Integral and quasi-abelian hearts of twin cotorsion pairs on extriangulated categories

Abstract

It was shown recently that the heart $\overline{\mathcal{H}}$ of a twin cotorsion pair $((\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V}))$ on an extriangulated category is semi-abelian. We provide a sufficient condition for the heart to be integral and another for the heart to be quasi-abelian. This unifies and improves the corresponding results for exact and triangulated categories. Furthermore, if $\mathcal{T}=\mathcal{U},$ then we show that the Gabriel-Zisman localization of $\overline{\mathcal{H}}$ at the class of its regular morphisms is equivalent to the heart of the single twin cotorsion pair $(\mathcal{S},\mathcal{T}).$ This generalizes and improves the known result for triangulated categories, thereby providing new insights in the exact setting.

Keywords:

• Extriangulated category
• heart
• integral category
• localization
• quasi-abelian category
• twin cotorsion pair

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 18E05
• Secondary 18E10
• 18E30
• 18E35
• 18E40
• 16G99

Acknowledgments

The authors would like to thank Thomas Brüstle and Robert J. Marsh for their support and guidance. This work began while the second author was visiting Sherbrooke, and he thanks the algebra group at Université de Sherbrooke for their hospitality and financial help. The authors are grateful to Dixy Msapato for spotting a typo in a previous version. The authors also thank the referee for comments and suggestions on an earlier version of the paper.

Additional information

Funding

The first author is supported by a “thésards étoiles” scholarship of the ISM, Bishop’s University and Université de Sherbrooke. The second author is grateful for financial support from the EPSRC grant EP/P016014/1 “Higher Dimensional Homological Algebra”. He also gratefully acknowledges financial support from the London Mathematical Society for his visit to Université de Sherbrooke. This study is also supported by Natural Sciences and Engineering Research Council of Canada. Some of this work was carried out during the second author’s Ph.D. at the University of Leeds.