# On the finiteness properties of local cohomology modules for regular local rings

## TL;DR

This paper investigates the finiteness of associated primes of certain Ext modules involving local cohomology over regular local rings, establishing finiteness results in dimensions up to 5 under specific conditions.

## Contribution

It proves that all homomorphic images of Ext modules involving local cohomology have finitely many associated primes in regular local rings of dimension up to 5, extending previous finiteness results.

## Key findings

- Finiteness of associated primes for Ext modules when dim R ≤ 4 or dim R/ a ≤ 3 with a field
- Finiteness of associated primes for Ext modules when dim R=5 and R contains a field
- Extension of finiteness properties to higher-dimensional regular local rings

## Abstract

Let $\frak a$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\frak a)$. The purpose of this paper is to show that all homomorphic images of the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$, whenever $\dim R \leq4$ or $\dim R/ \frak a \leq 3$ and $R$ contains a field. In addition, we show that if $\dim R=5$ and $R$ contains a field, then the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$.