Gorenstein injective modules and local cohomology
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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.References
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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
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2885-2891
- MSC (2000): Primary 13D05, 13D45
- DOI: https://doi.org/10.1090/S0002-9939-04-07461-1
- MathSciNet review: 2063107