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Artinian and Non-Artinian Local Cohomology Modules

  Published:2011-03-11
 Printed: Dec 2011
  • Mohammad T. Dibaei,
    Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran
  • Alireza Vahidi,
    Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran
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Abstract

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 $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the non-empty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M) $ is not artinian for some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.
Keywords: local cohomology modules, cohomological dimensions, Bass numbers local cohomology modules, cohomological dimensions, Bass numbers
MSC Classifications: 13D45, 13E10 show english descriptions Local cohomology [See also 14B15]
Artinian rings and modules, finite-dimensional algebras
13D45 - Local cohomology [See also 14B15]
13E10 - Artinian rings and modules, finite-dimensional algebras
 

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