A note on the use of Frobenius map and D-modules in local cohomology

TL;DR

This paper compares Frobenius depth and formal grade in local cohomology, reestablishes key results using formal local cohomology, and explores endomorphism rings of local cohomology modules to unify characteristic-dependent results.

Contribution

It introduces a comparison between Frobenius depth and formal grade, and rederives important results on annihilators of local cohomology modules across different characteristics.

Findings

F-depth and formal grade are comparable in local cohomology.

Reproves results of Lyubeznik and Huneke-Koh using formal local cohomology.

Analyzes endomorphism rings of local cohomology modules.

Abstract

The Frobenius depth denoted by F-depth defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal grade and reprove some results of Lyubeznik using formal local cohomology. Then the endomorphism rings of local cohomology modules will be considered. As an application, we reprove the results due to Huneke-Koh in positive characteristic and Lyubeznik in characteristic zero on the annihilators of local cohomology modules.