On the Cluster Category of a Marked Surface

Details On the Cluster Category of a Marked Surface On the Cluster Category of a Marked Surface

2010-05-13

arxiv.org/abs/1005.2422v2

Mathematics

Representation Theory

33 pages, we add a new corollary 1.6 which shows there is a bijection between triangulations of (S,M) and the cluster-tilting

Scientific paper

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.

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