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On the Cluster Category of a Marked Surface

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arxiv 1005.2422 v2 pith:ZQWEPFZ3 submitted 2010-05-13 math.RT

On the Cluster Category of a Marked Surface

classification math.RT
keywords categorycurvesclusterdescribemarkedobjectssurfacewithout
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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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    math.RT 2026-07 conditional novelty 6.0

    Indecomposable rigid objects in the derived category of the (2,2,2,2)-weighted projective line are shown to correspond to graded simple arcs on a sphere with four binaries, with Hom-dimensions given by oriented inters...