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arXiv · 2402.16974

Finite generation of split F-regular monoid algebras

Rankeya Datta·Karl Schwede·Kevin Tucker
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Abstract

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.

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Rankeya Datta, Karl Schwede, Kevin Tucker. 2024-02-26. Finite generation of split F-regular monoid algebras. https://arxiv.org/abs/2402.16974

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