Artinian and non-artinian local cohomology modules

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 $0\le r<{\dim }_R(M)$, any maximal element $\fq$ of the non-empty set of ideals $\{\fa$ : $\H_\fa^i(M)$ is not artinian for some $i$, $i\ge r$$\}$ is a prime ideal and that all Bass numbers of $\H_\fq^i(M)$ are finite for all $i\ge r$.

10 pages, to appear in Canadian Mathematical Bulletin

References in corpus (1)

• Local Cohomology and Matlis duality

Cited by in corpus (1)

• Some results on generalized local cohomology modules