From triangulated categories to module categories via localisation

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.

💡 Research Summary

The paper investigates the relationship between a Hom‑finite triangulated category 𝒯 and the module category over the endomorphism algebra of a rigid object inside it. Let T be a rigid object in 𝒯, i.e. Ext¹𝒯(T,T)=0, and set A = End𝒯(T)ᵒᵖ, a finite‑dimensional k‑algebra. In the classical 2‑Calabi‑Yau (2‑CY) setting, when T is a cluster‑tilting object, Keller and Reiten proved that the factor category 𝒯/ add T