Gorenstein Projective Modules Research Articles

Auslander‐type conditions and weakly Gorenstein algebras

AbstractLet be an Artin algebra. Under certain Auslander‐type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander‐type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if is left quasi‐Auslander, then is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if satisfies the Auslander condition, then is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander–Reiten's conjecture, which states that is Gorenstein if satisfies the Auslander condition.

A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

Frobenius pairs arising from Morita rings

This paper is to study how to construct Frobenius pairs over a Morita ring [Formula: see text]. Given Frobenius pairs [Formula: see text] and [Formula: see text] in [Formula: see text] and [Formula: see text], respectively, we construct Frobenius pairs [Formula: see text] and [Formula: see text] in [Formula: see text] under some conditions. As an application, we establish a sufficient and necessary condition of Gorenstein projective modules over the Morita ring [Formula: see text].

The N-stable category

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to N-complexes, one must find an appropriate candidate for the N-analogue of the stable category. We identify this “N-stable category” via the monomorphism category and prove Buchweitz’s theorem for N-complexes over a Grothendieck abelian category. We also compute the Serre functor on the N-stable category over a self-injective algebra and study the resultant fractional Calabi–Yau properties.

On the Self-injectivity and CM-free of an Artin Algebra with Radical Cubic Zero

In this paper, we succeed in proving that a connected Artin algebra whose Jacobson cubic radical is zero with each simple module and each Gorenstein projective module having the square radical of the cover projective of its first syzygy to be zero and verifying the coincidence covers, is either self-injective or CM-free.

The finitely presented torsion-free SG-projective modules are not necessarily projective

It is proved in [Strongly Gorenstein projective, injective and flat modules, J. Algebra 320 (2008) 2659–2674, Theorem 2.8] that a finitely presented torsion-free module is SG-projective if and only if it is projective. Let L⊆F be an extension of fields with [F:L]=2. Set R=L+XF[X], m=XF[X] and D=R[Y]. Via the study of w-invertibility over infra-Krull domains, we prove that the w-conductor m[Y] of D is SG-projective but not projective, and all maximal w-ideals other than m[Y] are w-invertible. It follows that a finitely presented torsion-free SG-projective module do not need to be projective.

Finiteness criteria for Gorenstein flat dimension and stability

Projectively coresolved Gorenstein flat modules were introduced recently by Saroch and Stovicek and were shown to be Gorenstein projective. While the relation between Gorenstein projective and Gorenstein flat modules is not well understood, the class of projectively coresolved Gorenstein flat modules is contained in the class of Gorenstein flat modules. This paper proves necessary and sufficient conditions for a module of finite Gorenstein flat dimension to be projectively coresolved Gorenstein flat, or of finite flat dimension. Stability results for the class of projectively coresolved Gorenstein flat modules are also established.

Eventually homological isomorphisms and Gorenstein projective modules

Eventually homological isomorphisms and Gorenstein projective modules

A note on Gorenstein projective and Gorenstein flat modules

We prove that all Gorenstein projective modules are Gorenstein flat if and only each Gorenstein projective module has finite Gorenstein flat dimension. As applications, some characterizations of strongly CM-free rings are established and an affirmative answer to a question posed by Wang [Colloq. Math. 170 (2022) 1–13] is obtained.

Triangle equivalences of Gorenstein defect categories induced by homological epimorphisms

We give a sufficient condition that a certain homological epimorphism between two finite dimensional algebras induces a triangle equivalence between their Gorenstein defect categories as well as between their stable categories of Gorenstein projective modules.

How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings

Let [Formula: see text] be a Morita ring with [Formula: see text]. We first study how to construct (complete) duality pairs of [Formula: see text]-modules using (complete) duality pairs of [Formula: see text]-modules and [Formula: see text]-modules, generalizing the result of Mao (Commun. Algebra 12 (2020) 5296–5310) about the duality pairs over a triangular matrix ring. Moreover, we construct Gorenstein projective modules relative to complete duality pairs of [Formula: see text]-modules. Finally, we give an application to Ding projective modules.

A bijection theorem for Gorenstein projective τ-tilting modules

We introduce the notions of Gorenstein projective [Formula: see text]-rigid modules, Gorenstein projective support [Formula: see text]-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi–Iyama–Reiten’s bijection theorem on support [Formula: see text]-tilting modules. More precisely, for an algebra [Formula: see text], we prove that there is a bijection between the set of Gorenstein projective support [Formula: see text]-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-[Formula: see text]-tilting finite algebras and show that [Formula: see text] is CM-[Formula: see text]-tilting finite if and only if [Formula: see text] is CM-[Formula: see text]-tilting finite. Moreover, we show that the Bongartz completion of a Gorenstein projective [Formula: see text]-rigid module need not be a Gorenstein projective [Formula: see text]-tilting module.

