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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, m) be a commutative Noetherian local ring, let I be an ideal of R and let M and N be finitely generated R-modules. Assume that pd(M) = d < ∞, dim N = n < ∞. First, we give the formula for the attached primes of the top generalized local cohomology module HId+n (M, N), later, we prove that if Att(HId+n (M, N)) = Att(HJd+n (M, N)), then HId+n (M, N) = HJd+n (M, N). [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]

We show that the i th generalized local cohomology module of finitely generated graded modules M and N over a standard positive graded commutative Noetherian ring R , with respect to the irrelevant ideal R + , is itself graded; all its graded components are finitely generated modules over R 0 , the component of R of degree zero. Also we prove that the n th component of of this generalized local cohomology module is zero for all n ≫ 0. Moreover, in this article, we study the asymptotic behaviour of as n → -∞ in the following cases: i is the least integer j for which is not finitely generated.The base ring R 0 is local of dimension at most one and for all i ≥ 0. [ABSTRACT FROM AUTHOR]