A NOTE ON ENDOMORPHISMS OF LOCAL COHOMOLOGY MODULES

Abstract

Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module <TEX>$H^c_I(M)$</TEX>, c = grade(I, M). In particular there is a natural homomorphism <TEX>$$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$</TEX>, <TEX>$where{\hat{\cdot}}^I$</TEX> 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 <TEX>$J{\subset}I$</TEX> with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).