1-Auslander-Gorenstein algebras which are tilted

From Auslander Algebras to Tilted Algebras

This paper has been withdrawn by the authors because it has been combined with "Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories" (arXiv:0903.0761) together. Please see the new version of the latter paper for the details.

Auslander-Gorenstein algebras and precluster tilting

We generalize the notions of $n$-cluster tilting subcategories and $τ$-selfinjective algebras into $n$-precluster tilting subcategories and $τ_n$-selfinjective algebras, where we show that a subcategory naturally associated to $n$-precluster tilting subcategories has a higher Auslander--Reiten theory. Furthermore, we give a bijection between $n$-precluster tilting subcategories and $n$-minimal Auslander--Gorenstein algebras, which is a higher dimensional analog of Auslander--Solberg correspondence (Auslander--Solberg, 1993) as well as a Gorenstein analog of $n$-Auslander correspondence (Iyama,…

支配 Auslander-Gorenstein 代数与混合 cluster tilting

Dominant Auslander-Gorenstein algebras and mixed cluster tilting

本文引入支配 Auslander-Gorenstein 代数与混合（预）cluster tilting 模，建立两者间的 Auslander 型双射对应，并证明平凡扩张代数与迭代 SGC-扩张均允许混合 tilting 模。

单项代数的 Auslander-Gorenstein 条件

The Auslander-Gorenstein condition for monomial algebras

本文研究了单项代数的 Auslander-Gorenstein 性质，通过将其归结为 Nakayama algebra 的分类并建立与双射 Auslander-Reiten 映射的等价关系，给出了新的同调刻画并证实了相关猜想。

Auslander-Gorenstein代数综述

A survey on Auslander-Gorenstein algebras

本文综述Auslander-Gorenstein代数，聚焦有限维代数，重点介绍特殊代数类的最新分类结果及Auslander-Reiten双射与其他已知双射间新发现的相互作用。

Recollements, Cohen-Macaulay Auslander algebras and Gorenstein projective conjecture

It is shown that a 4-recollement of derived categories of CM-finite algebras induces a 2-recollement of the corresponding Cohen-Macaulay Auslander algebras, which generalises the main theorem of Pan [S. Y. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra 216 (2012), 355{363]. Moreover, both Auslander- Reiten conjecture and Gorenstein projective conjecture are shown invariant under 3 (or 4)-recollement of unbounded derived categories of algebras.

Nakayama代数的解析有向图及其最小Auslander-Gorenstein性质

The resolution quiver of Nakayama algebras which are minimal Auslander-Gorenstein

本文通过Ringel的解析有向图，提出判断Nakayama代数是否为最小Auslander-Gorenstein的准则，核心方法基于自同态维数的奇偶性。

高维 Auslander–Iyama–Solberg 对应

A higher dimensional Auslander-Iyama-Solberg correspondence

本文将 Auslander–Iyama–Solberg 对应推广到高维情形，用任意具有有限投射与内射维数的自正交模替代极小忠实投射内射模，建立相对 Auslander–Gorenstein 对与推广的预簇倾斜模之间的新双射，并应用于一般线性群的模表示论。

Auslander-Gorenstein algebras, standardly stratified algebras and dominant dimensions

We give new properties of algebras with finite Gorenstein dimension coinciding with the dominant dimension $\geq 2$, which are called Auslander-Gorenstein algebras in the recent work of Iyama and Solberg, see \cite{IyaSol}. 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 Gorenstein dimen…

Homological behavior of Auslander's $k$-Gorenstein rings

In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $Λ$, we show that the right self-injective dimension of $Λ$ is at most $k$ if and only if each $M \in$mod $Λ$ satisfying the condition that Ext$_Λ^i(M, Λ)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $Λ$ are identical. In addition, we show that if $Λ$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $Λ$ with grade$M$ finite, then Ext$_…

0-Calabi-Yau Configurations and Finite Auslander-Reiten Quivers of Gorenstein Orders

We will revisit Wiedemann's classification of Auslander-Reiten quivers of representation-finite Gorenstein orders in this paper. We give a simpler proof of his result in which he described the Auslander-Reiten quiver of a representation-finite Gorenstein order in terms of a Dynkin diagram, a configuration and an automorphism group. A key notion in his result is configurations described in terms of Brauer relations with so-called Straßeneigenschaft. We show that configurations can be described in terms of $2$-Brauer relations very briefly.

