Ee. Enochs et al., FOXBY DUALITY AND GORENSTEIN INJECTIVE AND PROJECTIVE-MODULES, Transactions of the American Mathematical Society, 348(8), 1996, pp. 3223-3234
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.