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Quasi-abelian hearts of twin cotorsion pairs on triangulated categories

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We prove that, under a mild assumption, the heart H‾ of a twin cotorsion pair ((S,T),(U,V)) on a triangulated category C is a quasi-abelian category. If C is also Krull-Schmidt and T=U, we show that the heart of the cotorsion pair (S,T) is equivalent to the Gabriel-Zisman localisation of H‾ at the class of its regular morphisms. In particular, suppose C is a cluster category with a rigid object R and [XR] the ideal of morphisms factoring through XR=Ker(HomC(R,−)), then applications of our results show that C/[XR] is a quasi-abelian category. We also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of C.

Original languageEnglish
JournalJournal of Algebra
Pages (from-to)313-338
Number of pages26
Publication statusPublished - 15 Sept 2019
Externally publishedYes

Bibliographical note

Funding Information:
The author would like to thank Robert J. Marsh for his helpful guidance and support during the preparation of this article. The author is also grateful for financial support from the University of Leeds through a University of Leeds 110 Anniversary Research Scholarship. The author also thanks the referee for comments on an earlier version of the paper.

Publisher Copyright:
© 2019 Elsevier Inc.

Copyright 2019 Elsevier B.V., All rights reserved.

    Research areas

  • Cluster category, Heart, Localisation, Quasi-abelian category, Triangulated category, Twin cotorsion pair

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