Foxby duality and Gorenstein injective and projective modules

Abstract

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 CohenMacaulay 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.

DUALIZING MODULES AND n-PERFECT RINGS

“…In this section we characterize Gorenstein modules in terms of the so-called Auslander and Bass classes and generalize results obtained in [9]. We point out that similar results have been recently obtained for the commutative case in [4].…”

“…In this article we consider a general notion of a dualizing bimodule for a pair of rings S and R. We use this terminology since we want to generalize the main result of [9] to the non-commutative setting, that is, we want to characterize the Auslander and Bass classes which arise in this situation in terms of the Gorenstein injective and projective dimensions. We note that the language of cotilting theory is also appropriate in this setting.…”

“…and V R are finitely generated. If R is a Cohen-Macaulay local ring of Krull dimension d admitting a dualizing module Ω (see[9]), then Ω is a dualizing module in this sense. Let R and S be strongly graded rings over finite groups, right and left Noetherian, respectively, and let Se V Re be a dualizing module (for R e and S e ).…”

Ascent Properties of Auslander Categories

“…The relation between Auslander categories and Gorenstein dimensions is established in [19,24,46]. If D is a dualizing complex for R, then…”

“…One theme played in [8] is the following: (I) Results for Auslander categories have implications for Gorenstein dimensions. This is based on the realization that Auslander categories and Gorenstein homological dimensions are close kin [19,24]. The latter were introduced much earlier by Auslander and Bridger [3,4] and Enochs, Jenda, and Torrecillas [21,23].…”