On silting mutations preserving global dimension

TL;DR

This paper characterizes when silting mutations preserve the global dimension of derived endomorphism algebras, providing conditions in dg quivers and applying them to specific algebra classes, including a counterexample to a known conjecture.

Contribution

It offers an equivalent dg quiver condition for silting mutations to preserve $d$-siltingness and applies this to $ u_d$-finite algebras, including a counterexample to a conjecture.

Findings

Silting mutations preserve $d$-siltingness under a new dg quiver condition.

The mild assumption for preservation is always satisfied by $ u_d$-finite algebras.

Constructed a 2-representation tame algebra with a 2-cycle, countering a conjecture.

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 a counterexample to the open question by Herschend-Iyama-Oppermann: the quivers of higher hereditary algebras are acyclic. Our example is a $2$-representation tame algebra with a $2$-cycle which is derived equivalent to a toric Fano stacky surface.