Geodesic
Noncommutative numerical motives, Tannakian structures, and motivic Galois groups
Journal of the European Mathematical Society, Tome 18 (2016) no. 3, pp. 623-655.

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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 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.
DOI : 10.4171/jems/598
Classification : 14-XX, 18-XX, 19-XX
Keywords: Noncommutative algebraic geometry, noncommutative motives, periodic cyclic homology, Tannakian formalism, motivic Galois groups 
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     author = {Matilde Marcolli and Gon\c{c}alo Tabuada},
     title = {Noncommutative numerical motives, {Tannakian} structures, and motivic {Galois} groups},
     journal = {Journal of the European Mathematical Society},
     pages = {623--655},
     publisher = {mathdoc},
     volume = {18},
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     year = {2016},
     doi = {10.4171/jems/598},
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Matilde Marcolli; Gonçalo Tabuada. Noncommutative numerical motives, Tannakian structures, and motivic Galois groups. Journal of the European Mathematical Society, Tome 18 (2016) no. 3, pp. 623-655. doi : 10.4171/jems/598. http://geodesic.mathdoc.fr/articles/10.4171/jems/598/

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