Reductions of triangulated categories and simple‐minded collections

Abstract

AbstractSilting and Calabi–Yau reductions are important processes in representation theory to construct new triangulated categories from given ones, which are similar to Verdier quotient. In this paper, first we introduce a new reduction process of triangulated category, which is analogous to the silting (Calabi–Yau) reduction. For a triangulated category with a pre‐simple‐minded collection (pre‐SMC) , we construct a new triangulated category such that the SMCs in bijectively correspond to those in containing . Second, we give an analogue of Buchweitz's theorem for the singularity category of a SMC quadruple : the category can be realized as the stable category of an extriangulated subcategory of . Finally, we show the simple‐minded system (SMS) reduction due to Coelho Simões and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi–Yau reduction is the shadow of silting reduction due to Iyama and Yang.