Auslander-Gorenstein代数的新刻画

A new characterisation of Auslander-Gorenstein algebras

本文对Auslander-Gorenstein有限维代数给出新刻画，证实Marczinzik的猜想，还新证明具有Auslander-Gorenstein关联代数的有限格必为分配格。

Rings with Auslander Dualizing Complexes

A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to non-graded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing com…

Minimal Injective Resolutions and Auslander-Gorenstein Property for Path Algebras

Let $R$ be a ring and $\mathcal{Q}$ be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category $\rm{Rep} (\mathcal{Q} ,R)$ of representations of $\mathcal{Q}$ by left $R$-modules. We also extend our formula to all terms of the minimal injective resolution of $R\mathcal{Q}$. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra $R\mathcal{Q}$ is $k$-Gorenstein if and only if $\mathcal{Q}=\overrightarrow{A_{n}}$ and $R$ is a $k$-Gorenstein ring, w…

The Auslander-Reiten components of the bounded derived category for a cluster-tilted algebra of type Ã

We classify the Auslander-Reiten components of the bounded derived category of Λ, where Λ is a cluster-tilted of type Ã. The main tool is the combinatoric description of the indecomposable complexes in the bounded homotopy category via homotopy strings and homotopy bands.

Recollements of Cohen-Macaulay Auslander algebras and Gorenstein derived categories

Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a $\mathbb{D}^{\rm{b}}({\rm{mod\mbox{-}}} )$ level recollement, writing $A$ in terms of $B$ and $C$, then we get a $\mathbb{D}^-({\rm{Mod\mbox{-}}} )$ level recollement of certain functor categories, induces from the module categories of $A$, $B$ and $C$. As an application, we generalise the main theorem of Pan [Sh. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra, 216 (2012), 355-363] in terms of recollements of Gorenstein artin algebras. Moreover, we…

Auslander-type conditions and cotorsion pairs

We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $Λ$ satisfies the Auslander condition (that is, $Λ$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion pairs.

Tilting modules over Auslander-Gorenstein Algebras

For a finite dimensional algebra $Λ$ and a non-negative integer $n$, we characterize when the set $\tilt_nΛ$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalize results of Happel-Unger. Moreover, for an $n$-Gorenstein algebra $Λ$ with $n\geq 1$, we construct a minimal element in $\tilt_{n}Λ$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $Λ$ and its factor algebra $Γ=Λ/(e)$, we show that there is a bij…

On Cohen-Macaulay Auslander algebras

Cohen-Macaulay Auslander algebras are the endomorphism algebras of representation generators of the subcategory of Gorenstein projective modules over $\rm{CM}$-finite algebras. In this paper, we study Cohen-Macaulay Auslander algebras over $1$-Gorenstein algebras and $Ω_{\mathcal{G}}$-algebras. $1$-Gorenstein algebras are those of algebras with global Gorenstein projective dimension at most one and $Ω_{\mathcal{G}}$-algebras are a class of algebras introduced in this paper, including some important class of algebras for example Gentle algebras and more generally quadratic monomial algebras. It…

球模与Auslander-Yoneda代数的Auslander-Gorenstein条件

Spherical modules and the Auslander--Gorenstein condition for Auslander--Yoneda algebras

研究有限全局维数代数$A$的Auslander-Yoneda代数，证明其为Auslander-Gorenstein代数当且仅当每个不可分解左右$A$-模是Auslander-Bridger意义下的球模，进而刻画球代数并分类球Nakayama代数。

Finitistic Auslander algebras

Recently, Chen and Koenig in \cite{CheKoe} and Iyama and Solberg in \cite{IyaSol} independently introduced and characterised algebras with dominant dimension coinciding with the Gorenstein dimension and both dimensions being larger than or equal to two. In \cite{IyaSol}, such algebras are named Auslander-Gorenstein algebras. Auslander-Gorenstein algebras generalise the well known class of higher Auslander algebras, where the dominant dimension additionally coincides with the global dimension. In this article we generalise Auslander-Gorenstein algebras further to algebras having the property th…

Auslander-Reiten sequences for Gorenstein rings of dimension one

Let $R$ be a complete local Gorenstein ring of dimension one, with maximal ideal m. We show that if $M$ is a Cohen-Macaulay $R$-module which begins an AR-sequence, then this sequence is produced by a particular endomorphism of m corresponding to a minimal prime ideal. We apply this result to determining the shape of some components of Auslander-Reiten quivers.

Co-Gorenstein algebras

We review the theory of Co-Gorenstein algebras, which was introduced by Beligiannis in the article "The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, Auslander-Buchweitz Contexts and (Co-)Stabilization". We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture.

