Rigid objects, triangulated subfactors and abelian localizations

Abstract

We show that the abelian category $\mathsf{mod}\text{-}\mathcal{X }$ of coherent functors over a contravariantly finite rigid subcategory $\mathcal{X }$ in a triangulated category $\mathcal{T }$ is equivalent to the Gabriel–Zisman localization at the class of regular maps of a certain factor category of $\mathcal{T }$, and moreover it can be calculated by left and right fractions. Thus we generalize recent results of Buan and Marsh. We also extend recent results of Iyama–Yoshino concerning subfactor triangulated categories arising from mutation pairs in $\mathcal{T }$. In fact we give a classification of thick triangulated subcategories of a natural pretriangulated factor category of $\mathcal{T }$ and a classification of functorially finite rigid subcategories of $\mathcal{T }$ if the latter has Serre duality. In addition we characterize $2$-cluster tilting subcategories along these lines. Finally we extend basic results of Keller–Reiten concerning the Gorenstein and the Calabi–Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between $2$-cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors.