Characterizing When the Category of Gorenstein Projective Modules is an Abelian Category

We find sufficient and necessary conditions for the category of Gorenstein projective modules of an artin algebra being an abelian category, and give another proof for the Auslander–Solberg correspondence which demonstrates the concrete form of the category of Gorenstein projective modules. Then we find a characterization for this category of Gorenstein projective modules. At last we give an example of this correspondence.     

Gorenstein tilting pairs

Abstract

Let A be an n-Gorenstein ring. Employing the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we introduce the concept of Gorenstein tilting pair. Moreover, we give a simple characterization on Gorenstein tilting pair, which shows that Gorenstein cotilting and tilting modules are special examples of Gorenstein tilting pair.     

A note on relative tilting modules

Bazzoni had given a simple characterization of infinitely generated n-tilting modules. Though her method is even inapplicable to classical n-tilting modules over Artin algebras, we show in this note that a similar characterization does hold for (finitely generated) relative n-tilting modules introduced by Auslander and Solberg for Artin algebras, by using a different method. We also present some applications.     

Tilting Preenvelopes and Cotilting Precovers

We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes.     

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.     

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 Auslander's 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 Auslander's theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslander's theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslander's sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslander's theorem when R is Gorenstein.     

Tilting subcategories in extriangulated categories

Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.     

On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective–injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.     

ω-Gorenstein injective modules and dimensions

Gorenstein injective modules and dimensions have been studied extensively by many authors. In this paper, we investigate Gorenstein injective modules and dimensions relative to a Wakamatsu tilting module.     

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P₁P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rej_p-torsion parts, and the elements of L′ by their Tr_p-torsion-free parts.     

Foxby duality and Gorenstein injective and projective modules

In 1966, Auslander introduced the notion of the -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 -dimensions. It seemed appropriate to call the modules with -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 -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 -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.

     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.     

Selforthogonal modules with finite injective dimension

The category consisting of finitely generated modules which are left orthogonal with a cotilting bimodule is shown to be functorially finite. The notion of left orthogonal dimension is introduced, and then a necessary and sufficient condition of selforthogonal modules having finite injective dimension and a characterization of cotilting modules are given.     

Strict Mittag-Leffler Conditions and Gorenstein Modules

In this paper, firstly, we characterize some rings by strict Mittag-Leffler conditions. Then, we investigate when Gorenstein projective modules are Gorenstein flat by employing tilting modules and cotorsion pairs. Finally, we study the direct limits of Gorenstein projective modules.     

When are definable classes tilting and cotilting classes?

It is known that tilting classes are of finite type, while cotilting classes are not always of cofinite type. We investigate this phenomenon. By using a bijection between definable classes of left modules and definable classes of right modules, we prove that it reflects the asymmetry existing between the notions of covers and envelopes or, otherwise stated, right and left approximations.In particular we show that there exist definable torsion classes containing the injective modules which are not tilting classes.     

Maximal modifications and Auslander–Reiten duality for non-isolated singularities

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of \(\operatorname{Spec}R\) , in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.     

Relative syzygies and grade of modules

Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen-Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, we establish a relation between the injective dimension of a generalized tilting module ω and the finitistic dimension with respect to ω.     

Weak Morita Duality

Abstract

The notion of a weak Morita duality between subcategories of Mod-R and S-Mod is introduced and investigated. This concept includes dualities induced by cotilting modules over finite dimensional algebras, the R-dual for both noetherian rings of injective dimension ≤ 1 and the Matlis duality domains, as well as the first author's generalized Morita duality. Moreover it yields a “cotilting theorem” dual to the classical tilting theorem, and a characterization of generalized Morita duality.