Strong $F$-regularity and generating morphisms of local cohomology modules

TL;DR

This paper provides a criterion for strong F-regularity in certain Cohen-Macaulay rings and constructs explicit generating morphisms for top local cohomology modules, offering new insights into the structure of determinantal rings.

Contribution

It introduces a new criterion for strong F-regularity in non-Gorenstein Cohen-Macaulay rings and explicitly constructs generating morphisms for local cohomology modules, simplifying proofs of known results.

Findings

Established a criterion for strong F-regularity in specific rings.

Constructed explicit generating morphisms for local cohomology modules.

Provided a new, simplified proof that generic determinantal rings are strongly F-regular.

Abstract

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular.