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Let  be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd(M, N) denote the supremum of the i's such that [image omitted]. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for cd(M, N). Next, over a Cohen-Macaulay local ring (R, ), we show that [image omitted] provided 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. [ABSTRACT FROM AUTHOR]

Let (R,) be a Noetherian local ring, M a finitely generated R-module with finite projective dimension n, N an arbitrary R-module, and  be an ideal of R which is generated by s elements. In this article, we provide a surjective homomorphism from ordinary local cohomology module [image omitted] to top generalized local cohomology module [image omitted], where Pn, M is an nth syzygy of a projective resolution of M. Also, by using this epimorphism, we prove some results about the attached primes, coassociated primes, the Betti numbers, and Artinian properties of certain generalized local cohomology modules. [ABSTRACT FROM AUTHOR]