A Note on Endomorphisms of Local Cohomology Modules

TL;DR

This paper investigates the structure of endomorphism rings of local cohomology modules over local rings, establishing conditions for isomorphisms and extending known results from the case of the ring itself.

Contribution

It provides new sufficient conditions under which the natural homomorphism between endomorphism rings is an isomorphism, extending previous work beyond the case of the ring.

Findings

Identifies conditions for isomorphism of endomorphism rings

Extends known constructions from the ring case to modules

Analyzes homomorphisms between endomorphism rings for different ideals

Abstract

Let $I$ denote an ideal of a local ring $(R,\mathfrak{m})$ of dimension $n$. Let $M$ denote a finitely generated $R$-module. We study the endomorphism ring of the local cohomology module $H^c_I(M), c = \grade (I,M)$. In particular there is a natural homomorphism $\Hom_{\hat{R}^I}(\hat{M}^I, \hat{M}^I)\to \Hom_{R}(H^c_{I}(M),H^c_{I}(M))$, where $\hat{\cdot}^I$ denotes the $I$-adic completion functor. We prove sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J \subset I$ with the property $\grade(I,M) = \grade(J,M)$. Our results extends constructions known in the case of $M = R$ (see e.g. \cite{h1}, \cite{p7}, \cite{p1}).