Iyama's higher Auslander correspondence via the homological theory of idempotent ideals

TL;DR

This paper provides a concise, self-contained proof of Iyama's higher Auslander correspondence, linking $d$-cluster-tilting modules over Artin algebras to $d$-Auslander algebras using homological theory of idempotent ideals.

Contribution

It offers a new, streamlined proof of the higher Auslander correspondence leveraging the homological theory of idempotent ideals.

Findings

Establishes the correspondence between $d$-cluster-tilting modules and $d$-Auslander algebras.

Utilizes the homological theory of idempotent ideals for a simplified proof.

Clarifies the structure of endomorphism rings in higher Auslander theory.

Abstract

A celebrated result in representation theory is that of higher Auslander correspondence. Let $\Lambda$ an Artin algebra and $X$ a $d$-cluster-tilting module. Iyama has shown that the endomorphism ring $\Gamma$ of $X$ is a $d$-Auslander algebra, and moreover this gives a correspondence between $d$-cluster-tilting modules and $d$-Auslander algebras. We present a self-contained and concise proof using the homological theory of idempotent ideals of Auslander--Platzeck--Todorov.