Generalized local cohomology modules and homological Gorenstein dimensions

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Let be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let (M,N) denote the supremum of the i's such that Hⁱ(M,N)≠ 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for (M,N). Next, over a Cohen-Macaulay local ring (R,), we show that (M,N)= R-(ᵣN,M), 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.