Overtoun M. G. Jenda

Foxby duality and Gorenstein injective and projective modules

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 article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module. AMS 2000 Mathematics subject classification: Primary 16D20

Cotorsion Pairs in C(R-Mod)

In [8] Salce introduced the notion of a cotorsion pair (A,B) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proved useful in a variety of settings. In this article we will consider complete cotorsion pairs (C,D) in the category C(R-Mod) of complexes of left R-modules over some ring R. If (C,D) is such a pair, and if C is closed under taking suspensions, we will show when we regard K(C) and K(D) as subcategories of the homotopy category K(RMod), then the embedding functors K(C) → K(R-Mod) and K(D) → K(R-Mod) have left and right adjoints, respectively. In finding examples of such pairs, we will describe a procedure for using Hoveys results in [5] to find a new model structure on C(R-Mod).

Ω-Gorenstein Projective and Flat Covers and Ω-Gorenstein Injective Envelopes

Abstract In this paper, we show that if R is a local Cohen–Macaulay ring admitting a dualizing module Ω, then Ω-Gorenstein projective and flat covers and Ω-Gorenstein injective envelopes exist for certain modules. These results generalize the well known results for local Gorenstein rings.

COPURE INJECTIVE MODULES

Abstract A module is said to be copure injective if it is injective with respect to all modules A ⊂ B with B/A injective. We first characterize submodules that have the extension property with respect to copure injective modules. Then we characterize commutative rings with finite self injective dimension in terms of copure injective modules. Finally, we show that the quotient categories of reduced copure injective modules and reduced h- divisible modules are isomorphic.

Covers and Envelopes by V-Gorenstein Modules

ABSTRACT In this article, we study V-Gorenstein modules relatives to a dualizing module (Enochs et al., in press). These modules constitute a generalization of the well-known Gorenstein modules and at the same time an extension to the noncommutative case of Ω-Gorenstein modules (cf. Enochs and Jenda, 2000a, in press). We show that, under hypothesis of finiteness of projective dimension for flat modules (a property that holds for many rings), covers and envelopes for these classes of modules exist.

GORENSTEIN AND Ω-GORENSTEIN INJECTIVE COVERS AND FLAT PREENVELOPES

ABSTRACT The aim of this paper is to show that if R is a commutative local Cohen–Macaulay ring admitting a dualizing module Ω , then Gorenstein and Ω -Gorenstein injective covers and Gorenstein and Ω -Gorenstein flat preenvelopes exist for certain classes of R -modules.

A Generalization of Auslander's Last Theorem

Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen–Macaulay approximation. Yoshimo extended Auslanders result to local Cohen–Macaulay rings admitting a dualizing module.Over a local Gorenstein ring the finitely generated maximal Cohen–Macaulay modules are the finitely generated Gorenstein projective modules so in fact Auslanders theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslanders theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslanders sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslanders theorem when R is Gorenstein.

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapGn →Gn/HomR(M, G) is an injective precover if ExtR1(M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depthRR ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depthRR > 0.

Tensor and torsion products of injective modules

Yoneda raised the question of whether the tensor product of injective modules is injective. Ishikawa showed that for a commutative noetherian ring R this is the case if the injective envelope E(R) of R is flat. The latter condition is equivalent to R being generically Gorenstein (i.e., R>p Gorenstein for every minimal prime ideal p). We use a version of this characterization to prove the converse of Ishikawas result and raise the question of the injectivity of the higher torsion products of injective modules We show that for a commutative noetherian ring R these products are injective if and only if R is Gorenstein. More precisely, we prove that for a Gorenstein ring R and a prime ideal p ⊂ R, Tor, (E(R,p is 0 unless i = ht p and in this case is isomorphic to E(R/p).