Notes on endomorphisms, local cohomology and completion

Abstract

Let $M$ denote a finitely generated module over a Noetherian ring $R$. For an ideal $I\subset R$ there is a study of the endomorphisms of the local cohomology module $H_I^g(M),g=\mathrm{grade}(I,M),$ and related results. Another subject is the study of left derived functors of the $I$-adic completion ${\Lambda }_i^I(H_I^g(M))$, motivated by a characterization of Gorenstein rings given in the book by Simon and the author. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.