A note on the non-Artinianness of top local cohomology modules

TL;DR

This paper investigates whether top local cohomology modules over Noetherian rings are Artinian, providing specific conditions under which they are or are not, especially in the context of three-dimensional local UFDs.

Contribution

It establishes a criterion for the Artinianness of top local cohomology modules over three-dimensional local UFDs, linking it to the cohomological dimension.

Findings

Top local cohomology modules are Artinian if and only if the cohomological dimension is 3 in the specified setting.

Provides new insights into the non-Artinianness of certain top local cohomology modules.

Extends understanding of local cohomology modules in the context of Noetherian local UFDs.

Abstract

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this article, we examine the question of whether an arbitrary top local cohomology module, $\operatorname{H}^{\operatorname{cd}(I,M)}_I(M)$, is Artinian, or not. Several results related to this question are obtained; in particular, we prove that over a Noetherian local unique factorization domain $R$ of dimension three, for a finitely generated faithful module $M$, a top local cohomology module is Artinian if and only if $\operatorname{cd}(I,M)= 3$.