On the factorial case of Huneke's conjecture for local cohomology modules

TL;DR

This paper advances the understanding of Huneke's conjecture by proving it for factorial domains in dimensions 4 and 5, and for regular fields in dimension 6, using algebraic and geometric tools.

Contribution

It establishes the conjecture for factorial domains in dimensions 4 and 5, and for regular rings containing a field in dimension 6, extending previous partial results.

Findings

Confirmed Huneke's conjecture for factorial domains in dimension 4.

Extended the conjecture to dimension 5 under additional conditions.

Applied Hartshorne-Lichtenbaum vanishing theorem for dimension 6.

Abstract

A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional Noetherian local ring $R$, local cohomology modules of finitely generated $R$-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if $d\leq 3$, and witnessed some progress in special cases for higher $d$. In this paper, assuming that $R$ is a factorial domain, we establish the case $d=4$, and under different additional conditions (in a couple of results) also the case $d=5$. Finally, when $R$ is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$.