Some Results about Endomorphism Rings for Local Cohomology Defined by a Pair of Ideals

Abstract

Let $(R,\mathfrak{m},k)$ denote a local ring. For $I$ and $J$ ideals of $R$, for all integer $i$, let $H_{I,J}^i(-)$ denote the $i$-th local cohomology functor with respect to $(I,J)$. Here we give a generalized version of Local Duality Theorem for local cohomology defined by a pair of ideals. Also, for $M$ be a finitely generated $R$-module, we study the behavior of the endomorphism rings $H_{I,J}^t(M)$ and $D(H_{I,J}^t(M))$ where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is non-zero and $D(-):={\mathrm{Hom}}_R(-,E_R(k))$ is the Matlis dual functor. We show too that if $R$ be a $d$-dimensional complete Cohen-Macaulay and $H_{I,J}^i(R)=0$ for all $i\ne t$, the natural homomorphism $R\to {\mathrm{Hom}}_R(H_{I,J}^t(K_R),H_{I,J}^t(K_R))$ is an isomorphism and for all $i\ne t$, where $K_R$ denote the canonical module of $R$.