REPETITIVE HIGHER CLUSTER CATEGORIES OF TYPE An

We show that the repetitive higher cluster category of type An, defined as the orbit category [Formula: see text], is equivalent to a category defined on a subset of diagonals in a regular polygon. This generalizes the construction of Caldero–Chapoton–Schiffler [Quivers with relations arising from clusters (An case), Trans. Amer. Math. Soc.358(3) (2006) 1347–1364], which we recover when p = m = 1, and the work of Baur–Marsh, [A geometric description of the m-cluster categories, Trans. Amer. Math. Soc.360(11) (2008) 5789–5803], treating the case p = 1, m > 1. Our approach also leads to a geometric model of the bounded derived category in type A.

Acyclic Calabi–Yau categories

We prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. No direct combinatorial proof is known as yet. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.