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Intervals of s-torsion pairs in extriangulated categories with negative first extensions

Part of: Abelian categories

Published online by Cambridge University Press:  05 September 2022

TAKAHIDE ADACHI ,
HARUHISA ENOMOTO  and
MAYU TSUKAMOTO
TAKAHIDE ADACHI
Affiliation:
Faculty of Global and Science Studies, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8541, Japan. e-mail: tadachi@yamaguchi-u.ac.jp
HARUHISA ENOMOTO
Affiliation:
Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan. e-mail: henomoto@omu.ac.jp
MAYU TSUKAMOTO
Affiliation:
Graduate school of Sciences and Technology for Innovation, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan. e-mail: tsukamot@yamaguchi-u.ac.jp
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Abstract

As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to $\tau$-tilting reduction and brick labelling.

MSC classification

Primary: 18E40: Torsion theories, radicals

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 3 , May 2023 , pp. 451 - 469
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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