Artinian and non-artinian local cohomology modules

TL;DR

This paper investigates the relationships between cohomological dimensions of modules over noetherian rings, characterizes generalized Cohen-Macaulay modules via local cohomology finiteness, and analyzes artinian properties of local cohomology modules.

Contribution

It establishes new relations between cohomological dimensions, characterizes generalized Cohen-Macaulay modules through local cohomology finiteness, and describes properties of maximal ideals where local cohomology is not artinian.

Findings

Cohomological dimensions relate via intersections and sums of ideals.

Modules are generalized Cohen-Macaulay if certain local cohomology modules have finite length.

Maximal ideals where local cohomology is not artinian are prime, with finite Bass numbers.

Abstract

Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\fa$ and $\fb$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\fa, \fb$, $\fa\cap\fb$ and $\fa+ \fb$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\fa$ such that all local cohomology modules of $M$ with respect to $\fa$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$, any maximal element $\fq$ of the non-empty set of ideals $\{\fa$ : $\H_\fa^i(M)$ is not artinian for some $i$, $i\geq r$$\}$ is a prime ideal and that all Bass numbers of $\H_\fq^i(M)$ are finite for all $i\geq r$.