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Largest exact structures and almost split sequences on hearts of twin cotorsion pairs

Part of: Abelian categories

Published online by Cambridge University Press:  15 April 2024

Yu Liu ,
Wuzhong Yang  and
Panyue Zhou
Yu Liu
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, 710062 Xi’an, Shaanxi, P. R. China e-mail: recursive08@hotmail.com
Wuzhong Yang*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, 401331 Chongqing, P. R. China
Panyue Zhou
Affiliation:
School of Mathematics and Statistics, Changsha University of Science and Technology, 410114 Changsha, Hunan, P. R. China e-mail: panyuezhou@163.com
*
e-mail: mengxiankun061@qq.com
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Abstract

Hearts of cotorsion pairs on extriangulated categories are abelian categories. On the other hand, hearts of twin cotorsion pairs are not always abelian. They were shown to be semi-abelian by Liu and Nakaoka. Moreover, Hassoun and Shah proved that they are quasi-abelian under certain conditions. In this article, we first show that the heart of any twin cotorsion pair has a largest exact category structure and is always quasi-abelian. We also provide a sufficient and necessary condition for the heart of a twin cotorsion pair being abelian. Then by using the results we have got, we investigate the almost split sequences in the hearts of twin cotorsion pairs. Finally, as an application, we show that a Krull–Schmidt, Hom-finite triangulated category has a Serre functor whenever it has a cluster tilting object.

Keywords

Twin cotorsion pairheartquasi-abelianalmost split sequenceSerre functor

MSC classification

Primary: 18E10: Exact categories, abelian categories
Type
Article
Information
Canadian Journal of Mathematics , Volume 77 , Issue 4 , August 2025 , pp. 1347 - 1376
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Yu Liu is supported by the National Natural Science Foundation of China (Grant No. 12171397). Wuzhong Yang is supported by the National Natural Science Foundation of China (Grant No. 12271321) and the Scientific Research Foundation of Chongqing Normal University (Grant No. 24XLB008). Panyue Zhou is supported by the National Natural Science Foundation of China (Grant No. 12371034) and by the Hunan Provincial Natural Science Foundation of China (Grant No. 2023JJ30008). Wuzhong Yang is the corresponding author.

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