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*) over bar on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C-NC 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(dagger)(k)(F) by modifying its symmetry isomorphism constraints. By further assuming D-NC, we neutralize the Tannakian category Num(dagger) (k)(F) using (HP*) over bar. 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 to the classical ones as suggested by Kontsevich.