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In this paper, we study the cofiniteness of generalized local cohomology modules with respect to an ideal of a commutative Noetherian ring and the class of modules in dimension less than a fixed integer. As consequences, we present some results on the ordinary cofiniteness, weakly cofiniteness, and finiteness of the set of associated prime ideals of generalized local cohomology modules. [ABSTRACT FROM AUTHOR]

Let n and t be non-negative integers, R a commutative Noetherian ring with dim (R) ≤ n + 2 , a an ideal of R, M and N finite R-modules, and X an arbitrary R-module. We prove that if Ext R i (M / a M , X) is an FD < n R-module for all i ≤ t , then H a i (M , X) is an ( FD < n , a) -cofinite R-module for all i < t, Hom R (R / a , H a t (M , X)) is an FD < n R-module, and { p ∈ Ass R (H a i (M , X)) : dim (R / p) ≥ n } is a finite set for all i ≤ t. It shows that H a i (M , N) is ( FD < n , a) -cofinite and { p ∈ Ass R (H a i (M , N)) : dim (R / p) ≥ n } is finite for all i. In particular, H a i (M , N) is a -cofinite for all i whenever dim (R) ≤ 2. Also, Ass R (H a i (M , N)) is finite for all i when R is semi-local with dim (R) ≤ 3. [ABSTRACT FROM AUTHOR]

Let M and N be two finitely generated graded modules over a standard graded Noetherian ring R = ⊕ n≥0 Rn. In this paper we show that if R0 is semi-local of dimension ≤ 2 then, the set AssR0 ( HiR+ (M, N)n ) is asymptotically stable for n → −∞ in some special cases. Also, we study the torsion-freeness of graded generalized local cohomology modules HiR+ (M, N). Finally, the tame loci T i (M, N) of (M, N) will be considered and some sufficient conditions are proposed for the openness of these sets in the Zariski topology. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, a an ideal of R and M a finite R-modules. Let X be an arbitrary R-module (not necessary finite) and n be a nonnegative integer. It is shown that... We also study cominimaxness of generalized local cohomology modules in several cases. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring with non-zero identity, an ideal of R, M and X finite R–modules, and t a non-negative integer such that is –cofinite for all i < t. In this article, we show that and are finite, and is finite if and only if is finite. As a consequence, we observe that is a finite set. [ABSTRACT FROM AUTHOR]

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]