Abelian categories from triangulated categories via Nakaoka-Palu's localization

TL;DR

This paper extends the construction of abelian categories from triangulated categories by modifying Nakaoka-Palu's localization, unifying several existing approaches and introducing new phenomena like stable categories and recollements.

Contribution

It provides a generalized HTCP localization framework that encompasses Buan-Marsh's and Iyama-Yoshino's realizations, expanding the understanding of abelian categories from triangulated categories.

Findings

Unified framework for module categories over rigid objects

Inclusion of stable categories and recollements in the localization

Generalization of heart construction in triangulated categories

Abstract

The aim of this paper is to provide an expansion to Abe-Nakaoka's heart construction of the following two different realizations of the module category over the endomorphism ring of a rigid object in a triangulated category: Buan-Marsh's localization and Iyama-Yoshino's subfactor. Our method depends on a modification of Nakaoka-Palu's HTCP localization, a Gabriel-Zisman localization of extriangulated categories which is also realized as a subfactor of the original ones. Besides of the heart construction, our generalized HTCP localization involves the following phenomena: (1) stable category with respect to a class of objects; (2) recollement of triangulated categories; (3) recollement of abelian categories under a mild assumption.