高维 Auslander-Gorenstein 代数与 Gabriel 拓扑

Higher Auslander-Gorenstein Algebras And Gabriel Topologies

本文通过 Gabriel 拓扑刻画高维 Auslander(-Gorenstein) 代数，给出 Auslander-Buchsbaum-Serre 定理的高维版本，并揭示模的投射维数与零化子理想投射维数之间的联系。

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of Geiss-Leclerc-Schroer. In addition, we prove a general result about relative 3-Calabi-Yau duality ove…

刻画高阶 Auslander(-Gorenstein) 代数

Characterizing higher Auslander(-Gorenstein) Algebras

本文利用挠理论方法将经典 Auslander-Tachikawa 定理推广到高阶情形，用投射维数小于 n 的模范畴的 Abel 性质刻画 n-Auslander 代数，并给出高阶 Auslander-Gorenstein 代数的范畴刻画及应用。

Auslander-Reiten Triangles in Homotopy Categories

Let $A$ be an artin algebra. We show that the bounded homotopy category of finitely generated right $A$-modules has Auslander-Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [H2]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules admits Auslander-Reiten triangles, which improves a main result in [G].

Auslander-Gorenstein单则代数的分类：无环情况

Classification of Auslander-Gorenstein monomial algebras: The acyclic case

本文通过Coxeter矩阵的Bruhat分解对无环单则代数进行线性代数分类，证明了在温和假设下，单则无环有向图代数为Auslander正则当且仅当其Coxeter矩阵有Bruhat分解。

虚拟Gorenstein Artin代数是弱Gorenstein的

Virtually Gorenstein Artin algebras are weakly Gorenstein

该研究证明虚拟Gorenstein Artin代数上的半Gorenstein投射模均为Gorenstein投射模，解答了相关学者的问题，还验证了多个Gorenstein相关猜想对这类代数成立，并建立了判定模维数的新准则。

Auslander-Gorenstein algebras from Serre-formal algebras via replication

We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga-Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras - called the replicated algebras - given by certain subquotients of its repetitive algebra. We calculate…

Frobenius 函子与 $n$-挠自由对象

Frobenius functors and $n$-torsionfree objects

本文研究阿贝尔范畴中 $n$-挠自由对象，证明 Frobenius 函子保持且忠实函子反映该性质，并利用挠自由滤过的稳定化建立弱 Gorenstein 性，同时将 Auslander 型条件传递到 Frobenius 扩张，并构造了非 Gorenstein 代数的实例。

Auslander-Reiten duality for subcategories

Auslander-Reiten duality for module categories is generalized to some sufficiently nice subcategories. In particular, our consideration works for $\mathcal{P}^{<\infty}(Λ)$, the subcategory consisting of finitely generated modules with finite projective dimension over an artin algebra $Λ$, and also, the subcategory of Gorenstein projectove modules of $\rm{mod}\mbox{-}Λ$, denoted by $\rm{Gprj}\mbox{-}Λ$. In this paper, we give a method to compute the Auslander-Reiten translation in $\mathcal{P}^{<\infty}(Λ)$ whenever $Λ$ is a $1$-Gorenstein algebra. In addition, we characterize when the Ausland…

On syzygies over 2-Calabi-Yau tilted algebras

We characterize the syzygies and co-syzygies over 2-Calabi-Yau tilted algebras in terms of the Auslander-Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen-Macaulay modules, the representation dimension of algebras and the Igusa-Todorov functions. In particular, we prove that the Igusa-Todorov dimensions of d-Gorenstein algebras are equal to d. For cluster-tilted algebras of Dynkin type D, we give a geometric description of the stable Cohen-Macaulay category in terms of tagged arcs in the punctured disc. We also describe th…

Shapes of Auslander-Reiten Triangles

Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for constructing the connecting component of the derived category of any tilted algebra. In particular we obtain a technique for constructing the derived category of any tilted algebra of finite representation type.

Cluster-tilted algebras and slices

We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice.

On Auslander-Type Conditions of Modules

We prove that for a left and right Noetherian ring $R$, $_RR$ satisfies the Auslander condition if and only if so does every flat left $R$-module, if and only if the injective dimension of the $i$th term in a minimal flat resolution of any injective left $R$-module is at most $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of the $i$th term in a minimal injective coresolution (resp. flat resolution) of any left $R$-module $M$ is at most the flat (resp. injective) dimension of $M$ plus $i-1$ for any $i \geq 1$, if and only if the flat (resp. injective) dimension of…

From CM-finite to CM-free

The aim of this paper is twofold. On one hand, we prove a slight generalization of the stability for Gorenstein categories in [SWSW] and [Huang]; and show that the relative Auslander algebra of a CM-finite algebra is CM-free. On the other hand, we describe the bounded derived category, and the Gorenstein defect category introduced in [BJO], via Gorenstein-projective objects; and we show that the Gorenstein defect category of a CM-finite algebra is triangle-equivalent to the singularity category of its relative Auslander algebra.

Cluster-tilted algebras without clusters

Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study the module category of B. In particular, we introduce the notion of reflections of tilted algebras and define an algorithm that constructs the transjective component of the Auslander-Reiten quiver of cluster-tilted algebras of tree type.

