# A criterion for cofiniteness of modules

### Mohammad Khazaei

Urmia University, Iran### Reza Sazeedeh

Urmia University, Iran

## Abstract

Let $A$ be a commutative noetherian ring, $a$ be an ideal of $A$, and $m,n$ be non-negative integers. Let $M$ be an $A$-module such that $Ext_{A}(A/a,M)$ is finitely generated for all $i≤m+n$. We define a class $S_{n}(a)$ of modules and we assume that $H_{a}(M)∈S_{n}(a)$ for all $s≤m$. We show that $H_{a}(M)$ is $a$-cofinite for all $s≤m$ if either $n=1$ or $n≥2$ and $Ext_{A}(A/a,H_{a}(M))$ is finitely generated for all $1≤t≤n−1$, $i≤t−1$ and $s≤m$. If $A$ is a ring of dimension $d$ and $M∈S_{n}(a)$ for any ideal $a$ of dimension $≤d−1$, then we prove that $M∈S_{n}(a)$ for any ideal $a$ of $A$.

## Cite this article

Mohammad Khazaei, Reza Sazeedeh, A criterion for cofiniteness of modules. Rend. Sem. Mat. Univ. Padova 151 (2024), pp. 201–211

DOI 10.4171/RSMUP/128