Certain endomorphism rings of local cohomology modules and Lyubeznik numbers

TL;DR

This paper investigates the structure of local cohomology modules and their endomorphism rings, especially when classical vanishing theorems fail, and characterizes Lyubeznik tables for certain Cohen--Macaulay rings.

Contribution

It provides new expressions for endomorphism rings of local cohomology modules and describes the shape of Lyubeznik tables for partially sequentially Cohen--Macaulay rings.

Findings

New expressions for endomorphism rings of local cohomology modules.

Characterization of Lyubeznik tables for partially sequentially Cohen--Macaulay rings.

Insights into failures of the Hartshorne--Lichtenbaum Vanishing Theorem.

Abstract

The goal of this paper is twofold; on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne--Lichtenbaum Vanishing Theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules. On the other hand, building upon previous work by \`Alvarez Montaner, we exhibit the shape of Lyubeznik tables of the so--called partially sequentially Cohen--Macaulay rings as introduced by Sbarra and Strazzanti.