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^t_I,J ( M ) and 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 (−):= 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 ≠ t , the natural homomorphism R → 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.