Silting interval reduction and 0-Auslander extriangulated categories

Abstract

We give a reduction technique for silting intervals in extriangulated categories, which we call "silting interval reduction". It provides a reduction technique for tilting subcategories when the extriangulated categories are exact categories. In 0-Auslander extriangulated categories (a generalization of the well-known two-term category $K^{[-1,0]}(\mathsf{proj}\Lambda)$ for an Artin algebra $\Lambda$), we provide a reduction theory for silting objects as an application of silting interval reduction. It unifies two-term silting reduction and Iyama-Yoshino's 2-Calabi-Yau reduction. The mutation theory developed by Gorsky, Nakaoka and Palu recently can be deduced from it. Since there are bijections between the silting objects and the support $\tau$-tilting modules over certain finite dimensional algebras, we show it is compatible with $\tau$-tilting reduction. This compatibility theorem also unifies the two compatibility theorems obtained by Jasso in his work on $\tau$-tilting reduction. We give a new construction for 0-Auslander extriangulated categories using silting mutation, together with silting interval reduction, we obtain some results on silting quivers. Finally, we prove that $d$-Auslander extriangulated categories are related to a certain sequence of silting mutations.