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Artinian and Non-Artinian Local Cohomology Modules

Published online by Cambridge University Press:  20 November 2018

Mohammad T. Dibaei  and
Alireza Vahidi
Mohammad T. Dibaei
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran, andSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Irane-mail: dibaeimt@ipm.ir
Alireza Vahidi
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran, andPayame Noor University (PNU), Irane-mail: vahidi.ar@gmail.com
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Abstract

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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 $i\,\ge \,r$.

Keywords

13D4513E10local cohomology modulescohomological dimensionsBass numbers
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 4 , 01 December 2011 , pp. 619 - 629
Copyright
Copyright © Canadian Mathematical Society 2011

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