On endomorphism rings and dimensions of 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 $H_I^c(R)$ 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 $c\ge 2$ 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 $H_I^i(R).$

7 pages

References in corpus (1)

• On cohomologically complete intersections