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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 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]

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]

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 Φ be a system of ideals in a commutative Noetherian ring R, and let [InlineMediaObject not available: see fulltext.] be a Serre subcategory of R-modules. We set . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ [InlineMediaObject not available: see fulltext.]. It is shown that if the functor $$ D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) $$ is exact, then, for any $$ \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) $$ ∈ [InlineMediaObject not available: see fulltext.] for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that $$ H_\mathfrak{a}^i (M,N) $$ ∈ [InlineMediaObject not available: see fulltext.] for all i < t, then $$ Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) $$ ∈ [InlineMediaObject not available: see fulltext.], provided that [InlineMediaObject not available: see fulltext.] is contained in the class of weakly Laskerian R-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that $$ H_\mathfrak{a}^i (L) $$ ∈ [InlineMediaObject not available: see fulltext.], then $$ Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) $$ ∈ [InlineMediaObject not available: see fulltext.] if and only if $$ Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) $$ ∈ [InlineMediaObject not available: see fulltext.] for all j ≥ 0. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring with non-zero identity and α be a maximal ideal of R. An R-module M is called minimax if there is a finitely generated submodule N of M such that M/N is Artinian. Over a Gorenstein local ring R of finite Krull dimension, we proved that the Socle of Hαn (R) is a minimax R-module for each n ≥ 0. [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]