From triangulated categories to module categories via localisation

TL;DR

This paper establishes a connection between module categories over endomorphism algebras of rigid objects in triangulated categories and Gabriel-Zisman localisations, generalising a known 2-Calabi-Yau tilting theorem.

Contribution

It extends the 2-Calabi-Yau tilting theorem by showing the module category is equivalent to a localisation, broadening the understanding of rigid objects in triangulated categories.

Findings

Module category is equivalent to Gabriel-Zisman localisation.

Generalises the 2-Calabi-Yau tilting theorem.

Provides a new perspective on rigid objects in triangulated categories.

Abstract

We show that the category of finite-dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the 2-Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.