A Note on Endomorphisms of Local Cohomology Modules

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 $H_I^c(M),c=\text{grade}(I,M)$. 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$\hat{\cdot}^I$denotesthe$I$-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 $\text{grade}(I,M)=\text{grade}(J,M)$. Our results extends constructions known in the case of $M=R$ (see e.g. \cite{h1}, \cite{p7}, \cite{p1}).