FOXBY DUALITY AND GORENSTEIN INJECTIVE AND PROJECTIVE-MODULES

In 1966, Auslander introduced the notion of the G-dimension of a finit ely generated module over a Cohen-Macaulay noetherian ring and found t he basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Ausl ander and Bridger (Mem. Amer. Math. Sec., vol, 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dimensions. It seemed appro priate to call the modules with G-dimension 0 Gorenstein projective, s o the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611-633 and were shown to have pr operties predicted by Auslander's results. The way we define Gorenstei n injective mod;les can be dualized, and so we can define Gorenstein p rojective 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 To rrecillas. However, to get good results it was necessary to take the b ase 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 generat ed modules in one category are precisely those of finite G-dimension. We extend this result to modules which are not necessarily finitely ge nerated and also prove the dual result, i.e. we characterize the modul es 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 thos e having finite Gorenstein injective dimensions. We note that this dua lity then allows us to extend many of our results to the original Ausl ander setting.