A criterion for cofiniteness of modules

TL;DR

This paper establishes criteria for the cofiniteness of local cohomology modules over noetherian rings, linking Ext-finiteness conditions and module classes to cofiniteness properties.

Contribution

It introduces a new class of modules and provides conditions under which local cohomology modules are $rak a$-cofinite, extending previous results in cohomology theory.

Findings

Local cohomology modules are $rak a$-cofinite under certain Ext-finiteness conditions.

The class $ ext{S}_n(rak a)$ helps characterize cofiniteness of modules.

Results apply to modules over rings of bounded dimension.

Abstract

Let $A$ be a commutative noetherian ring, $\frak a$ 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 $i\leq m+n$. 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 $s\leq m$. We show that $H_{\frak a}^s(M)$ is $\frak a$-cofinite for all $s\leq m$ if either $n=1$ or $n\geq 2$ and $\Ext_A^{i}(A/\frak a,H_{\frak a}^{t+s-i}(M))$ is finitely generated for all $1\leq t\leq n-1$, $i\leq t-1$ and $s\leq m$. If $A$ is a ring of dimension $d$ and $M\in\cS_n(\frak a)$ for any ideal $\frak a$ of dimension $\leq d-1$, then we prove that $M\in\cS_n(\frak a)$ for any ideal $\frak a$ of $A$.