Abelian hearts of twin cotorsion pairs on extriangulated categories

TL;DR

This paper investigates conditions under which hearts of twin cotorsion pairs induced by $d$-cluster tilting subcategories in extriangulated categories are abelian, extending previous work and revealing hereditary properties.

Contribution

It provides a necessary and sufficient condition for these hearts to be abelian and demonstrates their hereditary nature, generalizing prior results to a triangulated setting.

Findings

Hearts are abelian if and only if certain conditions are met.

Such hearts are shown to be hereditary.

The results extend Liu's work from exact to triangulated categories.

Abstract

It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian. In this article, we consider a special kind of hearts of twin cotorsion pairs induced by $d$-cluster tilting subcategories in extriangulated categories. We give a necessary and sufficient condition for such hearts to be abelian. In particular, we also can see that such hearts are hereditary. As an application, this generalizes the work by Liu in an exact case, thereby providing new insights in a triangulated case.