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 $\mathcal{T}$ with a pre-simple-minded collection (=pre-SMC) $\mathcal{R}$, we construct a new triangulated category $\mathcal{U}$ such that the SMCs in $\mathcal{U}$ bijectively correspond to those in $\mathcal{T}$ containing $\mathcal{R}$. Secondly, we give an analogue of Buchweitz's theorem for the singularity category ${\mathcal{T}}_{\mathrm{sg}}$ of a SMC quadruple $(\mathcal{T},{\mathcal{T}}^{\mathrm{p}},\mathbb{S},\mathcal{S})$: the category ${\mathcal{T}}_{\mathrm{sg}}$ can be realized as the stable category of an extriangulated subcategory $\mathcal{F}$ of $\mathcal{T}$. Finally, we show the SMS (simple-minded system) 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.

26 pages, title changed, new sections 2.4, 3.2 added, an appendix attached