Annihilators of local cohomology modules and simplicity of rings of differential operators

TL;DR

This paper investigates the relationship between local cohomology modules and their annihilators, using differential operator rings to provide new insights and partial answers to longstanding questions in algebra.

Contribution

It introduces new results connecting local cohomology annihilators with differential operator rings, advancing understanding of their structure and properties.

Findings

Partial positive answers to open questions about local cohomology annihilators

New structural insights into local cohomology as modules over rings of differential operators

Enhanced understanding of the simplicity of rings of differential operators

Abstract

One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.