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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Foxby duality and Gorenstein injective
and projective modules


Authors: Edgar E. Enochs, Overtoun M. G. Jenda and Jinzhong Xu
Journal: Trans. Amer. Math. Soc. 348 (1996), 3223-3234
MSC (1991): Primary 13C10, 13C11; Secondary 13C99
MathSciNet review: 1355071
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Abstract: In 1966, Auslander introduced the notion of the $G$-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of $G$-dimensions. It seemed appropriate to call the modules with $G$-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of $G$-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite $G$-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.


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Additional Information

Edgar E. Enochs
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Overtoun M. G. Jenda
Affiliation: Department of Discrete Mathematics and Statistics, Auburn University, Auburn, Alabama 36849-5307

Jinzhong Xu
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01624-8
PII: S 0002-9947(96)01624-8
Received by editor(s): September 7, 1994
Received by editor(s) in revised form: October 2, 1995
Article copyright: © Copyright 1996 American Mathematical Society