From triangulated categories to module categories via localisation II: Calculus of fractions

TL;DR

This paper demonstrates that certain quotient categories derived from triangulated categories become abelian after localization, establishing a connection to module categories over endomorphism algebras, thus advancing the understanding of categorical localizations.

Contribution

It proves that the quotient of a Hom-finite triangulated category by a rigid object is preabelian and that its regular morphisms admit a calculus of fractions, leading to an abelian category equivalent to module categories.

Findings

The quotient category is preabelian.

Regular morphisms admit a calculus of fractions.

The localized category is equivalent to modules over an endomorphism algebra.

Abstract

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.