On an endomorphism ring of local cohomology

TL;DR

This paper investigates the structure and endomorphism ring of the top local cohomology module in cases where the Hartshorne-Lichtenbaum Vanishing Theorem does not hold, revealing new insights into its algebraic properties.

Contribution

It provides a detailed analysis of the endomorphism ring of local cohomology modules when the vanishing theorem fails, including a key isomorphism for complete rings.

Findings

Endomorphism ring structure characterized in non-vanishing cases.

Established isomorphism between $H^d_I(R)$ and $H^d_{rak m}(R/J)$ for complete rings.

Connectedness properties related to the structure of local cohomology modules.

Abstract

Let $I$ be an ideal of a local ring $(R,\mathfrak m)$ with $d = \dim R.$ For the local cohomology module $H^i_I(R)$ it is a well-known fact that it vanishes for $i > d$ and is an Artinian $R$-module for $i = d.$ In the case that the Hartshorne-Lichtenbaum Vanishing Theorem fails, that is $H^d_I(R) \not= 0,$ we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case $R$ is complete we prove - as a technical tool - that $H^d_I(R) \simeq H^d_{\mathfrak m}(R/J)$ for a certain ideal $J \subset R.$ Thus, properties of $H^d_I(R)$ and its Matlis dual might be described in terms of the local cohomology supported in the maximal ideal.