Commutative Gorenstein Rings Research Articles

Gorenstein Projective Coresolutions and Co-Tate Homology Functors

For a local commutative Gorenstein ring [Formula: see text], Enochs et al. in [Gorenstein projective resolvents, Comm. Algebra 44 (2016) 3989–4000] defined a functor [Formula: see text] and showed that this functor can be computed by taking a totally acyclic complex arising from a projective coresolution of the first component or a totally acyclic complex arising from a projective resolution of the second component. In order to define the functor [Formula: see text] over general rings, we introduce the right Gorenstein projective dimension of an [Formula: see text]-module [Formula: see text], [Formula: see text], via Gorenstein projective coresolutions, and give some equivalent characterizations for the finiteness of [Formula: see text]. Then over a general ring [Formula: see text] we define a co-Tate homology group [Formula: see text] for [Formula: see text]-modules [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], and prove that [Formula: see text] can be computed by complete projective coresolutions of the first variable or by complete projective resolutions of the second variable.

TILTING THEORY FOR GORENSTEIN RINGS IN DIMENSION ONE

In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category$\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$admits a tilting (respectively, silting) object for a$\mathbb{Z}$-graded commutative Gorenstein ring$R=\bigoplus _{i\geqslant 0}R_{i}$. Here$\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)$is the singularity category, and$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$is the stable category of$\mathbb{Z}$-graded Cohen–Macaulay (CM)$R$-modules, which are locally free at all nonmaximal prime ideals of$R$.In this paper, we give a complete answer to this problem in the case where$\dim R=1$and$R_{0}$is a field. We prove that$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$always admits a silting object, and that$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$admits a tilting object if and only if either$R$is regular or the$a$-invariant of$R$is nonnegative. Our silting/tilting object will be given explicitly. We also show that if$R$is reduced and nonregular, then its$a$-invariant is nonnegative and the above tilting object gives a full strong exceptional collection in$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$.

The cohomology annihilator of a curve singularity

The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to their double branched covers. We also show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization and in dimension one the cohomology annihilator of its completion. Finally, we investigate a relation between the cohomology annihilator of a Gorenstein ring and stable annihilators of its noncommutative resolutions.

PROGRESS ON THE AUSLANDER–REITEN CONJECTURE

Let $R$ be a commutative Gorenstein ring. A result of Araya reduces the Auslander–Reiten conjecture on the vanishing of self-extensions to the case where $R$ has Krull dimension at most one. In this paper we extend Araya’s result to certain $R$-algebras. As a consequence of our argument, we obtain examples of bound quiver algebras that satisfy the Auslander–Reiten conjecture.

On the Auslander–Reiten conjecture for algebras

A recent result of Araya asserts that if the Auslander–Reiten conjecture holds in codimension one for a commutative Gorenstein ring R, then it holds for R. This note extends this result to left Gorenstein R-algebras Λ, whenever R is a commutative Gorenstein ring. This, in particular, implies that any finitely generated self-orthogonal Gorenstein projective Λ-module is projective, provided Λ is an isolated singularity and dim⁡R≥2. Also, some examples of bound quiver algebras satisfying the Auslander–Reiten conjecture are presented.

Homotopy equivalences induced by balanced pairs

We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which a balanced pair of subcategories gives rise to a triangle-equivalence between two homotopy categories of complexes. As an application, we prove that for a left-Gorenstein ring, there exists a triangle-equivalence between the homotopy category of its Gorenstein projective modules and the homotopy category of its Gorenstein injective modules, which restricts to a triangle-equivalence between the homotopy category of projective modules and the homotopy category of injective modules. In the case of commutative Gorenstein rings we prove that up to a natural isomorphism our equivalence extends Iyengar–Krause's equivalence.

Gorenstein Orders Associated with Modules

Let R be a commutative Gorenstein ring. We call a Noether R-algebra Λ a Gorenstein R-order provided Λ has Gorenstein dimension zero as an R-module and Λ ≅ Hom R (Λ, R) as Λ-bimodules. Let Λ be a Gorenstein R-order. We provide several ways to extend Λ to a Gorenstein R-order Δ containing an idempotent e such that Λ ≅ eΔe. In particular, for a finitely generated right Λ-module M having Gorenstein dimension zero as an R-module, setting Δ = EndΛ(M ⊕ Λ), we show that EndΛ(Ω n M ⊕ Λ) is derived equivalent to Δ for all n ∈ ℤ, and hence if Δ is a Gorenstein R-order then so is every EndΛ(Ω n M ⊕ Λ), and that Δ is a Gorenstein R-order if M 𝔭 is projective as a right Λ𝔭-module for every prime ideal 𝔭 of R with ht 𝔭 ≤ 1 and , 1 ≤ i ≤ ht 𝔭 −2, for every prime ideal 𝔭 of R with ht 𝔭 ≥3.