The Representation Dimension of a Selfinjective Algebra of Wild Tilted Type

We prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three, and give an explicit construction of an Auslander generator of its module category. We also show that if a connected selfinjective algebra admits an acyclic generalised standard Auslander-Reiten component then its representation dimension is equal to three.

Representation theory of partial relation extensions

Let C be a finite dimensional algebra of global dimension at most two. A partial relation extension is any trivial extension of C by a direct summand of its relation C-C-bimodule. When C is a tilted algebra, this construction provides an intermediate class of algebras between tilted and cluster tilted algebras. The text investigates the representation theory of partial relation extensions. When C is tilted, any complete slice in the Auslander-Reiten quiver of C embeds as a local slice in the Auslander-Reiten quiver of the partial relation extension; Moreover, a systematic way of producing part…

Indecomposable pure-injective objects in stable categories of Gorenstein-projective modules over Gorenstein orders

We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.

Derived equivalences for Cohen-Macaulay Auslander algebras

Let $A$ and $B$ be Gorenstein Artin algebras of Cohen-Macaulay finite type. We prove that, if $A$ and $B$ are derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent.

Algebras with finite relative dominant dimension and almost n-precluster tilting modules

In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce the almost n-precluster tilting module and establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules.

On the stable Auslander-Reiten components of certain monomorphism categories

Let $Λ$ be an Artin algebra and let $\rm{Gprj}\mbox{-}Λ$ denote the class of all finitely generated Gorenstein projective $Λ$-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category $\mathcal{S}({\rm Gprj}\mbox{-}Λ)$ containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category $\underline{\rm{Gprj}}\mbox{-}Λ$. In particular, for the finite components, we show that under certain mild conditions t…

高阶 Auslander 代数的 Auslander–Buchsbaum 公式及其应用

An Auslander-Buchsbaum formula for higher Auslander algebras and applications

本文建立高阶 Auslander 代数的非交换 Auslander–Buchsbaum 公式，据此证明倾斜 Auslander 代数与整体维数不超过2的 QF-1 代数等同，并给出分类及向更高同调维数的推广。

On relative Auslander algebras

Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent.

关于\(n -\)戈伦斯坦环的塞尔型准则及其在中山代数中的应用

A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras

研究关于左右诺特环的塞尔型条件\((G_n)\)，证明\(R\)是\(n -\)戈伦斯坦环当且仅当满足\((G_n)\)。通过合冲过滤将该准则应用于中山代数，保持并反映奥斯兰德 - 戈伦斯坦性质，解决了相关猜想。

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories. We give homological characterizations of Auslander algebras, especially an answer to a question of M. Artin. They are also closely related to Auslander's representation dimension of artin algebras and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities.

Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type

Let $Λ$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of $Λ$ form a maximal forward hom-orthogonal sequence of $Λ$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $Λ$. Thus we give a proof of Igusa-Todorov's conjecture.

Constructing tilted algebras from cluster-tilted algebras

Any cluster-tilted algebra is the relation extension of a tilted algebra. We present a method to, given the distribution of a cluster-tilting object in the Auslander-Reiten quiver of the cluster category, construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.

On algebras of finite Cohen-Macaulay type

We study Artin algebras $A$ and commutative Noetherian complete local rings $R$ in connection with the following decomposition property of Gorenstein-projective modules: $(*)$ any Gorenstein-projective module is a direct sum of finitely generated modules. We show that this direct decomposition property is related to the property of the algebra $A$, or the ring $R$, being (virtually) Gorenstein of finite Cohen-Macaquly type. Along the way we generalize classical results of Auslander and Ringel-Tachikawa from the early seventies, and results of Chen and Yoshino on the structure of Gorenstein-pro…

Another characterization of tilted algebras

We give a new characterization of tilted algebras by the existence of certain special subquivers in their Auslander-Reiten quiver. This result includes the existent characterizations of this kind and yields a way to obtain more tilted quotient algebras from a given algebra.

Pro-p Iwahori-Hecke algebras are Gorenstein

Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F-rational points of a connected split reductive group over F. For k an arbitrary field, we study the homological properties of the Iwahori-Hecke k-algebra H' and of the pro-p Iwahori-Hecke k-algebra H of G. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of G. If G is semisimple, we also show that this upper bound is sharp, that both H and H' are Auslander-Gorenstein and that there is a duality functor on the finitely generated modu…

Monomial Gorenstein algebras and the stably Calabi--Yau property

A celebrated result by Keller--Reiten says that $2$-Calabi--Yau tilted algebras are Gorenstein and stably $3$-Calabi--Yau. This note shows that the converse holds in the monomial case: a $1$-Gorenstein monomial algebra with a $3$-Calabi--Yau singularity category is $2$-Calabi--Yau tilted. We study the case of other Goresntein monomial algebras with stably Calabi--Yau singularity categories.