In this paper we prove the following generalization of a result of Hartshorne: Let (S, n) be a regular local ring of dimension 4. Assume that x, y, u, v is a regular system of parameters for S and a := xu + yv. Then for each finitely generated S-module N withusing this result, for any commutative Noetherian complete local ring (R, m), 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 H i I (M ) are I-cofinite for all i ≥ 0. Hartshorne's example: (See [14, §3]) Let k be a field, R = k[[u, v]][x, y], I = (x, y)R, P = (u, v, x, y)R and f = ux + vy. Then Soc R P H 2 IR P (R P /f R P ) is infinite dimensional.