# A New Outlook on Cofiniteness

## TL;DR

This paper investigates conditions under which local cohomology modules are cofinite, establishing abelian properties of certain subcategories and extending results to rings with small dimension or cohomological dimension.

## Contribution

It proves the abelian nature of the subcategory of cofinite modules when cohomological dimension is at most 1 and extends cofinite properties of local cohomology modules to rings with small dimension.

## Key findings

- Subcategory of cofinite modules is abelian when $ ext{cd}(rak{a}, R) extless= 1.
- Local cohomology modules are cofinite under certain dimensional constraints.
- Answers to key questions about cofinite modules in specific cases.

## Abstract

Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the technique of way-out functors, we show that if $\cd(\mathfrak{a},R)\leq 1$, or $\dim(R/\mathfrak{a}) \leq 1$, or $\dim(R) \leq 2$, then the local cohomology module $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i \in \mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3.