Geodesic
Largest exact structures and almost split sequences on hearts of twin cotorsion pairs
Canadian journal of mathematics, Tome 77 (2025) no. 4, pp. 1347-1376

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DOI

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.
DOI : 10.4153/S0008414X2400035X
Mots-clés : Twin cotorsion pair, heart, quasi-abelian, almost split sequence, Serre functor 
Liu, Yu; Yang, Wuzhong; Zhou, Panyue. Largest exact structures and almost split sequences on hearts of twin cotorsion pairs. Canadian journal of mathematics, Tome 77 (2025) no. 4, pp. 1347-1376. doi: 10.4153/S0008414X2400035X
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     title = {Largest exact structures and almost split sequences on hearts of twin cotorsion pairs},
     journal = {Canadian journal of mathematics},
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     year = {2025},
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