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Melkersson condition on Serre subcategories

Volume 144 / 2016

Reza Sazeedeh, Rasul Rasuli Colloquium Mathematicum 144 (2016), 289-300 MSC: 13C60, 13D45. DOI: 10.4064/cm6384-9-2015 Published online: 14 April 2016

Abstract

Let $R$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $R$, and let $\mathcal {S}$ be a subcategory of the category of $R$-modules. The condition $C_{\mathfrak a}$, defined for $R$-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to $\mathfrak a$ belong to $\mathcal {S}$. In this paper, we define and study the class $\mathcal {S}_{\mathfrak a}$ consisting of all modules satisfying $C_{\mathfrak a}$. If $\mathfrak a$ and $\mathfrak b$ are ideals of $R$, we get a necessary and sufficient condition for $\mathcal {S}$ to satisfy $C_{\mathfrak a}$ and $C_{\mathfrak b}$ simultaneously. We also find some sufficient conditions under which $\mathcal {S}$ satisfies $C_{\mathfrak a}$. As an application, we investigate when local cohomology modules lie in a Serre subcategory.

Authors

  • Reza SazeedehDepartment of Mathematics
    Urmia University
    P.O. Box 165, Urmia, Iran
    and
    School of Mathematics
    Institute for Research in Fundamental Sciences (IPM)
    P.O. Box 19395-5746, Tehran, Iran
    e-mail
  • Rasul RasuliMathematics Department
    Faculty of Science
    Payame Noor University (PNU)
    Tehran, Iran
    e-mail

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