Notes on endomorphisms, local cohomology and completion

TL;DR

This paper investigates endomorphisms of local cohomology modules and their derived functors over Noetherian rings, providing new Cohen-Macaulay criteria and illustrating results with examples.

Contribution

It introduces novel insights into the structure of local cohomology modules and their endomorphisms, extending Cohen-Macaulay characterizations.

Findings

Endomorphisms of local cohomology modules are characterized.

Derived functors of $I$-adic completion offer new Cohen-Macaulay criteria.

Examples illustrate the theoretical results.

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 $H^g_I(M), g = \operatorname{grade} (I,M),$ and related results. Another subject is the study of left derived functors of the $I$-adic completion $\Lambda^I_i(H^g_I(M))$, 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.