Lo cal Cohomology with Resp ect to a Cohomologically Complete Intersection Pair of Ideals

Let $(R,\fm,k)$ be a local Gorenstein ring of dimension $n$. Let $\H_{I,J}^i(R)$ be the local cohomology with respect to a pair of ideals $I,J$ and $c$ be the $\inf\{i|\H_{I,J}^i(R)\neq0\}$. A pair of ideals $I, J$ is called cohomologically complete intersection if $\H_{I,J}^i(R)=0$ for all $i\neq c$. It is shown that, when $\H_{I,J}^i(R)=0$ for all $i\neq c$, (i) a minimal injective resolution of $\H_{I,J}^c(R)$ presents like that of a Gorenstein ring; (ii) $\Hom_R(\H_{I,J}^c(R),\H_{I,J}^c(R))\simeq R$, where $(R,\fm)$ is a complete ring. Also we get an estimate of the dimension of $\H_{I,J}^i(R)$.