Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives
I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable alge...
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| Format: | Article |
| Language: | en_US |
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Elsevier
2017
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| Online Access: | http://hdl.handle.net/1721.1/107796 https://orcid.org/0000-0001-5558-9236 |
| _version_ | 1826195468504268800 |
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| author | Trigo Neri Tabuada, Goncalo Jorge |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Trigo Neri Tabuada, Goncalo Jorge |
| author_sort | Trigo Neri Tabuada, Goncalo Jorge |
| collection | MIT |
| description | I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev–Panin's computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneous varieties. As a second application, we show that the noncommutative motive of a twisted projective homogeneous variety is trivial if and only if the Brauer classes of the associated central simple algebras are trivial. Along the way we construct a fully-faithful ⊗-functor from Merkurjev–Panin's motivic category to Kontsevich's category of noncommutative Chow motives, which is of independent interest. |
| first_indexed | 2024-09-23T10:13:09Z |
| format | Article |
| id | mit-1721.1/107796 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:13:09Z |
| publishDate | 2017 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/1077962022-09-30T19:43:24Z Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives Trigo Neri Tabuada, Goncalo Jorge Massachusetts Institute of Technology. Department of Mathematics Trigo Neri Tabuada, Goncalo Jorge I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev–Panin's computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneous varieties. As a second application, we show that the noncommutative motive of a twisted projective homogeneous variety is trivial if and only if the Brauer classes of the associated central simple algebras are trivial. Along the way we construct a fully-faithful ⊗-functor from Merkurjev–Panin's motivic category to Kontsevich's category of noncommutative Chow motives, which is of independent interest. 2017-03-31T14:55:53Z 2017-03-31T14:55:53Z 2014-07 2014-06 Article http://purl.org/eprint/type/JournalArticle 0021-8693 http://hdl.handle.net/1721.1/107796 Tabuada, Gonçalo. “Additive Invariants of Toric and Twisted Projective Homogeneous Varieties via Noncommutative Motives.” Journal of Algebra 417 (November 2014): 15–38. https://orcid.org/0000-0001-5558-9236 en_US http://dx.doi.org/10.1016/j.jalgebra.2014.06.028 Journal of Algebra Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier arXiv |
| spellingShingle | Trigo Neri Tabuada, Goncalo Jorge Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives |
| title | Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives |
| title_full | Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives |
| title_fullStr | Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives |
| title_full_unstemmed | Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives |
| title_short | Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives |
| title_sort | additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives |
| url | http://hdl.handle.net/1721.1/107796 https://orcid.org/0000-0001-5558-9236 |
| work_keys_str_mv | AT trigoneritabuadagoncalojorge additiveinvariantsoftoricandtwistedprojectivehomogeneousvarietiesvianoncommutativemotives |