The Integral Cluster Category

TL;DR

This paper studies integral cluster categories over principal ideal domains, revealing their favorable properties such as being describable as orbit categories, with consistent indecomposable rigid objects and transitive mutation operations.

Contribution

It demonstrates that integral cluster categories over principal ideal domains have better-behaved structural properties than previously expected.

Findings

Categories can be described as orbit categories

Indecomposable rigid objects are independent of the ground ring

Mutation operation is transitive

Abstract

Integral cluster categories of acyclic quivers have recently been used in the representation-theoretic approach to quantum cluster algebras. We show that over a principal ideal domain, such categories behave much better than one would expect: They can be described as orbit categories, their indecomposable rigid objects do not depend on the ground ring and the mutation operation is transitive.