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On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules

  Published:2016-02-03
 Printed: Jun 2016
  • Majid Rahro Zargar,
    Department of Advanced Technologies, University of Mohaghegh ardabili, Namin, Ardabil, Iran
  • Hossein Zakeri,
    Faculty of mathematical sciences and computer, Kharazmi University, 599 Taleghani Avenue, Tehran 15618, Iran
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Abstract

Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\mathfrak{d}_R X$, $\operatorname{\mathsf{Gfd}}_R X$ and $\operatorname{\mathsf{G_C-fd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$-module such that $\operatorname{H}_{\mathfrak{a}}^i(M)=0$ for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then $\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings.
Keywords: flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative Cohen-Macaulay module, semidualizing module flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative Cohen-Macaulay module, semidualizing module
MSC Classifications: 13D05, 13D45, 18G20 show english descriptions Homological dimension
Local cohomology [See also 14B15]
Homological dimension [See also 13D05, 16E10]
13D05 - Homological dimension
13D45 - Local cohomology [See also 14B15]
18G20 - Homological dimension [See also 13D05, 16E10]
 

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