Matlis Duality and Finiteness Properties of Generalized Local Cohomology Modules

Let a be an ideal of a commutative Noetherian local ring (R, m) and M, N be two finitely generated R-modules such that M is of finite projective dimension n. Let t be a positive integer. We show that if there exists a regular sequence x(1),... ,x(t) is an element of a with aN not equal N and the i-th local cohomology module H-a(i)(N) of N with respect to is zero for all i > t, then , H-a(t)(M, D(H-a(t)(N))) congruent to D(M circle times(R) N), where D(-) := Hom(R)(-, E). Also, we prove that if N is a Cohen-Macaulay R-module of dimension d, then the generalized local cohomology module H-m(n+d)(M,N) is co-Cohen-Macaulay of Noetherian dimension d. Finally, with an elementary proof, we show that Supp(R)(H-a(n+d-1)(M, N)) is finite.