On silting mutations preserving global dimension

Abstract

A $d$-silting object is a silting object whose derived endomorphism algebra has global dimension $d$ or less. We give an equivalent condition, which can be stated in terms of dg quivers, for silting mutations to preserve the $d$-siltingness under a mild assumption. Moreover, we show that this mild assumption is always satisfied by $\nu_d$-finite algebras. As an application, we give counterexamples to the open question by Herschend--Iyama--Oppermann: the quivers of higher hereditary algebras are acyclic. Our examples consist of a $2$-representation tame algebra with a $2$-cycle, and a $3$-homogeneous $2$-representation finite algebra with a cycle.