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 $i \not = c = \operatorname {height} I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$. Here we prove that this is true if and only if $H^i_I(R) = 0$ for $i = n, n-1$, 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, and an estimate of the dimension of $H^i_I(R).$