Gorenstein Projective Modules Research Articles

Another Gorenstein analogue of flat modules

ABSTRACTA right R-module M is called glat if any homomorphism from any finitely presented right R-module to M factors through a finitely presented Gorenstein projective right R-module. The concept of glat modules may be viewed as another Gorenstein analogue of flat modules. We first prove that the class of glat right R-modules is closed under direct sums, direct limits, pure quotients and pure submodules for arbitrary ring R. Then we obtain that a right R-module M is glat if and only if M is a direct limit of finitely presented Gorenstein projective right R-modules. In addition, we explore the relationships between glat modules and Gorenstein flat (Gorenstein projective) modules. Finally we investigate the existence of preenvelopes and precovers by glat and finitely presented Gorenstein projective modules.

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

For any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the monomorphism category as a full subcategory of the module category over the Morita ring with all entries the ring R and zero bimodule homomorphisms. This approach provides an interesting link between Morita rings and monomorphism categories. The aim of this paper is two-fold. First, we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.

Stable functors of derived equivalences and Gorenstein projective modules

From certain triangle functors, called nonnegative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of nonnegative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. In particular, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

ABSTRACTFor 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.

Totally acyclic complexes

It is known that over an Iwanaga–Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga–Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga–Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. The cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. 6. Every acyclic complex of flat modules is F-totally acyclic. 7. The cycles of every acyclic complex of Gorenstein flat modules are Gorenstein flat. Thus we improve slightly on a result of Iyengar and Krause; in [22] they proved that for a commutative noetherian ring R with a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if R is Gorenstein. We replace the dualizing complex hypothesis by the finiteness of the Krull dimension, and add more equivalent conditions.In the second part of the paper we focus on the noncommutative case. We prove that for a two sided noetherian ring R of finite finitistic flat dimension that satisfies the Auslander condition the following are equivalent: 1. Every complex of injective (left and respectively right) R-modules is totally acyclic. 2. R is Iwanaga–Gorenstein.

Pure-injectivity in the category of Gorenstein projective modules

In this paper, we introduce and study (weak) pure-injective Gorenstein projective modules. Let [Formula: see text] be an Artin algebra. We prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules coincides with the intersection of the category of pure-injective left [Formula: see text]-modules and that of Gorenstein projective left [Formula: see text]-modules. Then, we get an equivalent characterization of virtually Gorenstein algebras (being CM-finite). Furthermore, we prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules is enveloping in the category of left [Formula: see text]-modules; and if [Formula: see text] is virtually Gorenstein, then it is precovering in the category of pure-injective left [Formula: see text]-modules.

Some conditions for the existence of Gorenstein projective covers and preenvelopes

In this paper, we investigate the rings over which a module is Gorenstein flat if and only if it is Gorenstein projective. Some examples of such rings are given. We show that over such rings the class of Gorenstein projective modules is covering. We also characterize the rings over which the class of Gorenstein projective modules is preenveloping. As a conclusion, we obtain that a commutative ring is artinian if and only if every module has a Gorenstein projective preenvelope. The existence of pure injective Gorenstein injective preenvelopes over certain rings is also shown.

On Relative Derived Categories

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalize two results of Happel by proving the existence of AR-triangles in Gorenstein-derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent, and finally study relations between the Gorenstein-derived category of a quiver and its image under a reflection functor. Some interesting applications are provided.

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.

Gorenstein Modules and Auslander Categories

In this article, some new characterizations of Gorenstein projective, injective, and flat modules over commutative noetherian local rings are given.

Desingularization of Quiver Grassmannians for Gentle Algebras

In (Cerulli Irelli et al., Adv. Math. 245(1) 182–207 2013), Cerulli Irelli-Feigin-Reineke construct a desingularization of quiver Grassmannians for Dynkin quivers. Following them, a desingularization of arbitrary quiver Grassmannians for finite dimensional Gorenstein projective modules of 1-Iwanaga-Gorenstein gentle algebras is constructed in terms of quiver Grassmannians for their Cohen-Macaulay Auslander algebras.

Super Finitely Presented Modules and Gorenstein Projective Modules

Let R be a commutative ring. An R-module M is said to be super finitely presented if there is an exact sequence of R-modules where each Pi is finitely generated projective. In this article, it is shown that if R has the property (B) that every super finitely presented module has finite Gorenstein projective dimension, then every finitely generated Gorenstein projective module is super finitely presented. As an application of the notion of super finitely presented modules, we show that if R has the property (C) that every super finitely presented module has finite projective dimension, then R is K0-regular, i.e., K0(R[x1,…, xn]) ≅ K0(R) for all n ≥ 1.

