Factoriality and class groups of cluster algebras

TL;DR

This paper studies the class groups of cluster algebras, showing they are finitely generated free abelian groups with infinitely many height-1 prime ideals, and provides explicit descriptions for acyclic cases, extending factoriality classifications.

Contribution

It establishes that Krull domain cluster algebras have finitely generated free abelian class groups and describes these groups explicitly for acyclic seeds, extending factoriality results.

Findings

Class groups are finitely generated free abelian groups.

Every class contains infinitely many height-1 prime ideals.

Explicit class group descriptions for acyclic cluster algebras.

Abstract

Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-$1$ prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss--Leclerc--Schr\"oer.