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 iff 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 three is a cluster category. This entails the classification of the rigid Cohen-Macaulay modules first obtained by IyamaYoshino. As an application to the combinatorics of quiver mutation, we prove the nonacyclicity 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.