Top Local Cohomology Modules Are Almost Always Non-Artinian

TL;DR

This paper investigates when top local cohomology modules are Artinian or non-Artinian in Noetherian local rings, providing comprehensive results with specific exceptions and conditions.

Contribution

It offers a complete characterization of the Artinianness of top local cohomology modules for weakly finite or coatomic modules, resolving many cases and identifying key exceptions.

Findings

Top local cohomology modules are almost always non-Artinian.

Complete characterization of Artinianness for most cases.

Identifies specific conditions where Artinianness can occur.

Abstract

Let $(R,\mathfrak m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M$ a weakly finite or a coatomic $R$-module of dimension $n$. In this article, we resolve the Artinianness and non-Artinianness of top local cohomology modules, $H^{cd(I,M)}_I(M)$, in all cases except in the case $cd(I,M)=n-1$ and $dim(R/{I+Ann(M)})>1$ for which we have some sorter results under certain conditions.