TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 8, August 1996 FOXBY DUALITY AND GORENSTEIN INJECTIVE AND PROJECTIVE MODULES EDGAR E. ENOCHS, OVERTOUN M.G. JENDA, AND JINZHONG XU 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 at- tempted 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 Aus- lander’s results. The way we define Gorenstein injective modules can be du- alized, and so we can define Gorenstein projective modules (i.e. modules of G-dimension 0) whether the modules are finitely generated or not. The in- vestigation 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 hav- ing 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. 1. Introduction Unless stated otherwise, R will be a local Cohen-Macaulay ring of Krull dimen- sions d admitting a dualizing module D and with residue field k. The Matlis dual of M will be denoted M v . For any P ∈ Spec(R), we note that D P is a dualizing module for R P . Also, ˆ D is dualizing for ˆ R (the completion of R) and D/ID is dualizing for R/I if I is generated by an R-sequence. If the sequence is maximal, then dim R/I = 0, so D/ID ∼ = E R/I (k). Since D P = 0 for all P ∈ Spec(R), Hom(D, M ) = 0 implies M = 0. So then D ⊗ N = 0 implies N = 0 since (D ⊗ N ) v = Hom(D, N v ). Received by the editors September 7, 1994 and, in revised form, October 2, 1995. 1991 Mathematics Subject Classification. Primary 13C10, 13C11; Secondary 13C99. c 1996 American Mathematical Society 3223 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use