Auslander–Reiten theory revisited

Iyama, Osamu

doi:10.4171/062-1/8

Cited by 40 publications

(37 citation statements)

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“…To study higher Auslander algebras, the notion of n-cluster tilting subcategories (=maximal (n−1)-orthogonal subcategories) was introduced in [Iya3], and a higher analogue of Auslander-Reiten theory was developed in a series of papers [Iya1,Iya2,IO]; see also the survey paper [Iya4]. Recent results (in particular [Iya1], but also this paper and [HI,HZ1,HZ2,HZ3,IO]) suggest that n-cluster tilting modules behave very nicely if the algebra has global dimension n. In this paper, we call such algebras n-representation-finite and study them from the viewpoint of APR (=Auslander-Platzeck-Reiten) tilting theory (see [APR]).…”

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confidence: 99%

$n$-representation-finite algebras and $n$-APR tilting

Iyama

Oppermann

2011

Trans. Amer. Math. Soc.

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152

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$n$-representation-finite algebras and $n$-APR tilting

Iyama

Oppermann

2011

Trans. Amer. Math. Soc.

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152

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“…Higher Auslander–Reiten theory was introduced by Iyama in 2004 in (see also the survey article ). In addition to representation theory , it has exhibited connections to commutative algebra, commutative and non‐commutative algebraic geometry, and combinatorics, see for example .…”

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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“…This concept is used in the representation theory of finite F -algebras on vector spaces over a field F , such as the representation theory of quiver [15], and the representation theory of a group ring F [G] where G is a finite group( [6]. There are many mathematicians is developed the concept that Auslander given, such as Iyama [10], [11], and Oppermann [14]. From the previous paragraph, we have been knowing that the f -representation module of ring R is a module over R-algebra S. However, base on Example 5 and Proposition 1, the properties of the f -representation do not only depend on f -representation module of R but also on the properties of a ring homomorphism f .…”

Section: Introductionmentioning

confidence: 99%

Generalization of Schur's Lemma in Ring Representations on Modules over a Commutative Ring

Hijriati

Wahyuni

Wijayanti

2018

Eur. J. Pure Appl. Math.

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Abstract. Let R, S be rings with unity, M a module over S, where S a commutative ring, and f : R → S a ring homomorphism. A ring representation of R on M via f is a ring homomorphism µ : R → End S (M ), where End S (M ) is a ring of all S-module homomorphisms on M . One of the important properties in representation of ring is the Schur's Lemma. The main result of this paper is partly the generalization of Schur's Lemma in representations of ring on modules over a commutative ring 2010 Mathematics Subject Classifications: 16G10, 16W20, 12E15

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Auslander–Reiten theory revisited

Cited by 40 publications

References 100 publications

$n$-representation-finite algebras and $n$-APR tilting

$n$-representation-finite algebras and $n$-APR tilting

An introduction to higher Auslander-Reiten theory

Generalization of Schur's Lemma in Ring Representations on Modules over a Commutative Ring

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