# A note on Abelian categories of cofinite modules

## TL;DR

This paper proves that under certain conditions, the category of I-cofinite modules over a Noetherian ring forms an Abelian subcategory, answering longstanding questions in the theory of cofinite modules.

## Contribution

It establishes that the category of I-cofinite modules is Abelian when q(I,R) ≤ 1, resolving questions posed by Hartshorne and the author.

## Key findings

- Confirmed the Abelian property of I-cofinite modules under specific conditions.
- Extended understanding of the structure of cofinite modules in commutative algebra.
- Provided affirmative answers to open questions in the theory of cofinite modules.

## Abstract

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. In this article we answer affirmatively a question raised by the present author in \cite{B2}. Also, as an immediate consequence of this result it is shown that the category of all $I$-cofinite $R$-modules $\mathscr{C}(R, I)_{cof}$ is an Abelian subcategory of the category of all $R$-modules, whenever $q(I,R)\leq 1$. These assertions answer affirmatively a question raised by R. Hartshorne in [{\it Affine duality and cofiniteness}, Invent. Math. {\bf9}(1970), 145-164], in some special cases.