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

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.

💡 Research Summary

The paper investigates the relationship between a Hom‑finite triangulated category C equipped with a rigid object T (i.e. Ext¹(T,T)=0) and the module category over the opposite endomorphism algebra Γ = End_C(T)ᵒᵖ. The authors introduce the subcategory X_T = (Σ T)^⊥, consisting of objects annihilated by the functor Hom_C(T,–). They first form the additive quotient C/X_T, whose objects are those of C and whose morphisms are taken modulo those factoring through X_T.

A central result (Theorem 3.1) shows that C/X_T is pre‑abelian: every morphism possesses both a kernel and a cokernel. The proof relies on the triangulated structure of C and on a version of Wakamatsu’s Lemma, which connects minimal approximations by add T to the vanishing of certain triangles. Explicit constructions of kernels and cokernels are given via triangles involving minimal right add T‑approximations.

Next, the authors establish that C/X_T is integral. They prove that every object of add T becomes projective in the quotient and that every object of add Σ²T becomes injective. Consequently, C/X_T has enough projectives and enough injectives. Using results of Rump, they deduce that an integral pre‑abelian category is automatically left and right semi‑abelian, and they further show that the presence of enough projectives (resp. injectives) makes the category left (resp. right) integral.

The class R of regular morphisms in C/X_T (those that are both monic and epic) is then examined. Because the category is integral, R satisfies both a left and a right calculus of fractions. This is a non‑trivial improvement over the situation with the original class S of morphisms inverted by Hom_C(T,–), which does not admit such a calculus.

Localising C/X_T at R yields the Gabriel‑Zisman localisation (C/X_T)_R. By Rump’s theorem, this localisation is an abelian category. The authors identify its projective objects: up to isomorphism they are precisely the images of objects from add T. Since Hom_C(T,–) induces an equivalence between add T and the category of finitely generated projective Γ‑modules, the localisation (C/X_T)_R is equivalent to the category mod‑Γ of finite‑dimensional right Γ‑modules. In symbols, \