An introduction to higher Auslander-Reiten theory

Abstract

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.

25 pages, final version

References in corpus (2)

• Higher Auslander correspondence for dualizing $R$-varieties
• Right $n$-angulated categories arising from covariantly finite subcategories

Cited by in corpus (16)

• $d$-Auslander-Reiten sequences in subcategories
• Maximal ${\tau }_d$-rigid pairs
• Skew group algebras of Jacobian algebras
• Tensor products of n-complete algebras
• $d\mathbb{Z}$-cluster tilting subcategories of singularity categories
• On higher torsion classes
• Abelian quotients of the categories of short exact sequences
• Auslander's defect formula and a commutative triangle in an exact category
• The long $n$-exact sequence theorem in $n$-abelian categories
• Nakayama-type phenomena in higher Auslander--Reiten theory
• A characterisation of higher torsion classes
• Higher Auslander-Reiten sequences via morphisms determined by objects
• Higher Auslander's defect and classifying substructures of n-exangulated categories
• Modules determined by their composition factors in higher homological algebra
• Gabriel-Quillen embedding for $n$-exact categories
• Auslander-Reiten-Serre duality for n-exangulated categories