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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 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 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]

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 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, 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]

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]