Quasi-abelian hearts of twin cotorsion pairs on triangulated categories

TL;DR

This paper demonstrates that the heart of a twin cotorsion pair in a triangulated category is quasi-abelian under mild conditions, with specific results for cluster categories and localizations.

Contribution

It establishes the quasi-abelian nature of hearts of twin cotorsion pairs and relates them to Gabriel-Zisman localizations and subfactor categories in triangulated categories.

Findings

The heart of a twin cotorsion pair is quasi-abelian.

In Krull-Schmidt categories with T=U, the heart is equivalent to a localization of the original heart.

Application to cluster categories shows certain quotients are quasi-abelian.

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 [X_R] the ideal of morphisms factoring through X_R=Ker(Hom(R,-)), then applications of our results show that C/[X_R] 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.