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Noncommutative numerical motives, Tannakian structures, and motivic Galois groups
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Marcolli, Matilde and Tabuada, Gonçalo (2016) Noncommutative numerical motives, Tannakian structures, and motivic Galois groups. Journal of the European Mathematical Society, 18 (3). pp. 623-655. doi:10.4171/JEMS/598
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Official URL: http://dx.doi.org/10.4171/JEMS/598
Abstract
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.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science > Mathematics | ||||||
| Journal or Publication Title: | Journal of the European Mathematical Society | ||||||
| Publisher: | European Mathematical Society Publishing House | ||||||
| ISSN: | 1435-9855 | ||||||
| Official Date: | 16 February 2016 | ||||||
| Dates: |
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| Volume: | 18 | ||||||
| Number: | 3 | ||||||
| Page Range: | pp. 623-655 | ||||||
| DOI: | 10.4171/JEMS/598 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published |
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