# Modules cofinite and weakly cofinite with respect to an ideal

## TL;DR

This paper investigates the properties of modules cofinite and weakly cofinite relative to an ideal in a Noetherian ring, providing new characterizations, conditions for cofiniteness, and categorical structure insights.

## Contribution

It offers new criteria for cofiniteness and weak cofiniteness of modules, especially in dimension one, and establishes the category of weakly cofinite modules as a full Abelian subcategory.

## Key findings

- Characterization of cofinite modules via Ext modules for dimension one ideals.
- Conditions ensuring weakly cofinite modules are also cofinite.
- The category of all weakly cofinite modules forms a full Abelian subcategory.

## Abstract

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $\frak a$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $\frak a$, if and only if, $\Ext^i_R(R/\frak a,M)$ is finitely generated for all $i\leq {\rm cd}(\frak a,M)+1$, whenever $\dim R/\frak a=1$. In addition, we show that if $M$ is finitely generated and $H^i_{\frak a}(M)$ are weakly Laskerian for all $i\leq t-1$, then $H^i_{\frak a}(M)$ are ${\frak a}$-cofinite for all $i\leq t-1$ and for any minimax submodule $K$ of $H^{t}_{\frak a}(M)$, the $R$-modules $\Hom_R(R/{\frak a}, H^{t}_{\frak a}(M)/K)$ and $\Ext^{1}_R(R/{\frak a}, H^{t}_{\frak a}(M)/K)$ are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely for such ideals it suffices that the two first $\Ext$-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result we deduce that the category of all ${\frak a}$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.