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Keywords:
local cohomology; Matlis duality; endomorphism ring
Summary:
Let $\mathfrak {a}$, $I$, $J$ be ideals of a Noetherian local ring $(R,\mathfrak {m},k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H^t_{I,J}(M)$ and $D(H^t_{I,J}(M))$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):= {\rm Hom}_R(-,E_R(k))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H^i_{I,J}(R)=0$ for all $i\neq t$, the natural homomorphism $R\rightarrow {\rm Hom}_R(H^t_{I,J}(K_R), H^t_{I,J}(K_R))$ is an isomorphism, where $K_R$ denotes the canonical module of $R$. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.
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