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On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules

Published online by Cambridge University Press:  20 November 2018

Majid Rahro Zargar  and
Hossein Zakeri
Majid Rahro Zargar
Affiliation:
Department of Advanced Technologies, University of Mohaghegh ardabili, Namin, Ardabil, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran e-mail: m.zargar@uma.ac.ir
Hossein Zakeri
Affiliation:
Faculty ofMathematical Sciences and Computer, Kharazmi University, 599 Taleghani Ave., Tehran 15608, Iran e-mail: zargar9077@gmail.com e-mail: zakeri@khu.ac.ir
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Abstract

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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 $\text{f}{{\text{d}}_{R}}X,\,\text{Gf}{{\text{d}}_{R}}X$ and ${{\text{G}}_{\text{C}}}-\text{f}{{\text{d}}_{R}}\,X\,\text{by}\,\text{T}\left( X \right)$. Let $M$ be an $R$-module such that $\text{H}_{\mathfrak{a}}^{i}\left( M \right)\,=\,0$ for all $i\,\ne \,n$. It is proved that if $T\left( M \right)\,<\,\infty$, then $\text{T}\left( \text{H}_{\mathfrak{a}}^{n}\left( M \right) \right)\,\le \,\text{T}\left( M \right)\,+\,n$, 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

13D0513D4518G20flat dimensionGorenstein injective dimensionGorenstein flat dimensionlocal cohomologyrelative Cohen–Macaulay modulesemidualizing module
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 403 - 416
Copyright
Copyright © Canadian Mathematical Society 2016

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