A noncommutative generalization of Auslander's last theorem

2007

One of the main results of this paper is the characterization of the rings over which all modules are strongly Gorenstein projective. We show that these kinds of rings are very particular cases of the well-known quasi-Frobenius rings. We give examples of rings over which all modules are Gorenstein projective but not necessarily strongly Gorenstein projective.

Journal of Mathematics

Let Δ 0,0 = A A N B B M A B be a Morita ring such that the bimodule homomorphisms are zero. In this paper, we give sufficient conditions for a Δ 0,0 -module X , Y , f , g to be Gorenstein-projective. As an application, we give sufficient conditions when the algebras A and B inherit the strongly CM-freeness of Δ 0,0 .

Algebra Colloquium, 2009

Given a commutative Noetherian local ring (R, 𝔪), it is shown that R is Gorenstein if and only if there exists a system of parameters x1,…,xd of R which generates an irreducible ideal and [Formula: see text] for all t > 0. Let n be an arbitrary non-negative integer. It is also shown that for an arbitrary ideal 𝔞 of a commutative Noetherian (not necessarily local) ring R and a finitely generated R-module M, [Formula: see text] is finitely generated if and only if there exists an 𝔞-filter regular sequence x1,…,xn∈ 𝔞 such that [Formula: see text] for all t > 0.

Proceedings of the American Mathematical Society, 1973

Reiten has demonstrated that the trivial Hochschild extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein local ring. Here it is proved that any commutative extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein ring. Also Gorenstein extensions of a local Cohen-Macaulay ring by a module are studied. Introduction. Suppose A is a commutative ring and that M is an A-module. A commutative extension of A by M is an exact sequence of abelian groups 0-► M-i* E-^> A-*■ 0, where £ is a commutative ring, the map n is a ring homomorphism and the /4-module structure on M is related to (E, i,-n) by the equations ei(x)-i(tr(e)x), for all e e E and all x e M. The / identifies M with an ideal of square zero in E. (On the other hand if 3 is an ideal of square zero in E, then 3 is an F/3-module and 0-«^j-* E-*-E/3->0 is an extension of F/3 by 30 The trivial extension of A by M is the exact sequence 0-► M-U M x A-^U-A->0 where ; is the first coordinate map, where-n is the second projection and where M x A is a ring whose underlying additive structure is the direct sum of abelian groups and whose multiplication is given elementwise by (m, a)(m', a') = (ma' +m'a, aa') for all m,m' eM and all a, a'e A. This extension is denoted by Aix. M. Now suppose A is a commutative noetherian local ring with maximal ideal m. An ^-module of finite type M is a canonical module if it has the

Algebras and Representation Theory, 1999

Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen-Macaulay approximation. Yoshimo extended Auslander's result to local Cohen-Macaulay rings admitting a dualizing module.

Journal of Mathematics

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .

Transactions of the American Mathematical Society, 1996

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 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 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.

Journal of Algebra and Its Applications, 2012

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper, we study the amalgamated duplication ring R I which is introduced by D'Anna and Fontana. It is shown that if R satisfies Serre's condition (Sn) and Ip is a maximal Cohen-Macaulay Rp -module for every p ∈ Spec (R), then R I satisfies Serre's condition (Sn). Moreover if R I satisfies Serre's condition (Sn), then so does R. This gives a generalization of the same result for Cohen-Macaulay rings in [D'Anna, A construction of Gorenstein rings, J. Algebra 306 (2006) 507-519]. In addition it is shown that if R is a local ring and Ann R (I) = 0, then R I is quasi-Gorenstein if and only if b R satisfies Serre's condition (S 2 ) and I is a canonical ideal of R. This result improves the result of D'Anna which is corrected by Shapiro and states that if R is a Cohen-Macaulay local ring, then R I is Gorenstein if and only if the canonical ideal of R exists and is isomorphic to I, provided Ann R (I) = 0.

Czechoslovak Mathematical Journal, 2012

Let R be a commutative Noetherian ring. It is shown that the finitely generated R-module M with finite Gorenstein dimension is reflexive if and only if Mp is reflexive for p ∈ Spec(R) with depth(Rp) 1, and G-dim Rp (Mp) depth(Rp) − 2 for p ∈ Spec(R) with depth(Rp) 2. This gives a generalization of Serre and Samuel's results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for n 2 we give a characterization of n-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every R-module has a k-torsionless cover provided R is a k-Gorenstein ring.

Publications mathématiques de l'IHÉS, 1975

Minimal injective resolutions with applications to dualizing modules and Gorenstein modules Publications mathématiques de l'I.H.É.S., tome 45 (1975), p. 193-215 <http://www.numdam.org/item?id=PMIHES_1975__45__193_0> © Publications mathématiques de l'I.H.É.S., 1975, tous droits réservés. L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/