Gorenstein Homological Algebra Research Articles

Gorenstein Krull domains and their factor rings

The concept of Gorenstein Krull domains can be viewed as a Gorenstein analogue of Krull domains, which are not necessarily integrally closed. This allows us to study those domains from the point view of Gorenstein homological algebra. By the so-called w-operation, we prove that an integral domain is Gorenstein Krull if and only if for any nonzero nonunit, the Gorenstein global dimension of its w-factor ring is zero.

Some results on Gorenstein (weak) global dimension of rings

The principle “Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra” was given by Henrik Holm. We give support to the principle. We prove that the Gorenstein weak global dimension of a ring R equals the supremum of the Gorenstein flat dimensions of the finitely presented cyclic (left or right) R-modules. We also show that, if R is a coherent ring and x is a central nonzero-divisor in R contained in the Jacobson radical of R, then Finally, we give the Gorenstein counterpart of the well-known Hilbert’s Syzygy Theorem.

Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

Let R be a general ring. Duality pairs of R-modules were introduced by Holm-Jørgensen. Most examples satisfy further properties making them what we call semi-complete duality pairs in this paper. We attach a relative theory of Gorenstein homological algebra to any given semi-complete duality pair 𝔇=(ℒ,𝒜). This generalizes the homological theory of the AC-Gorenstein modules defined by Bravo–Gillespie–Hovey, and we apply this to other semi-complete duality pairs. The main application is that the Ding injective modules are the right side of a complete (perfect) cotorsion pair, over any ring. Completeness of the Gorenstein flat cotorsion pair over any ring arises from the same duality pair.

Cut cotorsion pairs

AbstractWe present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

The role of w-tilting modules in relative Gorenstein (co)homology

AbstractLetRRbe a ring,CCbe a leftRR-module andS=EndR(C)S={{\rm{End}}}_{R}\left(C). WhenCCis semidualizing, the Auslander classAC(S){{\mathcal{A}}}_{C}\left(S)and the Bass classℬC(R){{\mathcal{ {\mathcal B} }}}_{C}\left(R)associated withCChave been the subject of extensive investigations. It has been proved that these classes, also known as Foxby classes, are one of the central concepts of (relative) Gorenstein homological algebra. In this paper, we answer several natural questions which arise when we weaken the condition ofCCbeing semidualizing: if we letCCbe w-tilting (see Definition 2.1), we establish the conditions for the pair(AC(S),AC(S)⊥1)\left({{\mathcal{A}}}_{C}\left(S),{{\mathcal{A}}}_{C}{\left(S)}^{{\perp }_{1}})to be a perfect cotorsion theory and for the pair(BC⊥1(R),BC(R))\left({}^{{\perp }_{1}}B_{C}\left(R),{B}_{C}\left(R))to be a complete hereditary cotorsion theory. This tells us when the classes of Auslander and Bass are preenveloping and precovering, which generalizes a number of results disseminated in the literature. We investigate Gorenstein flat modules relative to a not necessarily semidualizing moduleCCand we find conditions for the class ofGC{G}_{C}-projective modules to be special precovering, the class ofGC{G}_{C}-flat modules to be covering, the one of GorensteinCC-projective modules to be precovering and that of GorensteinCC-injective modules to be preenveloping. We also find how to recover Foxby classes fromAddR(C){{\rm{Add}}}_{R}\left(C)-resolutions ofRR.

Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

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 .

Infinitely Generated Gorenstein Tilting Modules

The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing “Gorenstein tilting approximations” and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of complements to partial Gorenstein tilting modules as well as some connections between Gorenstein tilting modules and finitistic dimension conjectures.

Recollements associated to cotorsion pairs over upper triangular matrix rings

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M 0 & B \end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. Given two complete hereditary cotorsion pairs $(\mathcal{A}_{A},\mathcal{B}_{A})$ and $(\mathcal{C}_{B},\mathcal{D}_{B})$ in $A$-Mod and $B$-Mod respectively. We define two cotorsion pairs $(\Phi(\mathcal{A}_{A},\mathcal{C}_{B}), \mathrm{Rep}(\mathcal{B}_{A},\mathcal{D}_{B}))$ and $(\mathrm{Rep}(\mathcal{A}_{A},\mathcal{C}_{B}), \Psi(\mathcal{B}_{A},\mathcal{D}_{B}))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary. Given two cofibrantly generated model structures $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ on $A$-Mod and $B$-Mod respectively. Using the result above, we investigate when there exist a cofibrantly generated model structure $\mathcal{M}_{T}$ on $T$-Mod and a recollement of $\mathrm{Ho}(\mathcal{M}_{T})$ relative to $\mathrm{Ho}(\mathcal{M}_{A})$ and $\mathrm{Ho}(\mathcal{M}_{B})$. Finally, some applications are given in Gorenstein homological algebra.

Frobenius functors and Gorenstein flat dimensions

We prove that if the Frobenius functor F (from the category of left R-modules to the category of left S-modules) is faithful, then for any R-module X, the Gorenstein flat dimension of X is equal to the Gorenstein flat dimension of F(X), which is motivated by a result of Nakayama and Tsuzuku about relations between Frobenius extensions and flat dimension of modules. It is a well-known metatheorem of Holm that every result in homological algebra has a counterpart in Gorenstein homological algebra. Note that the flat dimension of X can be different from that of F(X). Our main result provides a counterexample to the converse of Holm’s metatheorem. In addition, some applications are given.

