In this paper we assume that R is a Gorenstein Noetherian ring. We show that if (R, m) is also a local ring with Krull dimension d that is less than or equal to 2, then for any nonzero ideal a of R , H d a (R) is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if R is a Gorenstein ring, then for any R-module M its local cohomology modules can be calculated by means of a resolution of M by Gorenstein injective modules. Also we prove that if R is d-Gorenstein, M is a Gorenstein injective and a is a nonzero ideal of R, then Γa(M) is Gorenstein injective.
Let R be a commutative Noetherian local ring and let a be a proper ideal of R. In this paper, as a main result, it is shown that if M is a Gorenstein R-module with c = ht M a, then H i a (M ) = 0 for all i = c is completely encoded in homological properties of H c a (M ), in particular in its Bass numbers. Notice that, this result provides a generalization of a result of Hellus and Schenzel which has been proved before, as a main result, in the case where M = R.
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