On class groups of upper cluster algebras

Abstract

We compute the class group of a full rank upper cluster algebra in terms of its exchange polynomials. As a corollary, we recover a theorem by Cao, Keller, and Qin from 2023 characterizing the UFDs among these algebras. Furthermore, under the additional hypothesis of acyclicity, we obtain a result by Garcia Elsener, Lampe, and Smertnig from 2019. Moreover, we show that all cluster and upper cluster algebras are finite factorization domains, meaning that every non-unit factors as a product of atoms (or irreducibles) and, for each element, there are only finitely many such factorizations up to order and associates. This strengthens another result by Cao, Keller, and Qin showing that cluster and upper cluster algebras are atomic.