A Gabriel-type theorem for cluster tilting

Mizuno, Yuya

doi:10.1112/plms/pdt046

Cited by 12 publications

(9 citation statements)

References 35 publications

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“…(We do not assume gldim Λ ≤ n in this paper in contrast with several earlier papers [18,24,25]. ) See for example [1,19,20,26,28,29,30,31,34,36] for further results in higher Auslander-Reiten theory.…”

Section: Introductionmentioning

confidence: 97%

Auslander–Gorenstein algebras and precluster tilting

Iyama

Solberg²

2018

Advances in Mathematics

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Abstract. We generalize the notions of n-cluster tilting subcategories and τ -selfinjective algebras into n-precluster tilting subcategories and τn-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 AuslanderSolberg correspondence (Auslander-Solberg, 1993) as well as a Gorenstein analog of n-Auslander correspondence (Iyama, 2007). The Auslander-Reiten theory associated to an n-precluster tilting subcategory is used to classify the n-minimal Auslander-Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander-Solberg.

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Section: Introductionmentioning

confidence: 97%

Auslander–Gorenstein algebras and precluster tilting

Iyama

Solberg²

2018

Advances in Mathematics

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Section: Introductionmentioning

confidence: 99%

“…The question thus occurs if there are generalizations of the fundamental results about representation-finite algebras to algebras possessing a d-cluster-tilting module, that is, to d-representation-finite algebras. Significant progess in this and related directions has been made in recent years; see, for example, [5,25,28,29,31,32,39,40,41,43,44,45,46,50,52,53].The fundamental idea of this paper is to construct d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of certain algebras Λ of finite global dimension, called ν d -finite algebras. This class of algebras includes d-representation-finite algebras of global dimension d (which are a higher-dimensional analogue of representation-finite hereditary algebras) and, more generally, twisted fractionally Calabi-Yau algebras.…”

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confidence: 99%

d-Representation-finite self-injective algebras

Darpö

Iyama

2020

Advances in Mathematics

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We give a systematic method to construct self-injective algebras which are d-representation-finite in the sense of higher-dimensional Auslander-Reiten theory. Such algebras are given as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most d. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for k-linear categories to the setting of higher-dimensional Auslander-Reiten theory.Applications include n-fold trivial extensions and higher preprojective algebras, which are shown to be d-representation-finite in many cases. We also get a complete classification of all d-representation-finite self-injective Nakayama algebras for arbitrary d.An autoequivalence φ of a Krull-Schmidt category C is said to be admissible if φ i acts freely on isomorphism classes of indecomposable objects in C for all i > 0. For C = Λ, this is equivalent to Λ/φ being a finite-dimensional algebra (see Lemma 3.4).The aim of the present paper is to generalize Riedmann's construction from the point of view of higher-dimensional Auslander-Reiten theory.After Gabriel's and Riedtmann's seminal results, the theory of representation-finite algebras over algebraically closed fields had two culmination points. One is the paper [14] by Bautista-Gabriel-Roiter-Salmeron showing that it is a finite problem, concerning the combinatorics of ray categories, to determine whether or not a given algebra is representation finite (see also [24, Chapters 13,14]). The other one is the characterization of Auslander-Reiten quivers of representation-finite algebras by Igusa-Todorov [34] and Brenner [16] (see also [35]). In view of Auslander's bijection [9] between representation-finite algebras and Auslander algebras (that is, algebras of global dimension at most two and dominant dimension at least two), the latter result can be seen as a structure theorem for Auslander algebras.Seeking to generalize Auslander's result, one is naturally led to consider d-Auslander-algebras, i.e., algebras of global dimension at most d + 1 and dominant dimension at least d + 1. Such algebras correspond bijectively to equivalence classes of so-called d-cluster-tilting modules [37] (see Definition 2.1) below. The question thus occurs if there are generalizations of the fundamental results about representation-finite algebras to algebras possessing a d-cluster-tilting module, that is, to d-representation-finite algebras. Significant progess in this and related directions has been made in recent years; see, for example, [5,25,28,29,31,32,39,40,41,43,44,45,46,50,52,53].The fundamental idea of this paper is to construct d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of certain algebras Λ of finite global dimension, called ν d -finite algeb...

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“…Higher Auslander-Reiten theory was introduced by Iyama in 2004 in [18,19] (see also the survey article [20]). In addition to representation theory [15,23,31], it has exhibited connections to commutative algebra, commutative and non-commutative algebraic geometry, and combinatorics, see for example [1,16,17,24,32].…”

Section: Introductionmentioning

confidence: 99%

An introduction to higher Auslander-Reiten theory

Jasso

Kvamme

2018

Bull. London Math. Soc.

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This article consists of an introduction to Iyama's higher Auslander–Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander–Reiten theory, including the existence of d‐almost‐split sequences in d‐cluster‐tilting subcategories, following the approach to classical Auslander–Reiten theory due to Auslander, Reiten, and Smalø. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for d‐exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in d‐cluster‐tilting subcategories.

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A Gabriel-type theorem for cluster tilting

Cited by 12 publications

References 35 publications

Auslander–Gorenstein algebras and precluster tilting

Auslander–Gorenstein algebras and precluster tilting

d-Representation-finite self-injective algebras

An introduction to higher Auslander-Reiten theory

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