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 HP* 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 HP*. 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.