# On Class Groups of Upper Cluster Algebras

Mara Pompili  University of Graz, Department of Mathematics and Scientific
Computing, NAWI Graz, Heinrichstrasse 36, 8010 Graz, Austria
<EMAIL_ADDRESS>
###### Abstract.

We compute the class groups of full rank upper cluster algebras in terms of
the exchange polynomials. This characterizes the UFDs among these algebras.
Our results simultaneously generalize theorems of Garcia Elsener, Lampe, and
Smertnig from 2019 and of Cao, Keller, and Qin from 2023. Furthermore we show
that every (upper) cluster algebra is a finite factorization domain.

###### Key words and phrases:

Cluster algebras. Upper cluster algebras. UFDs. Factorization theory. Krull
domains. Class groups.

###### 2020 Mathematics Subject Classification:

Primary 13F60. Secondary 13F05, 13F15

The author was partially supported by the Austrian Science Fund (FWF),
projects W1230 and P36742-N

## Introduction

In recent years, cluster algebras have emerged as a powerful tool in various
branches of mathematics, including algebraic geometry, representation theory,
and mathematical physics. These structures provide deep insights into the
interplay between algebraic and combinatorial phenomena. Strongly related to a
cluster algebra is its upper cluster algebra. Upper cluster algebras are
overrings of cluster algebras that are in general better behaved from a ring-
theoretic perspective and often even coincide with the cluster algebra, for
instance, in the locally acyclic case [BFZ05, Mul14].

Cluster algebras were introduced by Fomin and Zelevinsky [FZ02] in the early
2000s, with the intent of investigating dual canonical bases and total
positivity within semisimple Lie groups. Quickly, cluster algebra theory has
evolved into a distinct and self-sustained domain, transcending its initial
motivations, cf. for example the surveys [FZ03, Kel12, Lec10, LW14, Nak22,
Zel05]. It has found numerous applications, particularly in fields like
representation theory, Poisson geometry, and Teichmüller theory. In this paper
we focus on ring-theoretic properties of (upper) cluster algebras, following
in the footsteps of [CKQ23, GELS19, GLS13, Mul13].

The Laurent phenomenon (Theorem 1.9) implies that every seed gives rise to an
embedding of the cluster algebra into a Laurent polynomial ring. The upper
cluster algebra is defined as the intersection of all these Laurent polynomial
rings. Therefore upper cluster algebras are an upper bound for cluster
algebras. The intrinsic characteristics of upper cluster algebras simplify the
exploration of their factorization properties. An even more accessible
scenario emerges within the case of full rank cluster algebras, where upper
cluster algebras take the form of a finite intersection of Laurent polynomial
rings.

Locally acyclic cluster algebras and full rank upper cluster algebras are
Krull domains. Krull domains [Fos73] are one of the central objects in the
study of non-unique factorizations. They form an important generalization of
unique factorization domains (UFDs), because they possess unique factorization
on the level of divisorial ideals, which allows one to systematically study
factorizations of elements. A key invariant of Krull domains are their class
groups, which measure the failure of unique factorization in these domains. In
particular, a domain is a UFD if and only if it is a Krull domain with trivial
class group. In the present paper, we study class groups of upper cluster
algebras that satisfy the starfish condition at one seed (see Definition
1.13), in particular this includes all full rank upper cluster algebras.

The exploration of factorization properties within cluster algebras was
initially undertaken by Geiss, Leclerc, and Schröer [GLS13]. Subsequently,
Garcia Elsener, Lampe, and Smertnig achieved a breakthrough by computing class
groups and their ranks for acyclic cluster algebras [GELS19]. Very recently,
Cao, Keller, and Qin [CKQ23] switched the focus to upper cluster algebras.
They provided a complete characterization of full rank upper cluster algebras
that are UFDs and they also exhibited local factorization properties using
valuation pairs.

In this paper, we simultaneously generalize results of Garcia Elsener, Lampe,
and Smertnig and Cao, Keller, and Qin, computing the class group of full rank
upper cluster algebras in terms of the exchange polynomials. We show that the
class group $\mathcal{C}(\mathcal{U})$ of a full rank upper cluster algebra
$\mathcal{U}$ is a finitely generated free abelian group with rank $r=t-n$,
where $t$ is the number of irreducible factors of the exchange polynomials and
$n$ is the number of exchangeable variables (see Theorem 3.8). Furthermore, we
show that in this case the upper cluster algebra $\mathcal{U}$ contains
infinitely many height-1 prime divisors in each class. This leads to a
fascinating dichotomy between those upper cluster algebras that are UFDs and
those that are not: in the latter case every finite set
$L\subseteq\mathbb{Z}_{\geq 2}$ can be realized as a length set of some
element. This result even applies to a larger class of upper cluster algebras
than the full rank ones (see Section 3).

We remark that compared to the results of Garcia Elsener, Lampe, and Smertnig
[GELS19] we need the extra assumption on full rank; on the other hand we do
not need the acyclicity assumption present in their work. Thus, while our
results do not fully generalize results of [GELS19] in every case, they do so
in a large and important subclass.

Whereas factorization properties in cluster algebras that are Krull domains
have been studied, so far it has not been considered which factorization
properties hold in full generality in arbitrary (upper) cluster algebras. For
instance, the Markov cluster algebra is not a Krull domain [GELS19, Section
6]. Interestingly, its upper cluster algebra is however a UFD. Indeed, the
question of whether upper cluster algebras that are not Krull domains exist
remains an open problem. In the setting of (upper) cluster algebras without
any extra conditions, we establish that every (upper) cluster algebra is a
finite factorization domain (see Theorem 2.7), hence in particular atomic.
That is, every element has a factorization into atoms (i.e., irreducible
elements) and for a given element there are only finitely many such
factorizations up to permutation and associativity. In particular, the
existence of factorizations has been tacitly used in a few places in the
literature, but no proof of it has been published.

