Abelian Categories Arising from Cluster Tilting Subcategories

Abstract

For a triangulated category \({\mathcal {T}}\), if \({\mathcal {C}}\) is a cluster-tilting subcategory of \({\mathcal {T}}\), then the factor category \({\mathcal {T}}{/}{\mathcal {C}}\) is an abelian category. Under certain conditions, the converse also holds. This is a very important result of cluster-tilting theory, due to Koenig–Zhu and Beligiannis. Now let \({\mathcal {B}}\) be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory \({\mathcal {C}}\) of \({\mathcal {B}}\), which is a generalization of cluster tilting subcategory. We show that \({\mathcal {C}}\) is cluster tilting if and only if the factor category \({\mathcal {B}}{/}{\mathcal {C}}\) is abelian. Our result generalizes the related results on a triangulated category and is new for an exact category case.

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Acknowledgements

The authors would like to thank the referee for reading the paper carefully and for many suggestions on mathematics and English expressions.

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Correspondence to Panyue Zhou.

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Panyue Zhou: The authors wish to thank Professor Bin Zhu for his helpful advices.

Yu Liu is supported by the Fundamental Research Funds for the Central Universities (Grants No. 2682018ZT25) and the National Natural Science Foundation of China (Grants No. 11901479). Panyue Zhou is supported by the Hunan Provincial Natural Science Foundation of China (Grants No. 2018JJ3205) and the National Natural Science Foundation of China (Grants Nos. 11901190 and 11671221), and by the Scientific Research Fund of Hunan Provincial Education Department (Grants No. 19B239).

Communicated by Wendy Lowen.

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Liu, Y., Zhou, P. Abelian Categories Arising from Cluster Tilting Subcategories. Appl Categor Struct 28, 575–594 (2020). https://doi.org/10.1007/s10485-019-09590-w

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Keywords

  • Extriangulated categories
  • Cluster tilting subcategories
  • Abelian categories

Mathematics Subject Classification

  • 18E30
  • 18E10