A sufficient condition for strong $F$-regularity

TL;DR

This paper establishes a new sufficient condition involving local cohomology and canonical ideals that guarantees an $F$-finite Noetherian local ring is strongly $F$-regular, implying it is Cohen-Macaulay and normal.

Contribution

It provides a novel criterion linking $F$-regularity to the simplicity of a specific local cohomology module under $S_2$ conditions.

Findings

Rings satisfying the conditions are strongly $F$-regular.

Such rings are Cohen-Macaulay and normal.

The criterion involves the simplicity of $H^{d-1}_{rak{m}}(R/I)$ as an $R ext{-}F$-module.

Abstract

Let $(R,\mathfrak{m},K)$ be an $F$-finite Noetherian local ring which has a canonical ideal $I \subsetneq R$. We prove that if $R$ is $S_2$ and $H^{d-1}_{\mathfrak{m}}(R/I)$ is a simple $R\{F\}$-module, then $R$ is a strongly $F$-regular ring. In particular, under these assumptions, $R$ is a Cohen-Macaulay normal domain.