Cluster tilting for higher Auslander algebras

The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor $\tau_n$ plays an important role in the study of $n$-cluster tilting subcategories. We study the category $\MM_n$ of preinjective-like modules obtained by applying $\tau_n$ to injective modules repeatedly. We call a finite dimensional algebra $\Lambda$ \emph{$n$-complete} if $\MM_n=\add M$ for an $n$-cluster tilting object $M$. Our main result asserts that the endomorphism algebra $\End_\Lambda(M)$ is $(n+1)$-complete. This gives an inductive construction of $n$-complete algebras. For example, any representation-finite hereditary algebra $\Lambda^{(1)}$ is 1-complete. Hence the Auslander algebra $\Lambda^{(2)}$ of $\Lambda^{(1)}$ is 2-complete. Moreover, for any $n\ge1$, we have an $n$-complete algebra $\Lambda^{(n)}$ which has an $n$-cluster tilting object $M^{(n)}$ such that $\Lambda^{(n+1)}=\End_{\Lambda^{(n)}}(M^{(n)})$. We give the presentation of $\Lambda^{(n)}$ by a quiver with relations. We apply our results to construct $n$-cluster tilting subcategories of derived categories of $n$-complete algebras.