Quasi-abelian hearts of twin cotorsion pairs on triangulated categories

We prove that, under a mild assumption, the heart (H) over bar 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) over bar 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(c)(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. (C) 2019 Elsevier Inc. All rights reserved.