On endomorphism rings and dimensions of local cohomology modules

TL;DR

This paper investigates the conditions under which the endomorphism rings of local cohomology modules over Gorenstein rings are isomorphic to the ring itself, providing new criteria and vanishing results for these modules.

Contribution

It establishes new equivalences for the isomorphism of endomorphism rings of local cohomology modules, extending previous results, and offers vanishing theorems and dimension estimates.

Findings

Endomorphism ring isomorphic to R iff certain local cohomology modules vanish

Provides criteria involving isolated singularities and complete intersections

Includes vanishing results and dimension bounds for local cohomology modules

Abstract

Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not= c = \height I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$ (cf. \cite{HSt} and \cite{HS}). Here we prove that this is true if and only if $H^i_I(R) = 0, i = n, n -1$ provided $c \geq 2$ and $R/I$ has an isolated singularity resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of $H^i_I(R)$ for all $i > m, m$ a given integer, resp. an estimate of the dimension of $H^i_I(R).$