Strict Mittag–Leffler Conditions on Gorenstein Modules

In this article, we investigate the relations between Gorenstein projective modules and Gorenstein flat modules in terms of strict Mittag–Leffler condition. We give some conditions under which Gorenstein projectives are Gorenstein flat, and discuss when the direct limits of Gorenstein projective modules are Gorenstein flat. Moreover, we study the dual of Gorenstein injective modules with strict Mittag–Leffler condition.

A Characterization of Gorenstein Projective Modules

In this article, we give a new characterization of Gorenstein projective modules. As applications of our result, we prove that a strongly Gorenstein projective module of countable type is Gorenstein flat, and each left R-module has a special Gorenstein projective precover whenever all projective left R-modules have finite injective dimension.

APPROXIMATIONS AND ADJOINTS FOR CATEGORIES OF COMPLEXES OF GORENSTEIN PROJECTIVE MODULES

In the paper, it is proven that every object in C(R-GProj) has a special C(R-Proj)-preenvelope, and then some adjoints in homotopy categories related to Gorenstein projective modules are given, where C(R-Proj) is the subcategory of complexes of projective R-modules, and C(R-GProj) is the subcategory of complexes of Gorenstein projective R-modules.

Gorenstein categories $\mathcal G(\mathscr X,\mathscr Y,\mathscr Z)$ and dimensions

Let $\mathscr {A}$ be an abelian category and $\mathscr {X},\mathscr {Y},\mathscr {Z}$ additive full subcategories of $\mathscr {A}$. We introduce and study the Gorenstein category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$ as a common generalization of some known modules such as Gorenstein projective (injective) modules \cite {EJ95}, strongly Gorenstein flat modules \cite {DLM} and Gorenstein FP-injective modules \cite {DM}, and prove the stability of $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$. We also establish Gorenstein homological dimensions in terms of the category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$.

Gorenstein homological theory for differential modules

We show that a differential module is Gorenstein projective (injective, respectively) if and only if its underlying module is Gorenstein projective (injective, respectively). We then relate the Ringel–Zhang theorem on differential modules to the Avramov–Buchweitz–Iyengar notion of projective class of differential modules and prove that for a ring R there is a bijective correspondence between projectively stable objects of split differential modules of projective class not more than 1 and R-modules of projective dimension not more than 1, and this is given by the homology functor H and stable syzygy functor ΩD. The correspondence sends indecomposable objects to indecomposable objects. In particular, we obtain that for a hereditary ring R there is a bijective correspondence between objects of the projectively stable category of Gorenstein projective differential modules and the category of all R-modules given by the homology functor and the stable syzygy functor. This gives an extended version of the Ringel–Zhang theorem.

Duality Pairs Induced by Gorenstein Projective Modules with Respect to Semidualizing Modules

Let C be a semidualizing module over a commutative Noetherian ring R. We investigate duality pairs induced by C-Gorenstein projective modules. It is proven that R is Artinian if and only if \((\mathcal {GPC},\mathcal {GIC})\) is a duality pair if and only if \((\mathcal {GIC},\mathcal {GPC})\) is a duality pair and \(M^{+}\in {\mathcal {GIC}}\) whenever \(M\in {\mathcal {GPC}}\), where \(\mathcal {GPC}\) (\(\mathcal {GIC}\)) is the class of \(\mathcal {GC}\)-Gorenstein projective (\(\mathcal {GC}\)-Gorenstein injective) R-modules. In particular, we give a necessary and sufficient condition for a commutative Artinian ring to be virtually Gorenstein. Moreover, we get that R is Artinian if and only if the class \(\mathcal {GP}\) of Gorentein projective R-modules is preenveloping. As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a commutative Artinian ring.

($\mathcal {X},\mathcal {Y}$)-Gorenstein projective and injective modules

This paper introduces and studies (X,Y)-Gorenstein projective and injective modules, which are a generalization of Enochs' Gorenstein projective and injective modules, respectively. Our main aim is to investigate the relations among various (X,Y)-Gorenstein projective modules.

On homotopy categories of Gorenstein modules: Compact generation and dimensions

Let $A$ be a virtually Gorenstein algebra of finite CM-type. We establish a duality between the subcategory of compact objects in the homotopy category of Gorenstein projective left $A$-modules and the bounded Gorenstein derived category of finitely generated right $A$-modules. Let $R$ be a two-sided noetherian ring such that the subcategory of Gorenstein flat modules $R\mbox{-}\mathcal{GF}$ is closed under direct products. We show that the inclusion $K(R\mbox{-}\mathcal{GF})\to K(R\mbox{-}{\rm Mod})$ of homotopy categories admits a right adjoint. We introduce the notion of Gorenstein representation dimension for an algebra of finite CM-type, and establish relations among the dimension of its relative Auslander algebra, Gorenstein representation dimension, the dimension of the bounded Gorenstein derived category, and the dimension of the bounded homotopy category of its Gorenstein projective modules.