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 | References | Similar Articles | Additional Information

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

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.