A new version of Local-Global Principle for annihilations of local cohomology modules

K. Khashyarmanesh; M. Yassi; A. Abbasi

Displaying similar documents to “A new version of Local-Global Principle for annihilations of local cohomology modules”

Some results on the cofiniteness of local cohomology modules

Sohrab Sohrabi Laleh, Mir Yousef Sadeghi, Mahdi Hanifi Mostaghim (2012)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$ , $M$ an $R$ -module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $𝒜ℱ$ modules. The main result is that if the $R$ -module ${Ext}_{R}^{t} (R / 𝔞, M)$ is finite (finitely generated), $H_{𝔞}^{i} (M)$ is $𝔞$ -cofinite for all $i < t$ and $H_{𝔞}^{t} (M)$ is minimax then $H_{𝔞}^{t} (M)$ is $𝔞$ -cofinite. As a consequence we show that if $M$ and $N$ are finite $R$ -modules and $H_{𝔞}^{i} (N)$ is minimax for all $i < t$ then the set of associated prime ideals of the generalized local cohomology...

Local-global principle for annihilation of general local cohomology

J. Asadollahi, K. Khashyarmanesh, Sh. Salarian (2001)

Colloquium Mathematicae

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Let A be a Noetherian ring, let M be a finitely generated A-module and let Φ be a system of ideals of A. We prove that, for any ideal in Φ, if, for every prime ideal of A, there exists an integer k(), depending on , such that $^{k ()}$ kills the general local cohomology module $H_{Φ_{}}^{j} (M_{})$ for every integer j less than a fixed integer n, where $Φ_{} : =_{} : \in Φ$ , then there exists an integer k such that $^{k} H_{Φ}^{j} (M) = 0$ for every j < n.

Artinianness of formal local cohomology modules

Shahram Rezaei (2019)

Commentationes Mathematicae Universitatis Carolinae

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Let $𝔞$ be an ideal of Noetherian local ring $(R, 𝔪)$ and $M$ a finitely generated $R$ -module of dimension $d$ . In this paper we investigate the Artinianness of formal local cohomology modules under certain conditions on the local cohomology modules with respect to $𝔪$ . Also we prove that for an arbitrary local ring $(R, 𝔪)$ (not necessarily complete), we have ${Att}_{R} (𝔉_{𝔞}^{d} (M)) = Min V ({Ann}_{R} 𝔉_{𝔞}^{d} (M)) .$

Associated primes of local cohomology modules of generalized Laskerian modules

Dawood Hassanzadeh-Lelekaami, Hajar Roshan-Shekalgourabi (2019)

Czechoslovak Mathematical Journal

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Let $ℐ$ be a set of ideals of a commutative Noetherian ring $R$ . We use the notion of $ℐ$ -closure operation which is a semiprime closure operation on submodules of modules to introduce the class of $ℐ$ -Laskerian modules. This enables us to investigate the set of associated prime ideals of certain $ℐ$ -closed submodules of local cohomology modules.

Matlis reflexive and generalized local cohomology modules

Amir Mafi (2009)

Czechoslovak Mathematical Journal

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Let $(R, 𝔪)$ be a complete local ring, $𝔞$ an ideal of $R$ and $N$ and $L$ two Matlis reflexive $R$ -modules with $Supp (L) \subseteq V (𝔞)$ . We prove that if $M$ is a finitely generated $R$ -module, then ${Ext}_{R}^{i} (L, H_{𝔞}^{j} (M, N))$ is Matlis reflexive for all $i$ and $j$ in the following cases: (a) $\dim R / 𝔞 = 1$ ; (b) $cd (𝔞) = 1$ ; where $cd$ is the cohomological dimension of $𝔞$ in $R$ ; (c) $\dim R \leq 2$ . In these cases we also prove that the Bass numbers of $H_{𝔞}^{j} (M, N)$ are finite.

Cohomological dimension filtration and annihilators of top local cohomology modules

Ali Atazadeh, Monireh Sedghi, Reza Naghipour (2015)

Colloquium Mathematicae

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Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration $= {M_{i}}_{i = 0}^{c}$ , where c = cd(,M) and $M_{i}$ denotes the largest submodule of M such that $c d (, M_{i}) \leq i$ . Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module $H_{}^{c} (M)$ , namely $A n n_{R} (H_{}^{c} (M)) = A n n_{R} (M / M_{c - 1})$ . As a consequence, there exists an ideal of R such that $A n n_{R} (H_{}^{c} (M)) = A n n_{R} (M / H ⁰_{} (M))$ . This generalizes the...

