Localisation and colocalisation of triangulated categories at thick subcategories

TL;DR

This paper introduces a framework for localising and colocalising triangulated categories at thick subcategories, establishing long exact sequences that relate the original, localised, and colocalised categories, and connecting homological functors with their derived versions.

Contribution

It develops a new approach to localising and colocalising triangulated categories at thick subcategories, including natural long exact sequences and relations between homological functors and their derived counterparts.

Findings

Defined colocalisation for triangulated categories at thick subcategories

Constructed natural long exact sequences linking categories and functors

Connected homological functors with their total derived functors

Abstract

Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a natural long exact sequence containing the canonical map between a homological functor and its total derived functor with respect to a thick subcategory.