Periodic Modules and Acyclic Complexes

We study the behaviour of modules $M$ that fit into a short exact sequence $0\to M\to C\to M\to 0$, where $C$ belongs to a class of modules $\mathcal C$, the so-called $\mathcal C$-periodic modules. We find a rather general framework to improve and generalize some well-known results of Benson and Goodearl and Simson. In the second part we will combine techniques of hereditary cotorsion pairs and presentation of direct limits, to conclude, among other applications, that if $M$ is any module and $C$ is cotorsion, then $M$ will be also cotorsion. This will lead to some meaningful consequences in the category $\textrm{Ch}(R)$ of unbounded chain complexes and in Gorenstein homological algebra. For example we show that every acyclic complex of cotorsion modules has cotorsion cycles, and more generally, every map $F\to C$ where $C$ is a complex of cotorsion modules and $F$ is an acyclic complex of flat cycles, is null-homotopic. In other words, every complex of cotorsion modules is dg-cotorsion.

A Generalization of the Nakayama Functor

In this paper we introduce a generalization of the Nakayama functor for finite-dimensional algebras. This is obtained by abstracting its interaction with the forgetful functor to vector spaces. In particular, we characterize the Nakayama functor in terms of an ambidextrous adjunction of monads and comonads. In the second part we develop a theory of Gorenstein homological algebra for such Nakayama functor. We obtain analogues of several classical results for Iwanaga-Gorenstein algebras. One of our main examples is the module category Λ-Mod of a k-algebra Λ, where k is a commutative ring and Λ is finitely generated projective as a k-module.

Duality pairs and stable module categories

Let R be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair (F,I) where F is the class of flat R-modules and I is the class of injective R-modules. For a general R, the AC-Gorenstein homological algebra of Bravo–Gillespie–Hovey is the one coming from the duality pair (L,A) where L is the class of level R-modules and A is class of absolutely clean R-modules. Indeed we show here that the work in [1] can be extended to obtain similar abelian model structures on R-Mod from any a complete duality pair (L,A). It applies in particular to the original duality pairs constructed by Holm–Jørgensen.

Ding w-Flat Modules and Dimensions

The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-flat module if [Formula: see text] is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules … → P1 → P0 → P0 → P1 → … such that M ≅ Im(P0 → P0) and such that the functor HomR(−, F) leaves the sequence exact whenever F is w-flat. Several wellknown classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension (of modules and rings) and the existence of Ding w-flat precovers are also studied.

Frobenius pairs in abelian categories

In this work, we revisit Auslander-Buchweitz Approximation Theory and find some relations with cotorsion pairs and model category structures. From the notions of relatives generators and cogenerators in Approximation Theory, we introduce the concept of left Frobenius pairs $(\mathcal{X},\omega)$ in an abelian category $\mathcal{C}$. We show how to construct from $(\mathcal{X},\omega)$ a projective exact model structure on $\mathcal{X}^\wedge$, as a result of Hovey-Gillespie Correspondence applied to two compatible and complete cotorsion pairs in $\mathcal{X}^\wedge$. These pairs can be regarded as examples of what we call cotorsion pairs relative to a thick subcategory of $\mathcal{C}$. We establish some correspondences between Frobenius pairs, relative cotorsion pairs, exact model structures and Auslander-Buchweitz contexts. Finally, some applications of these results are given in the context of Gorenstein homological algebra by generalizing some existing model structures on the categories of modules over Gorenstein and Ding-Chen rings, and by encoding the stable module category of a ring as a certain homotopy category. We also present some connections with perfect cotorsion pairs, covering classes, and cotilting modules.

On minimal Auslander–Gorenstein algebras and standardly stratified algebras

We give new properties of algebras with finite self-injective dimension coinciding with the dominant dimension d≥2, which are called minimal Auslander–Gorenstein algebras in the recent work of Iyama and Solberg, see [22]. In particular, when those algebras are standardly stratified, we give criteria when the category of (properly) (co)standardly filtered modules has a nice homological description using tools from the theory of dominant dimensions and Gorenstein homological algebra. We give some examples of standardly stratified algebras having dominant dimension equal to the self-injective dimension, including examples having an arbitrary natural number as the self-injective dimension and blocks of finite representation type of Schur algebras.

Relative Homological Algebra via Truncations

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R -modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category \mathcal{A} and fix a class of “injective objects” \mathcal{I} . We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(\mathcal{A};\mathcal{I}) , we need more: the split error term must vanish. This is the case when \mathcal I is the class of all injective R -modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4 ^\ast - n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop a machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman's Brown-Adams representability theorem for compactly generated categories.

Gorenstein Projective Precovers

The existence of Gorenstein projective precovers is one of the main open problems in Gorenstein homological algebra. We prove that the class of Gorenstein projective modules is a special precovering over any right coherent and left n-perfect ring. This is joint work with S. Estrada and S. Odabasi, and more details can be found in Estrada et al. (Gorenstein flat and projective (pre)covers. Preprint [2]).

G-Gorenstein Complexes

We present in the context of Gorenstein homological algebra the notion of a "G-Gorenstein complex" as the counterpart of the classical notion of a Gorenstein complex. In particular, we investigate equivalences between the category of G-Gorenstein complexes of fixed dimension and the G-class of modules.

Stability of Gorenstein objects in triangulated categories

Let $\mathcal{C}$ be a triangulated category with a proper class $\xi$ of triangles. Asadollahi and Salarian introduced and studied $\xi$-Gorenstein projective and $\xi$-Gorenstein injective objects, and developed Gorenstein homological algebra in $\mathcal{C}$. In this paper, we further study Gorenstein homological properties for a triangulated category. First, we discuss the stability of $\xi$-Gorenstein projective objects, and show that the subcategory $\mathcal{GP}(\xi)$ of all $\xi$-Gorenstein projective objects has a strong stability. That is, an iteration of the procedure used to define the $\xi$-Gorenstein projective objects yields exactly the $\xi$-Gorenstein projective objects. Second, we give some equivalent characterizations for $\xi$-Gorenstein projective dimension of object in $\mathcal{C}$.