Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

TL;DR

This paper explores the relationship between Frobenius actions on local cohomology and tight closure of ideals, introducing relative Frobenius actions and characterizing F-nilpotent singularities.

Contribution

It develops the concept of relative Frobenius actions on local cohomology and links F-nilpotent singularities to tight closure and Frobenius closure of parameter ideals.

Findings

F-nilpotent singularities characterized by tight and Frobenius closure equality

Introduction of relative Frobenius actions on local cohomology

Establishment of new connections between Frobenius actions and tight closure

Abstract

The study of Frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure. This paper establishes new connections by developing the notion of relative Frobenius actions on local cohomology. As an application, we show that a ring has $F$-nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its Frobenius closure.