Genus of division algebras over fields with infinite transcendence degree
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929519968911360 |
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| author | Tikhonov, Sergey V. |
| author_facet | Tikhonov, Sergey V. |
| contents | We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let $K$ be a finite field extension of a field which is a purely transcendental extension of infinite transcendence degree of some subfield. We show that if $D$ is a central division $K$-algebra, then ${\bf gen}(D)$ consists of Brauer classes $[D']$ such that $[D]$ and $[D']$ generate the same subgroup of $Br(K)$. In particular, the genus of any division $K$-algebra of exponent 2 is trivial. Note that the family of such fields is closed under finitely generated extensions. Moreover, if $char(K) \ne 2$, we prove that the genus of a simple algebraic group of type $\mathrm{G}_2$ over such a field $K$ is trivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Genus of division algebras over fields with infinite transcendence degree Tikhonov, Sergey V. Rings and Algebras Algebraic Geometry We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let $K$ be a finite field extension of a field which is a purely transcendental extension of infinite transcendence degree of some subfield. We show that if $D$ is a central division $K$-algebra, then ${\bf gen}(D)$ consists of Brauer classes $[D']$ such that $[D]$ and $[D']$ generate the same subgroup of $Br(K)$. In particular, the genus of any division $K$-algebra of exponent 2 is trivial. Note that the family of such fields is closed under finitely generated extensions. Moreover, if $char(K) \ne 2$, we prove that the genus of a simple algebraic group of type $\mathrm{G}_2$ over such a field $K$ is trivial. |
| title | Genus of division algebras over fields with infinite transcendence degree |
| topic | Rings and Algebras Algebraic Geometry |
| url | https://arxiv.org/abs/2409.19321 |