Cotorsion pairs in the cluster category of a marked surface

JOURNAL OF ALGEBRA

卷:391

Article

Zhang, Jie¹  Zhou, Yu¹  Zhu, Bin¹ 

[1] Tsinghua Univ, Dept Math Sci, Beijing 100084, Peoples R China

关键词: Marked surface;    Cluster category;    Cotorsion pair;    t-Structure;    Painting;    Mutation;    Rotation;   

DOI  :  10.1016/j.jalgebra.2013.06.014

来源: Elsevier

【 摘 要 】

We study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surface and the indecomposable objects in the associated cluster category, we prove that the dimension of extension space of two indecomposable objects in the cluster categories equals to the intersection number of the corresponding curves. By using this result, we prove that there are no non-trivial t-structures in the cluster categories when the surface is connected. Based on this result, we give a classification of cotorsion pairs in these categories. Moreover we define the notion of paintings of a marked surface without punctures and their rotations. They are a geometric model of cotorsion pairs and of their mutations respectively. (C) 2013 Elsevier Inc. All rights reserved.

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