Characterizing higher Auslander(-Gorenstein) Algebras

TL;DR

This work develops abelianness-based characterizations for higher Auslander(-Gorenstein) algebras by leveraging torsion and cotorsion theories, extending the classic Auslander–Tachikawa framework. It proves that, for an $n$-minimal Auslander–Gorenstein algebra $ ext{Γ}$, the category ${ m Gproj}^{oldsymbol{ullet} ext{≤}n-1}( ext{Γ})$ is abelian and closely tied to a two-step subcategory ${ m Sub}^2(oldsymbol{ ext{Q}})$, yielding torsion-pair and tilting/cotilting structures, and establishing Auslander–Iyama correspondences for $n$-precluster tilting modules. The paper also develops a parallel theory for $ au_n$-selfinjective algebras and connects these results to cotorsion pairs and torsion-cotorsion triples, providing categorical descriptions of semisimplicity and self-injectivity and clarifying the relationship to higher Auslander algebras. Overall, it unifies and extends prior work (Kong, Iyama–Solberg, and the Auslander–Iyama correspondence) and offers explicit module-category descriptions via submodule constructions and torsion theories.

Abstract

It is well known that for Auslander algebras, the category of all (finitely generated) projective modules is an abelian category and this property of abelianness characterizes Auslander algebras by Tachikawa theorem in 1974. Let $n$ be a positive integer. In this paper, by using torsion theoretic methods, we show that $ n $-Auslander algebras can be characterized by the abelianness of the category of modules with projective dimension less than $ n $ and a certain additional property, extending the classical Auslander-Tachikawa theorem. By Auslander-Iyama correspondence a categorical characterization of the class of Artin algebras having $ n $-cluster tilting modules is obtained. Since higher Auslander algebras are a special case of higher Auslander-Gorenstein algebras, the results are given in the general setting as extending previous results of Kong. Moreover, as an application of some results, we give categorical descriptions for the semisimplicity and selfinjectivity of an Artin algebra. Higher Auslander-Gorenstein Algebras are also studied from the viewpoint of cotorsion pairs and, as application, we show that they satisfy in two nice equivalences.

Theorems & Definitions (55)

• Theorem 1: Auslander-Tachikawa Theorem
• Theorem 2: Higher Auslander-Tachikawa Theorem
• Theorem 3: Theorem \ref{['Thm: Characterizing Auslander-Gorenstein algebras I']}
• Lemma 2.1
• proof
• Definition 2.2: Tilting and Cotilting Modules
• Definition 2.3: Ext-projectives and Ext-injectives
• Lemma 2.4
• proof
• Lemma 2.5