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

A celebrated result in representation theory is that of higher Auslander correspondence. Let $\mathrm{\Lambda }$ an Artin algebra and $X$ a $d$-cluster-tilting module. Iyama has shown that the endomorphism ring $\mathrm{\Gamma }$ 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.