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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]

LetRbe a commutative Noetherian ring with nonzero identity, 𝔞 an ideal ofR,Ma finiteR–module,Xan arbitraryR–module, andna non-negative integer. Here, we show that, in the Serre subcategories of the category ofR–modules, how the generalized local cohomology modules, the ordinary local cohomology modules, and the extension modules behave similarly at the initial pointsi ≤ n. We conclude with some Artinianness and cofiniteness results for, and some finiteness results forand. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, 픞 an ideal of R, and M, N two nonzero finitely generated R-modules. Let t be a non-negative integer. It is shown that dim Supp for all i < t if and only if there exists an ideal 픟 of R such that dim R/픟 ≤ 1 and for all i < t. As a consequence all Bass numbers and all Betti numbers of generalized local cohomology modules are finite for all i < t, provided that the projective dimension (M) is finite. [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]