ABSTRACT. Let a be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd a (M, N) denote the supremum of the i's such that H i a (M, N) = 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for cd a (M, N). Next, over a Cohen-Macaulay local ring (R, m), we show thatprovided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.

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Abstract. Let a be an ideal of a commutative Noetherian ring R, M a finitely generated R-module and N a weakly Laskerian R-module. We show that if N has finite dimension d, then Ass R (H d a (N )) consists of finitely many maximal ideals of R. Also, we find the least integer i, such that H i a (M, N ) is not consisting of finitely many maximal ideals of R.

Abstract. Let a be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Our main result asserts that if dim R/a ≤ 1, then all generalized local cohomology modules H i a (M, N) are a-cofinite.