# Cofiniteness over Noetherian complete local rings

## TL;DR

This paper generalizes a result of Hartshorne on socle dimensions of local cohomology modules over regular local rings and characterizes ideals with cofinite local cohomology over Noetherian complete local rings.

## Contribution

It extends Hartshorne's theorem to a broader class of modules and provides a characterization of ideals with cofinite local cohomology in complete local rings.

## Key findings

- Socle of H^2_{(u,v)S}(N) is infinite dimensional for certain modules.
- Characterization of ideals with all local cohomology modules cofinite.
- Generalization of Hartshorne's result to regular local rings of dimension 4.

## Abstract

In this paper we prove the following generalization of a result of Hartshorne: Let $(S,\n)$ be a regular local ring of dimension $4$. Assume that $x,y,u,v$ is a regular system of parameters for $S$ and $a:=xu+yv$. Then for each finitely generated $S$-module $N$ with $\Supp N=V(aS)$ the socle of $H^2_{(u,v)S}(N)$ is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring $(R,\m)$, we characterize the class of all ideals $I$ of $R$ with the property that, for every finitely generated $R$-module $M$, the local cohomology modules $H^i_I(M)$ are $I$-cofinite for all $i\geq 0$.