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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Gorenstein injective modules and local cohomology


Author: Reza Sazeedeh
Journal: Proc. Amer. Math. Soc. 132 (2004), 2885-2891
MSC (2000): Primary 13D05, 13D45
Published electronically: May 21, 2004
MathSciNet review: 2063107
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Abstract: In this paper we assume that $R$ is a Gorenstein Noetherian ring. We show that if $(R,\mathfrak{m})$ is also a local ring with Krull dimension $d$ that is less than or equal to 2, then for any nonzero ideal $\mathfrak{a}$of $R$ , $H_{\mathfrak{a}}^d(R)$ is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if $R$is a Gorenstein ring, then for any $R$-module $M$ its local cohomology modules can be calculated by means of a resolution of $M$ by Gorenstein injective modules. Also we prove that if $R$ is $d$-Gorenstein, $M$ is a Gorenstein injective and $\mathfrak a$is a nonzero ideal of $R$, then ${\Gamma}_{\mathfrak{a}}(M)$ is Gorenstein injective.


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Additional Information

Reza Sazeedeh
Affiliation: Institute of Mathematics, University for Teacher Education, 599, Taleghani Avenue, Tehran 15614, Iran – and – Department of Mathematics, Urmia University, Iran

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07461-1
PII: S 0002-9939(04)07461-1
Keywords: Cover, Gorenstein injective, Gorenstein projective, local cohomology
Received by editor(s): December 5, 2002
Received by editor(s) in revised form: June 21, 2003
Published electronically: May 21, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society