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Letbe a positively graded commutative Noetherian ring with irrelevant idealand let ɑ be a graded ideal contained inR+. LetM, Nbe two finitely generated gradedR-modules. In this paper, we study the finiteness and vanishing of then-th graded componentof thei-th generalized local cohomology module ofMandNwith respect to ɑ. Also, we show that the leastisuch thatis equal to the least integerifor whichis not finitely graded, where a graded module is finitely graded which is non-zero in only finitely many graded pieces. [ABSTRACT FROM PUBLISHER]

Let be a commutative Noetherian ring with identity and be an ideal of . Assume that is a finite -module and and are minimax -modules such that . In this paper, among other things, we show that is minimax for all and when one of the following conditions holds: As a consequence, we obtain that the Bass numbers and Betti numbers of are finite for all when one of the above conditions holds. [ABSTRACT FROM AUTHOR]

We consider two finitely generated graded modules over a homogeneous Noetherian ring $$R = \oplus _{n \in \mathbb{N}_0 } R_n$$ with a local base ring ( R, m) and irrelevant ideal R of R. We study the generalized local cohomology modules H ( M,N) with respect to the ideal b = b + R, where b is an ideal of R. We prove that if dim R/b ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H( M,N)/a H( M,N) is Artinian, where $$\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0$$; for all i ≥ 0, the set $$Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$$ is asymptotically stable as n→−∞. Moreover, if H( M,N) is a finitely generated R-module for all n ≤ n and all j < i, where n ∈ ℤ and i ∈ ℕ, then for all n ≤ n, the set $$Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$$ is finite. [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]