Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We show in this... more Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We show in this paper that, for an integer $t$, if the local cohomology module $H^{i}_\mathfrak{a}(M)$ with respect to an ideal $\frak a$ is finitely generated for all $i<t$, then $$H^{i}_\mathfrak{a}(M/xM)\cong H^{i}_\mathfrak{a}(M)\oplus H^{i+1}_\mathfrak{a}(M)$ for all $\frak a$-filter regular elements $x$ containing in a enough large power of $\frak a$ and all $i<t-1$. As consequences we obtain generalizations, by very short proofs, of the main results of M. Brodmann and A.L. Faghani (A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(2000), 2851-2853) and of H.L. Truong and the first author (Asymptotic behavior of parameter ideals in generalized Cohen-Macaulay module, J. Algebra, 320(2008),158-168).
1 . Let (R, m) be a Noetherian local ring and M a finitely generated R-module. Following I. G. Ma... more 1 . Let (R, m) be a Noetherian local ring and M a finitely generated R-module. Following I. G. Macdonald [Mac], the set of all attached primes of the Artinian local cohomology module H i m (M) is denoted by Att R (H i m (M)). In [Sh, Theorem 3.7], R. Y. Sharp proved that if R is a quotient of a Gorenstein local ring then the shifted localization principle always holds true, i.e.
Proceedings of the American Mathematical Society, 2010
Let R be a Noetherian ring, a be an ideal of R and M be a finitely generated R-module. The aim of... more Let R be a Noetherian ring, a be an ideal of R and M be a finitely generated R-module. The aim of this paper is to show that if t is the least integer such that neither H t a (M ) nor supp(H t a (M)) is non-finite, then H t a (M ) has finitely many associated primes. This combines the main results of Brodmann and Faghani and independently Khashyarmanesh and Salarian.
Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. The fin... more Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. The finiteness dimension of M relative to a is defined by
On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomia... more On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomial type is at most one 1
Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We show in this... more Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We show in this paper that, for an integer $t$, if the local cohomology module $H^{i}_\mathfrak{a}(M)$ with respect to an ideal $\frak a$ is finitely generated for all $i<t$, then $$H^{i}_\mathfrak{a}(M/xM)\cong H^{i}_\mathfrak{a}(M)\oplus H^{i+1}_\mathfrak{a}(M)$ for all $\frak a$-filter regular elements $x$ containing in a enough large power of $\frak a$ and all $i<t-1$. As consequences we obtain generalizations, by very short proofs, of the main results of M. Brodmann and A.L. Faghani (A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(2000), 2851-2853) and of H.L. Truong and the first author (Asymptotic behavior of parameter ideals in generalized Cohen-Macaulay module, J. Algebra, 320(2008),158-168).
1 . Let (R, m) be a Noetherian local ring and M a finitely generated R-module. Following I. G. Ma... more 1 . Let (R, m) be a Noetherian local ring and M a finitely generated R-module. Following I. G. Macdonald [Mac], the set of all attached primes of the Artinian local cohomology module H i m (M) is denoted by Att R (H i m (M)). In [Sh, Theorem 3.7], R. Y. Sharp proved that if R is a quotient of a Gorenstein local ring then the shifted localization principle always holds true, i.e.
Proceedings of the American Mathematical Society, 2010
Let R be a Noetherian ring, a be an ideal of R and M be a finitely generated R-module. The aim of... more Let R be a Noetherian ring, a be an ideal of R and M be a finitely generated R-module. The aim of this paper is to show that if t is the least integer such that neither H t a (M ) nor supp(H t a (M)) is non-finite, then H t a (M ) has finitely many associated primes. This combines the main results of Brodmann and Faghani and independently Khashyarmanesh and Salarian.
Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. The fin... more Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. The finiteness dimension of M relative to a is defined by
On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomia... more On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomial type is at most one 1
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