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Journal of the European Mathematical Society

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Volume 18, Issue 3, 2016, pp. 623–655
DOI: 10.4171/JEMS/598

Published online: 2016-02-16

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

Matilde Marcolli[1] and Gonçalo Tabuada[2]

(1) California Institute of Technology, Pasadena, United States
(2) 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$(k)_F$ 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 $\overline{HP_\ast}$ 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$^\dagger(k)_F$ by modifying its symmetry isomorphism constraints. By further assuming $D_{NC}$, we neutralize the Tannakian category Num$^\dagger(k)_F$ using $\overline{HP_\ast}$. 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.

Keywords: Noncommutative algebraic geometry, noncommutative motives, periodic cyclic homology, Tannakian formalism, motivic Galois groups

Marcolli Matilde, Tabuada Gonçalo: Noncommutative numerical motives, Tannakian structures, and motivic Galois groups. J. Eur. Math. Soc. 18 (2016), 623-655. doi: 10.4171/JEMS/598