Silting reduction and Calabi–Yau reduction of triangulated categories

Abstract

We study two kinds of reduction processes of triangulated categories, that is, silting reduction and Calabi–Yau reduction. It is shown that the silting reduction $\mathcal {T}/\mathsf {thick}\mathcal {P}$ of a triangulated category $\mathcal {T}$ with respect to a presilting subcategory $\mathcal {P}$ can be realized as a certain subfactor category of $\mathcal {T}$, and that there is a one-to-one correspondence between the set of (pre)silting subcategories of $\mathcal {T}$ containing $\mathcal {P}$ and the set of (pre)silting subcategories of $\mathcal {T}/\mathsf {thick}\mathcal {P}$. This result is applied to show that the Amiot–Guo–Keller construction of $d$-Calabi–Yau triangulated categories with $d$-cluster-tilting objects takes silting reduction to Calabi–Yau reduction.