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 τn plays an important role in the study of n-cluster tilting subcategories. We study the category Mn of preinjective-like modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ n-complete if Mn = addM for an n-cluster tilting object M . Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)-complete. This gives an inductive construction of n-complete algebras. For example, any representation-finite hereditary algebra Λ(1) is 1complete. Hence the Auslander algebra Λ(2) of Λ(1) is 2-complete. Moreover, for any n ≥ 1, we have an n-complete algebra Λ(n) which has an n-cluster tilting object M (n) such that Λ(n+1) = EndΛ(n) (M (n)). We give the presentation of Λ(n) by a quiver with relations. We apply our results to construct n-cluster tilting subcategories of derived categories of n-complete algebras.