Absolute, Gorenstein, and Tate Torsion Modules

We show that there is an Avramov–Martsinkovsky type exact sequence with , Gtor, and Tor. We prove that if R is a Gorenstein ring, then the modules , n ≥ 1 can be computed using either a complete resolution of MR or using a complete resolution of RN. We show that over a Gorenstein ring a left R-module N is Gorenstein flat if and only if . We also show that over commutative Gorenstein rings the modules can be computed by the combined use of a flat resolution and a Gorenstein flat resolution of M.

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R -algebras over a commutative Gorenstein ring R . Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R -algebra A to have a Gorenstein R -algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R -algebra A whose endomorphism algebra is not a Gorenstein R -algebra.

Liaison classes of modules

We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules. As a consequence, we show that the intermediate cohomology modules and properties like being perfect, Cohen–Macaulay, Buchsbaum, or surjective-Buchsbaum are preserved in even module liaison classes. Furthermore, we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one.

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ⊃ R 2 ⊃ J ⊃ Rn (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R 2 such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = Gk = (2) × ⃛ × (2). Commutative Gorenstein rings appeared at first in the paper [3]. Torsion-free modules over commutative Gorenstein domains were investigated in [1]. Noncommutative Gorenstein orders were considered in [2] and [10]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5]. Arbitrary tiled orders were considered in [4], 11-14.

Tate–Vogel Completions of Half-Exact Functors

We provide a general method to construct the Tate–Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen–Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (S n and S n ), we define F ∨(X)=lim → S n S n F(X) and F ∧(X)=lim ← S n S n F(X), and call F ∨ and F ∧ the Tate–Vogel completions of F. We provide several properties of F ∨ and F ∧, and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate–Vogel completions with ordinary Tate–Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovsky's works, we can define the invariants ξ(F) and η(F) of F. If F=Ext R i (M, ), then they coincide with Martsinkovsky's ξ-invariants and Auslander's delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.

An Example of Bernstein Duality

We study local duality in a non-commutative framework which extends well known local dualities in commutative Gorenstein rings and in enveloping algebras of finite dimensional solvable Lie algebras. We show examples of non-commutative algebras having a local duality in our sense. In order to study these examples we will need a non-commutative version of Groebner bases.

Extension closed reflexive modules

Throughout this note, R stands for a ring with identity, and all modules are unital R-modules. In this note, we ask when the class of all finitely generated reflexive (torsionless) left modules is closed under extensions. This is not always the case, even if R is left and right noetherian (cf. Bass [2, Lemma 3.2] and [3, p. 12]). As will be shown, in case R is a commutat ive noetherian local domain of Krull dimension two, R is a CohenMacaulay ring whenever the class of the finitely generated reflexive modules is closed under extensions. Our main aim is to show that if R is a left and right noetherian ring then both the class of all finitely generated torsionless left modules and the class of all finitely generated reflexive left modules are closed under extensions if and only if for i = 1, 2 the functor Ext , 1 (Ext~ ( , R), R) vanishes on the finitely generated right modules. Note that the latter condition is satisfied if R is a commutat ive Gorenstein ring (see Bass [3]). We will show also that if R is a left and right noetherian ring with a minimal injective resolution 0 ~ RR ~ Eo ~ E1 , . . . such that weak dim E l < i + 1 for i = 0, 1 then both the class of finitely generated torsionless left modules and the class of finitely generated reflexive left modules are closed under extensions. In the proof, it will be shown that if R is a left and right noetherian ring with inj dim RR _--< 2 then the class of the finitely generated reflexive left modules is closed under extensions. On the other hand, even if R is a left and right noetherian ring with inj dim RR = inj dim R R < 2, the class of all finitely generated torsionless left modules is not necessarily closed under extensions. In what follows, we denote by ( )* both the R-dual functors, and for a given module M we denote by eu: M ~ M** the usual evaluation map and by E(M) the injective envelope of M. Recall that a module M is said to be torsionless if eu is a monomorph i sm and to be reflexive if eu is an isomorphism. Note that a module M is torsionless if and only if it is cogenerated by R. We denote by mod R (resp. mod R ~ the category of the finitely presented left (resp. right) modules. Note that if R is left noetherian then every finitely generated left module is finitely presented.