On an endomorphism ring of local cohomology

Abstract

Let $I$ be an ideal of a local ring $(R,\mathfrak{m})$ with $d=\dim R.$ For the local cohomology module $H_I^i(R)$ it is a well-known fact that it vanishes for $i>d$ and is an Artinian $R$-module for $i=d.$ In the case that the Hartshorne-Lichtenbaum Vanishing Theorem fails, that is $H_I^d(R)\ne 0,$ we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case $R$ is complete we prove - as a technical tool - that $H_I^d(R)\simeq H_{\mathfrak{m}}^d(R/J)$ for a certain ideal $J\subset R.$ Thus, properties of $H_I^d(R)$ and its Matlis dual might be described in terms of the local cohomology supported in the maximal ideal.

8 pages, The paper will appear in Journal "Communications in Algebra"