Simple-minded reductions of triangulated categories

Haibo Jin

Published 2019-07-11Version 1

We will introduce a new reduction process of triangulated category, which is analogue to the silting reduction and 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.