On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals

Let a, I, J be ideals of a Noetherian local ring (R, 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 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 nonzero and D(−):= HomR(−,ER(k)) is the Matlis dual functor. We show that if R is a d-dimensional complete Cohen-Macaulay ring and H I, J i (R) = 0 for all i ≠ t, the natural homomorphism R → HomR(H I, J t (KR), H I, J t (KR)) is an isomorphism, where KR 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.