On the Cluster Category of a Marked Surface

Abstract

We study in this paper the cluster category C(S,M) of a marked surface (S,M). We explicitly describe the objects in C(S,M) as direct sums of homotopy classes of curves in (S,M) and one-parameter families related to closed curves in (S,M). Moreover, we describe the Auslander-Reiten structure of the category C(S,M) in geometric terms and show that the objects without self-extensions in C(S,M) correspond to curves in (S,M) without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.

33 pages, we add a new corollary 1.6 which shows there is a bijection between triangulations of (S,M) and the cluster-tilting objects of C(S,M), and every rigid indecomposable object is reachable from an initial triangulation

References in corpus (2)

• Mutation in triangulated categories and rigid Cohen-Macaulay modules
• Quivers with potentials associated to triangulated surfaces

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