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

 

Canonical filtrations of Gorenstein injective modules


Authors: Edgar E. Enochs and Zhaoyong Huang
Journal: Proc. Amer. Math. Soc. 139 (2011), 2415-2421
MSC (2010): Primary 13D07, 16E30
Published electronically: December 9, 2010
MathSciNet review: 2784806
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Abstract: The principle ``Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra'' was given by Henrik Holm. There is a remarkable body of evidence supporting this claim. Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective.

Our main result is in support of the principle. Over commutative, noetherian rings injective modules have direct sum decompositions into indecomposable modules. We will show that Gorenstein injective modules over Gorenstein rings of finite Krull dimension have filtrations analogous to those provided by these decompositions. This result will then provide us with the tools to prove that all tensor products of Gorenstein injective modules over these rings are Gorenstein injective.


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Additional Information

Edgar E. Enochs
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: enochs@ms.uky.edu

Zhaoyong Huang
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, People’s Republic of China
Email: huangzy@nju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10686-X
PII: S 0002-9939(2010)10686-X
Keywords: Gorenstein injective modules, torsion products, filtrations
Received by editor(s): August 7, 2009
Received by editor(s) in revised form: July 5, 2010
Published electronically: December 9, 2010
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.