On the Matlis duals of local cohomology modules

Abstract.

Let (${R, \mathfrak{m}}$) be a commutative Noetherian local ring with non-zero identity, ${\mathfrak{a}}$ an ideal of R and M a finitely generated R-module with ${\mathfrak{a}M \neq M}$ . Let D(–) := Hom _R (–, E) be the Matlis dual functor, where $E := E(R/ \mathfrak{m})$ is the injective hull of the residue field $R/ \mathfrak{m}$ . We show that, for a positive integer n, if there exists a regular sequence ${x_1, . . . , x_n \, \in \, \mathfrak{a}}$ and the i-th local cohomology module H ⁱ _a (M) of M with respect to ${\mathfrak{a}}$ is zero for all i with i > n then ${H^{n}_{\mathfrak{a}}(D(H^{n}_{\mathfrak{a}}(M))) = E.}$