Melkersson condition on Serre subcategories

Reza Sazeedeh, Rasul Rasuli (2016)

Colloquium Mathematicae

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Let R be a commutative noetherian ring, let be an ideal of R, and let be a subcategory of the category of R-modules. The condition $C_{}$ , defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to belong to . In this paper, we define and study the class $_{}$ consisting of all modules satisfying $C_{}$ . If and are ideals of R, we get a necessary and sufficient condition for to satisfy $C_{}$ and $C_{}$ simultaneously. We also...

Some results on the local cohomology of minimax modules

Ahmad Abbasi, Hajar Roshan-Shekalgourabi, Dawood Hassanzadeh-Lelekaami (2014)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$ . It is shown that, if $M$ is a non-zero minimax $R$ -module such that $dim Supp H_{I}^{i} (M) \leq 1$ for all $i$ , then the $R$ -module $H_{I}^{i} (M)$ is $I$ -cominimax for all $i$ . In fact, $H_{I}^{i} (M)$ is $I$ -cofinite for all $i \geq 1$ . Also, we prove that for a weakly Laskerian $R$ -module $M$ , if $R$ is local and $t$ is a non-negative integer such that $dim Supp H_{I}^{i} (M) \leq 2$ for all $i < t$ , then ${Ext}_{R}^{j} (R / I, H_{I}^{i} (M))$ and ${Hom}_{R} (R / I, H_{I}^{t} (M))$ are weakly Laskerian for all $i < t$ and all $j \geq 0$ . As a consequence, the set of associated primes of $H_{I}^{i} (M)$ is finite for all $i \geq 0$ , whenever...

Quintasymptotic primes, local cohomology and ideal topologies

A. A. Mehrvarz, R. Naghipour, M. Sedghi (2006)

Colloquium Mathematicae

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Let Φ be a system of ideals on a commutative Noetherian ring R, and let S be a multiplicatively closed subset of R. The first result shows that the topologies defined by ${I_{a}}_{I \in Φ}$ and ${S (I_{a})}_{I \in Φ}$ are equivalent if and only if S is disjoint from the quintasymptotic primes of Φ. Also, by using the generalized Lichtenbaum-Hartshorne vanishing theorem we show that, if (R,) is a d-dimensional local quasi-unmixed ring, then $H_{Φ}^{d} (R)$ , the dth local cohomology module of R with respect to Φ, vanishes if and only if there...

Some bounds for the annihilators of local cohomology and Ext modules

Ali Fathi (2022)

Czechoslovak Mathematical Journal

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Let $𝔞$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a nonnegative integer. Let $M$ and $N$ be two finitely generated $R$ -modules. In certain cases, we give some bounds under inclusion for the annihilators of ${Ext}_{R}^{t} (M, N)$ and $H_{𝔞}^{t} (M)$ in terms of minimal primary decomposition of the zero submodule of $M$ , which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.

On the associated prime ideals of local cohomology modules defined by a pair of ideals

Maryam Jahangiri, Zohreh Habibi, Khadijeh Ahmadi Amoli (2016)

Discussiones Mathematicae General Algebra and Applications

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Let I and J be two ideals of a commutative Noetherian ring R and M be an R-module. For a non-negative integer n it is shown that, if the sets $A s s_{R} (E x t_{R}^{n} (R / I, M))$ and $S u p p_{R} (E x t_{R}^{i} (R / I, H_{I, J}^{j} (M)))$ are finite for all i ≤ n+1 and all j < n, then so is $A s s_{R} (H o m_{R} (R / I, H_{I, J}^{n} (M)))$ . We also study the finiteness of $A s s_{R} (E x t_{R}^{i} (R / I, H_{I, J}^{n} (M)))$ for i = 1,2.

CF-modules over commutative rings

Ahmed Najim, Mohammed Elhassani Charkani (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a commutative ring with unit. We give some criterions for determining when a direct sum of two CF-modules over $R$ is a CF-module. When $R$ is local, we characterize the CF-modules over $R$ whose tensor product is a CF-module.