Matlis reflexive and generalized local cohomology modules

Amir Mafi

Displaying similar documents to “Matlis reflexive and generalized 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...

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

K. Khashyarmanesh, M. Yassi, A. Abbasi (2004)

Colloquium Mathematicae

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Let R be a commutative Noetherian ring. Let and be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the -finiteness dimension of $f_{}^{} (N)$ relative to in the context of generalized local cohomology modules as $f_{}^{} (M, N) : = i n f i \geq 0 | \subseteq \sqrt (0 :_{R} H_{}^{i} (M, N))$ , where M is an R-module. We also show that $f_{}^{} (N) \leq f_{}^{} (M, N)$ for any R-module M. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.

Cofiniteness of generalized local cohomology modules

Kamran Divaani-Aazar, Reza Sazeedeh (2004)

Colloquium Mathematicae

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Let denote an ideal of a commutative Noetherian ring R, and M and N two finitely generated R-modules with pd M < ∞. It is shown that if either is principal, or R is complete local and is a prime ideal with dim R/ = 1, then the generalized local cohomology module $H_{}^{i} (M, N)$ is -cofinite for all i ≥ 0. This provides an affirmative answer to a question proposed in [13].

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...

Artinian cofinite modules over complete Noetherian local rings

Behrouz Sadeghi, Kamal Bahmanpour, Jafar A'zami (2013)

Czechoslovak Mathematical Journal

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Let $(R, 𝔪)$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$ -module. In this paper it is shown that if $𝔭$ is a prime ideal of $R$ such that $dim R / 𝔭 = 1$ and $(0 :_{M} 𝔭)$ is not finitely generated and for each $i \geq 2$ the $R$ -module ${Ext}_{R}^{i} (M, R / 𝔭)$ is of finite length, then the $R$ -module ${Ext}_{R}^{1} (M, R / 𝔭)$ is not of finite length. Using this result, it is shown that for all finitely generated $R$ -modules $N$ with $Supp (N) \subseteq V (I)$ and for all integers $i \geq 0$ , the $R$ -modules ${Ext}_{R}^{i} (N, M)$ are of finite length, if and only if, for all finitely generated $R$ -modules...

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.