Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

  • Matilde Marcolli

    California Institute of Technology, Pasadena, United States
  • Gonçalo Tabuada

    Massachusetts Institute of Technology, Cambridge, USA


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(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)F(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP\overline{HP_\ast} on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues CNCC_{NC} and DNCD_{NC} of Grothendieck's standard conjectures CC and DD. Assuming CNCC_{NC}, we prove that NNum(k)F(k)_F can be made into a Tannakian category NNum(k)F^\dagger(k)_F by modifying its symmetry isomorphism constraints. By further assuming DNCD_{NC}, we neutralize the Tannakian category Num(k)F^\dagger(k)_F using HP\overline{HP_\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.

Cite this article

Matilde Marcolli, Gonçalo Tabuada, Noncommutative numerical motives, Tannakian structures, and motivic Galois groups. J. Eur. Math. Soc. 18 (2016), no. 3, pp. 623–655

DOI 10.4171/JEMS/598