Noetherianity of polynomial rings up to group actions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915456314507264 |
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| author | Li, Liping Peng, Yinhe Yuan, Zhengjun |
| author_facet | Li, Liping Peng, Yinhe Yuan, Zhengjun |
| contents | Let $k$ be a commutative Noetherian ring, and $k[S]$ the polynomial ring whose indeterminates are parameterized by elements in a set $S$. We show that $k[S]$ is Noetherian up to highly homogenous actions of groups. In particular, there is a special linear order $\leqslant$ on infinite $S$ such that $k[S]$ is Noetherian up to actions of $\mathrm{Aut}(S, \leqslant)$, and the existence of such a linear order for every infinite set is equivalent to the axiom of choice. These Noetherian results are proved via a sheaf theoretic approach based on Artin's theorem, the work of Nagel-Römer, and a classification of highly homogenous groups by Cameron. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noetherianity of polynomial rings up to group actions Li, Liping Peng, Yinhe Yuan, Zhengjun Representation Theory Commutative Algebra Rings and Algebras Let $k$ be a commutative Noetherian ring, and $k[S]$ the polynomial ring whose indeterminates are parameterized by elements in a set $S$. We show that $k[S]$ is Noetherian up to highly homogenous actions of groups. In particular, there is a special linear order $\leqslant$ on infinite $S$ such that $k[S]$ is Noetherian up to actions of $\mathrm{Aut}(S, \leqslant)$, and the existence of such a linear order for every infinite set is equivalent to the axiom of choice. These Noetherian results are proved via a sheaf theoretic approach based on Artin's theorem, the work of Nagel-Römer, and a classification of highly homogenous groups by Cameron. |
| title | Noetherianity of polynomial rings up to group actions |
| topic | Representation Theory Commutative Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2502.14306 |