Rigid objects, triangulated subfactors and abelian localizations

TL;DR

This paper explores the structure of triangulated categories through abelian localizations and subfactor categories, extending key results on Gorenstein and Calabi-Yau properties, and characterizing 2-cluster tilting subcategories.

Contribution

It generalizes recent results on abelian localizations and triangulated subcategories, extending foundational theorems to broader contexts involving rigid and non-cluster tilting subcategories.

Findings

Generalized abelian localizations of triangulated categories.

Extended Gorenstein and Calabi-Yau properties to new subcategory classes.

Characterized 2-cluster tilting subcategories in this framework.

Abstract

We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent results of Buan-Marsh and Iyama-Yoshino. We also extend basic results of Keller-Reiten concerning the Gorenstein and the Calabi-Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between 2-cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors. Finally we characterize 2-cluster tilting subcategories along these lines.