Cofiniteness over Noetherian complete local rings

In this article, we prove the following generalization of a result of Hartshorne: Let be a regular local ring of dimension 4. Assume that is a regular system of parameters for S and . Then for each finitely generated S-module N with the socle of is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring , we characterize the class of all ideals I of R with the property that, for every finitely generated R-module M, the local cohomology modules are I-cofinite for all .