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Quasi-abelian hearts of twin cotorsion pairs on triangulated categories

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Quasi-abelian hearts of twin cotorsion pairs on triangulated categories. / Shah, Amit.

In: Journal of Algebra, Vol. 534, 15.09.2019, p. 313-338.

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Shah, Amit. / Quasi-abelian hearts of twin cotorsion pairs on triangulated categories. In: Journal of Algebra. 2019 ; Vol. 534. pp. 313-338.

Bibtex

@article{080f94b1303543819ece1165438f36ec,
title = "Quasi-abelian hearts of twin cotorsion pairs on triangulated categories",
abstract = "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.",
keywords = "Cluster category, Heart, Localisation, Quasi-abelian category, Triangulated category, Twin cotorsion pair",
author = "Amit Shah",
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: {\textcopyright} 2019 Elsevier Inc. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.",
year = "2019",
month = sep,
day = "15",
doi = "10.1016/j.jalgebra.2019.06.011",
language = "English",
volume = "534",
pages = "313--338",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "Academic Press",

}

RIS

TY - JOUR

T1 - Quasi-abelian hearts of twin cotorsion pairs on triangulated categories

AU - Shah, Amit

N1 - 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: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/9/15

Y1 - 2019/9/15

N2 - 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.

AB - 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.

KW - Cluster category

KW - Heart

KW - Localisation

KW - Quasi-abelian category

KW - Triangulated category

KW - Twin cotorsion pair

UR - http://www.scopus.com/inward/record.url?scp=85068063564&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2019.06.011

DO - 10.1016/j.jalgebra.2019.06.011

M3 - Journal article

AN - SCOPUS:85068063564

VL - 534

SP - 313

EP - 338

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -