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Cotorsion pairs in the cluster category of a marked surface

Jie Zhang Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, PR China Yu Zhou Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, PR China Bin Zhu Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, PR China

Representation Theory mathscidoc:2204.30003

Journal of Algebra, 391, 209-226, 2013.10
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.
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@inproceedings{jie2013cotorsion,
  title={Cotorsion pairs in the cluster category of a marked surface},
  author={Jie Zhang, Yu Zhou, and Bin Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220429170703596269180},
  booktitle={Journal of Algebra},
  volume={391},
  pages={209-226},
  year={2013},
}
Jie Zhang, Yu Zhou, and Bin Zhu. Cotorsion pairs in the cluster category of a marked surface. 2013. Vol. 391. In Journal of Algebra. pp.209-226. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220429170703596269180.
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