On connectedness and indecomposibility of local cohomology modules

TL;DR

This paper explores the relationship between the connectedness of certain geometric sets and the indecomposability of local cohomology modules in Gorenstein rings, extending known results and analyzing endomorphism rings.

Contribution

It extends Hochster and Huneke's result on indecomposability of local cohomology modules to more general ideals in Gorenstein rings and analyzes related connectedness properties.

Findings

$H^c_I(R)$ is indecomposable iff $V(I_d)$ is connected in codimension one.

The endomorphism ring of $H^c_I(R)$ is local Noetherian when $ ext{dim } R/I=1$.

Connectedness properties influence the structure of local cohomology modules.

Abstract

Let $I$ denote an ideal of a local Gorenstein ring $(R, \mathfrak m)$. Then we show that the local cohomology module $H^c_I(R), c = \height I,$ is indecomposable if and only if $V(I_d)$ is connected in codimension one. Here $I_d$ denotes the intersection of the highest dimensional primary components of $I.$ This is a partial extension of a result shown by Hochster and Huneke in the case $I$ the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of $H^c_I(R)$ is a local Noetherian ring if $\dim R/I = 1.$