An introduction to higher Auslander-Reiten theory

TL;DR

This paper introduces higher Auslander-Reiten theory for Artin algebras, providing alternative proofs of key results and extending classical concepts to higher homological algebra.

Contribution

It offers new proofs of fundamental results in higher Auslander-Reiten theory and adapts classical methods to the higher setting.

Findings

Existence of $d$-almost-split sequences in $d$-cluster-tilting subcategories

Adaptation of Auslander's defect formula to $d$-exact sequences

Establishment of morphisms determined by objects in $d$-cluster-tilting subcategories

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 Smal{\o}. 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.