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 nonco...
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| Format: | Article |
| Language: | en_US |
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European Mathematical Society Publishing House
2016
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| Online Access: | http://hdl.handle.net/1721.1/104797 https://orcid.org/0000-0001-5558-9236 |
| _version_ | 1826189715666108416 |
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| author | Marcolli, Matilde Trigo Neri Tabuada, Goncalo Jorge |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Marcolli, Matilde Trigo Neri Tabuada, Goncalo Jorge |
| author_sort | Marcolli, Matilde |
| collection | MIT |
| description | 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 [over-bar 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 C[subscript NC], we prove that NNum(k)F can be made into a Tannakian category NNum[superscript †](k)F by modifying its symmetry isomorphism constraints. By further assuming D[subscript NC], we neutralize the Tannakian category Num†(k)F using [over-bar 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. |
| first_indexed | 2024-09-23T08:19:54Z |
| format | Article |
| id | mit-1721.1/104797 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:19:54Z |
| publishDate | 2016 |
| publisher | European Mathematical Society Publishing House |
| record_format | dspace |
| spelling | mit-1721.1/1047972022-09-23T12:23:41Z Noncommutative numerical motives, Tannakian structures, and motivic Galois groups Marcolli, Matilde Trigo Neri Tabuada, Goncalo Jorge Massachusetts Institute of Technology. Department of Mathematics Trigo Neri Tabuada, Goncalo Jorge 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 [over-bar 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 C[subscript NC], we prove that NNum(k)F can be made into a Tannakian category NNum[superscript †](k)F by modifying its symmetry isomorphism constraints. By further assuming D[subscript NC], we neutralize the Tannakian category Num†(k)F using [over-bar 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. National Science Foundation (U.S.) (NSF grant DMS-0901221) National Science Foundation (U.S.) (NSF grant DMS-1007207) National Science Foundation (U.S.) (NSF grant DMS-1201512) National Science Foundation (U.S.) (NSF grant PHY-1205440) National Science Foundation (U.S.) (CAREER Award #1350472) Fundação para a Ciência e a Tecnologia (Portugal) (project grant UID/MAT/00297/2013) 2016-10-12T20:55:40Z 2016-10-12T20:55:40Z 2016 2015-08 Article http://purl.org/eprint/type/JournalArticle 1435-9855 1435-9863 http://hdl.handle.net/1721.1/104797 Marcolli, Matilde, and Gonçalo Tabuada. “Noncommutative Numerical Motives, Tannakian Structures, and Motivic Galois Groups.” J. Eur. Math. Soc. 18, no. 3 (2016): 623–655. https://orcid.org/0000-0001-5558-9236 en_US http://dx.doi.org/10.4171/jems/598 Journal of the European Mathematical Society Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf European Mathematical Society Publishing House arXiv |
| spellingShingle | Marcolli, Matilde Trigo Neri Tabuada, Goncalo Jorge Noncommutative numerical motives, Tannakian structures, and motivic Galois groups |
| title | Noncommutative numerical motives, Tannakian structures, and motivic Galois groups |
| title_full | Noncommutative numerical motives, Tannakian structures, and motivic Galois groups |
| title_fullStr | Noncommutative numerical motives, Tannakian structures, and motivic Galois groups |
| title_full_unstemmed | Noncommutative numerical motives, Tannakian structures, and motivic Galois groups |
| title_short | Noncommutative numerical motives, Tannakian structures, and motivic Galois groups |
| title_sort | noncommutative numerical motives tannakian structures and motivic galois groups |
| url | http://hdl.handle.net/1721.1/104797 https://orcid.org/0000-0001-5558-9236 |
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