Cofiniteness with respect to two ideals and local cohomology
type: Item
instance of: scholarly article
title: Cofiniteness with respect to two ideals and local cohomology
zbMATH Open document ID: 1412.13022
author: Malyha Nazari
author: Reza Sazeedeh
published in: Algebras and Representation Theory
publication date: 2019-03-27T00:00:00Z
review text: Let $A$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $A$, and $n$ a non-negative integer. In this paper, the authors say that $A$ admits $\text{P}_n(\mathfrak{a})$ if for any $A$-module $M$ with $\text{Supp}_A(M)\subseteq \text{V}(\mathfrak{a})$, finiteness of modules $\text{Ext}^i_A(A/\mathfrak{a}, M)$ for all $i\leq n$ implies that $M$ is $\mathfrak{a}$-cofinite. Recall that an $A$-module $M$ is said to be $\mathfrak{a}$-cofinite if $\text{Supp}_A(M)\subseteq \text{V}(\mathfrak{a})$ and $\text{Ext}^i_A(A/\mathfrak{a}, M)$ is finite for all $i$. They prove that if $d:= \dim(A)\geq 1$ and $A$ admits $\text{P}_{d- 1}(\mathfrak{a})$ for all ideals $\mathfrak{a}$ of $A$ with $\dim(A/\mathfrak{a})\leq d- 1$, then $A$ admits $\text{P}_{d- 1}(\mathfrak{a})$ for all ideals $\mathfrak{a}$ of $A$. As a consequence, $A$ satisfies $\text{P}_{1}(\mathfrak{a})$ (resp. $\text{P}_{2}(\mathfrak{a})$) for all ideals $\mathfrak{a}$ of $A$ if $\dim(A)= 2$ (resp. $A$ is local and $\dim(A)= 3$). In the case that $\dim(A)\leq 3$ (resp. $A$ is local and $\dim(A/\mathfrak{a})= 2$), it is shown that the full subcategory of $\mathfrak{a}$-cofinite $A$-modules of the category of $A$-modules is Abelian if and only if $(0:_{\text{coker}f}\mathfrak{a})$ (resp. $\text{Ext}^2_A(A/\mathfrak{a}, \text{ker}f)$) is finite for any homomorphism of $\mathfrak{a}$-cofinite $A$-modules $f$. Let $M$ be an $A$-module and $\mathfrak{b}$ an ideal of $A$ with $\mathfrak{b}\subseteq \mathfrak{a}$. The authors prove that $\text{Ext}^i_A(A/\mathfrak{a}, M)$ is finite for all $i\leq n$ when $\text{Ext}^i_A(A/\mathfrak{b}, M)$ is $\mathfrak{a}$-cofinite for all $i\leq n$. In particular, $M$ is $\mathfrak{a}$-cofinite whenever $A$ admits $\text{P}_n(\mathfrak{a})$, $\text{Supp}_A(M)\subseteq \text{V}(\mathfrak{a})$, and $\text{Ext}^i_A(A/\mathfrak{b}, M)$ is $\mathfrak{a}$-cofinite for all $i\leq n$. When $\text{Ext}^i_A(A/\mathfrak{a}, M)$ is finite for all $i\leq n+ 1$, they show that if $\dim(A/\ma…
reviewed by: Alireza Vahidi
Mathematics Subject Classification ID: 13D45
zbMATH DE Number: 7042791
zbMATH Keywords: cofinite module
zbMATH Keywords: local cohomology
MaRDI profile type: Publication
full work available at URL: s10468-018-9771-y
OpenAlex ID: W2793105845
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DOI: 10.1007/S10468-018-9771-Y
description: scientific article; zbMATH DE number 7042791