A note on endomorphisms of local cohomology modules
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Let $I$ denote an ideal of a Noetherian local ring $(R,\mathfrak{m})$. 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 provide 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}).
Abstract
Let $I$ denote an ideal of a Noetherian local ring $(R,\mathfrak{m})$. 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 provide 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}).
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