Finite generation of split F-regular monoid algebras
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909555520176128 |
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| author | Datta, Rankeya Schwede, Karl Tucker, Kevin |
| author_facet | Datta, Rankeya Schwede, Karl Tucker, Kevin |
| contents | Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or equivalently, that $S$ is a finitely generated monoid. Split $F$-regular rings are possibly non-Noetherian or non-$F$-finite rings that satisfy the defining property of strongly $F$-regular rings from the theories of tight closure and $F$-singularities. Our finite generation result provides evidence in favor of the conjecture that split $F$-regular rings in function fields over $k$ have to be Noetherian. The key tool is Diophantine approximation from convex geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16974 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite generation of split F-regular monoid algebras Datta, Rankeya Schwede, Karl Tucker, Kevin Commutative Algebra Algebraic Geometry 13A35, 14M25, 52B20, 20M25 Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or equivalently, that $S$ is a finitely generated monoid. Split $F$-regular rings are possibly non-Noetherian or non-$F$-finite rings that satisfy the defining property of strongly $F$-regular rings from the theories of tight closure and $F$-singularities. Our finite generation result provides evidence in favor of the conjecture that split $F$-regular rings in function fields over $k$ have to be Noetherian. The key tool is Diophantine approximation from convex geometry. |
| title | Finite generation of split F-regular monoid algebras |
| topic | Commutative Algebra Algebraic Geometry 13A35, 14M25, 52B20, 20M25 |
| url | https://arxiv.org/abs/2402.16974 |