Finite generation of split- F -regular $\text{split-}F\text{-regular}$ monoid algebras

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 $k\left[S\right]$ is $\text{split-}F\text{-regular}$, then $k\left[S\right]$ 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 $\text{split-}F\text{-regular}$ rings in function fields over $k$ have to be Noetherian. The key tool is Diophantine approximation from convex geometry.