A new characterization of silting subcategories in the stable category of a Frobenius extriangulated category

We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di, Liu, Wang and Wei (J. Algebra 525 (2019) 42–63) about the Auslander-Reiten type correspondence for silting subcategories over triangulated categories. More specifically, for any Frobenius extriangulated category , we establish a bijective correspondence between silting subcategories of the stable category and certain covariantly finite subcategories of . As a consequence, a characterization of silting subcategories in the stable category of a Frobenius exact category is given. This result is applied to homotopy categories over abelian categories with enough projectives, derived categories over Grothendieck categories with enough projectives, and to the stable category of Gorenstein projective modules over a ring R.

Lifting recollements of abelian categories and model structures

We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwanaga-Gorenstein rings.

Monic modules and semi-Gorenstein-projective modules

The category gp ( Λ ) of Gorenstein-projective modules over tensor algebra Λ = A ⊗ k B can be described as the monomorphism category mon ( B , gp ( A ) ) of B over gp ( A ) . In particular, Gorenstein-projective Λ-modules are monic. In this paper, we find the similar relation between semi-Gorenstein-projective Λ-modules and A -modules, via monic modules, namely, mon ( B , A ⊥ ) = mon ( B , A ) ∩ Λ ⊥ . Using this, it is proved that if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only each semi-Gorenstein-projective Λ-modules are monic; and that if B = k Q with Q a finite acyclic quiver, then Λ is weakly Gorenstein if and only if A is weakly Gorenstein. However, this relation itself does not answer the question whether there exist double semi-Gorenstein-projective Λ-modules which are not monic. Using the recent discovered examples of double semi-Gorenstein-projective A -modules which are not torsionless, we positively answer this question, by explicitly constructing a class of double semi-Gorenstein-projective T 2 ( A ) -modules with one parameter such that they are not monic, and hence not torsionless. The corresponding results are obtained also for the monic modules and semi-Gorenstein-projective modules over the triangular matrix algebras given by bimodules .

M-Periodic Gorenstein objects

We present and study the concept of m-periodic Gorenstein objects relative to a pair (A,B) of classes of objects in an abelian category, as a generalization of m-strongly Gorenstein projective modules over associative rings. We prove several properties when (A,B) satisfies certain homological conditions, like for instance when (A,B) is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established.

Virtually Gorenstein rings and relative homology of complexes

We extend the notion of virtually Gorenstein rings to the setting of arbitrary rings, and prove that all rings R of finite Gorenstein weak global dimension are virtually Gorenstein such that all Gorenstein projective R -modules are Gorenstein flat. For such a ring R , we introduce the notion of relative homology functors of complexes with respect to Gorenstein projective (resp., flat) modules, and establish a balanced and a vanishing result for the homology functor.

Generalized Gorenstein Modules

We introduce a generalization of the Gorenstein injective modules: the Gorenstein [Formula: see text]-injective modules (denoted by [Formula: see text]). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor [Formula: see text], with [Formula: see text] any [Formula: see text]-injective module. Thus, [Formula: see text] is the class of classical Gorenstein injective modules, and [Formula: see text] is the class of Ding injective modules. We prove that over any ring [Formula: see text], for any [Formula: see text], the class [Formula: see text] is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For [Formula: see text] we show that [Formula: see text] (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring [Formula: see text] is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein [Formula: see text]-projectives (denoted by [Formula: see text]). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for any[Formula: see text] the class [Formula: see text] is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.

Frobenius functors, stable equivalences and K-theory of Gorenstein projective modules

Owing to the difference in K-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for the Frobenius pair of faithful functors between two abelian categories to be a Quillen equivalence, which is also equivalent to that the Frobenius functors induce mutually inverse equivalences between stable categories of Gorenstein projective objects.We show that the category of Gorenstein projective objects is a Waldhausen category, then Gorenstein K-groups are introduced and characterized. As applications, we show that stable equivalences of Morita type preserve Gorenstein K-groups, CM-finiteness and CM-freeness. Two specific examples of path algebras are presented to illustrate the results, for which the Gorenstein K0 and K1-groups are calculated.

Construction of Gorenstein-projective modules over Morita rings

In this paper, we obtain necessary and sufficient conditions for all complete projective resolutions over a Morita ring [Formula: see text]. As special cases, we get a class of complete projective resolutions over [Formula: see text], from the ones over [Formula: see text] and [Formula: see text]. We also obtain necessary and sufficient conditions for all Gorenstein-projective modules over a Morita ring [Formula: see text]. As special cases, we get a class of Gorenstein-projective modules over [Formula: see text], from the ones over [Formula: see text] and [Formula: see text]. Furthermore, we determine all complete projective resolutions as well as all Gorenstein-projective modules over the [Formula: see text] matrix algebra [Formula: see text] over [Formula: see text].