On silting mutations preserving global dimension

TL;DR

This work identifies precise, dg-quiver–level criteria for when silting mutations preserve $d$-siltingness, linking mutation theory to global-dimension constraints and ν_d-finiteness. It shows that in the ν_d-finite setting, no cycles of degree $-d+1$ arise, which implies strong compatibility with cluster-tilting mutations and clarifies mutation behavior in higher representation theory. The paper also proves that $d$-representation infinite algebras remain so under suitable silting mutations and provides a concrete counterexample to the open question that higher hereditary quivers are acyclic, using a 2-representation infinite algebra with a 2-cycle derived-equivalent to a toric stacky surface. Overall, it advances understanding of when silting mutations preserve homological finiteness properties and how these mutations interact with cluster-tilting theory and higher homological algebra.

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 $ν_d$-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.

Theorems & Definitions (71)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 6
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• proof