Reductions of triangulated categories and simple-minded collections

Abstract

Silting and Calabi-Yau reductions are important process in representation theory to construct new triangulated categories from given one, 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 $\cal T$ with a pre-simple-minded collection (=pre-SMC) $\cal R$, we construct a new triangulated category $\cal U$ such that the SMCs in $\cal U$ bijectively correspond to those in $\cal T$ containing $\cal R$. Secondly, we give an analogue of Buchweitz's theorem for the singularity category $\cal T_{\rm sg}$ of a SMC quadruple $(\cal T,\cal T^{\rm p},\mathbb S, \cal S)$: the category $\cal T_{\rm sg}$ can be realized as the stable category of an extriangulated subcategory $\cal F$ of $\cal T$. Finally, we show the SMS (simple-minded system) reduction due to Coelho Sim\~oes 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.