Top Local Cohomology Modules Are Almost Always Non-Artinian

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.