Noncommutative Numerical Motives, Tannakian Structures, and Motivic Galois Groups

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 CNC and DNC of Grothendieck’s standard conjectures C and D. Assuming CNC , 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 DNC , we neutralize the Tannakian category Num(k)F using HP∗. Via the (super)Tannakian formalism, we then obtain well-defined noncommutative motivic (super-)Galois groups. Finally, making use of Deligne-Milne’s theory of Tate triples, we construct explicit homomorphisms relating these new noncommutative motivic (super-)Galois groups with the classical ones.