From triangulated categories to module categories via localization II: calculus of fractions

We show that the quotient of a Hom-finite triangulated category 𝒷 by the kernel of the functor Hom𝒷(T,−), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admits a calculus of left and right fractions. It follows that the Gabriel–Zisman localization 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 opposite of the endomorphism algebra of T in 𝒷.