Acta Mathematica Vietnamica

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Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings

Asghar Farokhi , $icon-email$ Alireza Nazari

Abstract

Let $(R, m)$ be a Noetherian local ring, $M$ a non-zero finitely generated $R$ -module, and let $I$ be an ideal of $R$ . In this paper, we establish some new properties of local cohomology modules $H_{I}^{i} (M)$ , $i \geq 0$ . In particular, we show that if $R$ is catenary, $M$ an equidimensional $R$ -module of dimension $d$ , and $x_{1}, x_{2}, \dots, x_{t}$ is an $I$ -filter regular sequence on $M$ , then $(0 :_{H_{I}^{d - j} (\frac{M}{⟨ x_{1}, x_{2}, \dots, x_{i - 1} ⟩ M})} x_{i})$ is $I$ -cofinite for all $i = 1, 2, \dots, t$ and all $i \leq j \leq t$ if and only if $H_{I}^{d - j} (\frac{M}{⟨ x_{1}, x_{2}, \dots, x_{i - 1} ⟩ M})$ is $I$ -cofinite for all $i = 1, 2, \dots, t$ and all $i \leq j \leq t$ . Also we study the cofiniteness of local cohomology modules over homomorphic image of Cohen-Macaulay rings and we show that $\frac{H_{I}^{W (I, M)} (M)}{I H_{I}^{W (I, M)} (M)}$ has finite support, where $W (I, M) := Max {i : H_{I}^{i} (M) ~is not weakly Laskerian} .$

Acta Mathematica Vietnamica

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Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings

Abstract