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor $\overline{H{P}_{\ast }}$ on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues $C_{NC}$ and $D_{NC}$ of Grothendieck's standard conjectures $C$ and $D$. Assuming $C_{NC}$, we prove that NNum$(k{)}_F$ can be made into a Tannakian category NNum${}^{†}(k{)}_F$ by modifying its symmetry isomorphism constraints. By further assuming $D_{NC}$, we neutralize the Tannakian category Num${}^{†}(k{)}_F$ using $\overline{H{P}_{\ast }}$. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.