Artinian and Non-Artinian Local Cohomology Modules

Dibaei, Mohammad T. ; Vahidi, Alireza

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 619-629

Résumé

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

Détail

DOI : 10.4153/CMB-2011-042-5

Mots-clés : 13D45, 13E10, local cohomology modules, cohomological dimensions, Bass numbers