A criterion for cofiniteness of modules

Abstract

Let $A$ be a commutative noetherian ring, $\mathfrak{a}$ be an ideal of $A$, and $m,n$ be non-negative integers. Let $M$ be an $A$-module such that ${\mathrm{Ext}}_A^i(A/\mathfrak{a},M)$ is finitely generated for all $i\le m+n$. We define a class ${\mathcal{S}}_n(\mathfrak{a})$ of modules and we assume that $H_{\mathfrak{a}}^s(M)\in {\mathcal{S}}_n(\mathfrak{a})$ 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 ${\mathrm{Ext}}_A^i(A/\mathfrak{a},H_{\mathfrak{a}}^{t+s-i}(M))$ 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 $M\in {\mathcal{S}}_n(\mathfrak{a})$ for any ideal $\mathfrak{a}$ of dimension $\le d-1$, then we prove that $M\in {\mathcal{S}}_n(\mathfrak{a})$ for any ideal $\mathfrak{a}$ of $A$.