A criterion for cofiniteness of modules

Abstract

Let $A$ be a commutative noetherian ring, $\mathfrak{a}$ be an ideal of $A$, $m,n$ be non-negative integers and let $M$ be an $A$-module such that $\Ext^i_A(A/\frak a,M)$ is finitely generated for all $i\le m+n$. We define a class $\cS_n(\frak a)$ of modules and we assume that $H_{\frak a}^s(M)\in\cS_{n}(\frak a)$ for all $s\le m$. We show that $H_{\mathfrak{a}}^s(M)$ is $\mathfrak{a}$-cofinite for all $s\le m$ if either $n=1$ or $n\ge 2$ and $\Ext_A^{i}(A/\frak a,H_{\frak a}^{t+s-i}(M))$ is finitely generated for all $1\le t\le n-1$, $i\le t-1$ and $s\le m$. If $A$ is a ring of dimension $d$ and $M\in\cS_n(\frak a)$ for any ideal $\mathfrak{a}$ of dimension $\le d-1$, then we prove that $M\in\cS_n(\frak a)$ for any ideal $\mathfrak{a}$ of $A$.