The paper is organized as follows. In Section 1, we recall basic definitions
and results on cluster algebras, factorization theory, and Krull domains. In
Section 2, we study some factorization properties of (upper) cluster algebras,
proving that cluster algebras and upper cluster algebras are finite
factorization domains. The proof we will not necessitate preliminaries on
Krull domains. In Section 3, we focus on full rank upper cluster algebras,
computing their class groups. Finally in Section 4, we give an interpretation
in terms of multiplicative ideal theory of the notion of valuation pairing and
local factorization introduced in [CKQ23].

### Notations and assumptions

Throughout the paper we consider cluster algebras of geometric type, allowing
frozen variables. However, we always assume that all frozen variables are
invertible. Moreover, $K$ will denote a field of characteristic zero, or the
ring $\mathbb{Z}$ of integers. If the base ring $K$ is a field, we assume that
the underlying ice quiver $\Gamma(B)$ of the exchange matrix $B$ of our
cluster algebra has no isolated exchangeable vertices.

A domain is a non-zero commutative ring $A$ without non-zero zero-divisors. We
denote by $A^{\times}$ its group of units, by $A^{\bullet}=A\setminus\\{0\\}$
its monoid of non-zero elements, and by $\mathbf{q}(A)$ its quotient field. We
denote by $\mathbb{N}$ the semigroup of positive integers and by
$\mathbb{N}_{0}$ the monoid $\mathbb{N}\cup\\{0\\}$. Moreover, if
$n\in\mathbb{N}$, we denote by $[1,n]$ the set $\\{1,\dots,n\\}$.

##  1\. Preliminaries

###  1.1. Quivers

A quiver is a finite directed graph. Thus, it is a tuple
$\mathcal{Q}=(\mathcal{Q}_{0},\mathcal{Q}_{1},s,t)$ where $\mathcal{Q}_{0}$
(the set of vertices) and $\mathcal{Q}_{1}$ (the set of arrows) are finite
sets and $s,t\colon\mathcal{Q}_{1}\to\mathcal{Q}_{0}$ are maps (the sources
and the targets). We write $\alpha\colon i\to j$ to indicate that $\alpha$ is
an arrow in $\mathcal{Q}_{1}$ with $s(\alpha)=i$ and $t(\alpha)=j.$

Through the entire paper, a quiver has no oriented cycles of length one or
two.

######  Definition 1.1 (Ice quivers).

An ice quiver is a quiver $\mathcal{Q}=(\mathcal{Q}_{0},\mathcal{Q}_{1},s,t)$
together with a partition of $\mathcal{Q}_{0}$ into exchangeable and frozen
vertices, with the assumption that there are no arrows between two frozen
vertices. The exchangeable part of $\mathcal{Q}$ is the subquiver on the set
of exchangeable vertices. We say that $\mathcal{Q}$ is acyclic if its
exchangeable part is an acyclic quiver, i.e., if it does not contain any
oriented cycles.

Let $n,m\in\mathbb{N}_{0}$ such that $n+m>0$. Let
$\mathcal{Q}=(\mathcal{Q}_{0},\mathcal{Q}_{1},s,t)$ be an ice quiver with
exchangeable vertices $[1,n]$ and frozen vertices $[n+1,n+m].$ We can
associate to $\mathcal{Q}$ a matrix
$B=B(\mathcal{Q})=(b_{ij})\in\mathcal{M}_{(n+m)\times n}(\mathbb{Z})$ defined
by

$b_{ij}=|\,\alpha\colon i\to j\,|-|\,\alpha\colon j\to i\,|.$

Given an $(n+m)\times n$ matrix, its principal part is the submatrix supported
on the first $n$ rows. Notice that the principal part of $B(\mathcal{Q})$ is
skew-symmetric.

######  Definition 1.2 (Exchange matrices).

A matrix $B=(b_{ij})\in\mathcal{M}_{n\times n}(\mathbb{Z})$ is skew-
symmetrizable if there exists a diagonal matrix $D\in\mathcal{M}_{n\times
n}(\mathbb{N})$ such that $DB$ is skew-symmetric. An $(n+m)\times n$ integer
matrix is an exchange matrix if its principal part is skew-symmetrizable.

For a matrix $B=(b_{ij})\in\mathcal{M}_{n\times n}(\mathbb{Z})$ being skew-
symmetrizable is equivalent to the existence of positive integers
$d_{1},\dots,d_{n}$ such that $d_{i}b_{ij}=-d_{j}b_{ji}$ for every
$i,j\in[1,n],$ hence if $B$ is a skew-symmetrizable matrix, either
$b_{ij}=b_{ji}=0$ or $b_{ij}b_{ji}<0$, so in particular $b_{ii}=0$.

######  Remark 1.3.

Let $\mathcal{Q}$ be an ice quiver, then the matrix $B(\mathcal{Q})$ is an
exchange matrix with skew-symmetric principal part. Conversely, we can
associate to any exchange matrix $B$ an ice quiver $\Gamma(B)$. The
exchangeable and frozen vertices of $\Gamma(B)$ are $[1,n]$ and $[n+1,n+m],$
respectively. The arrows of $\Gamma(B)$ are defined as follows: if $b_{ij}>0$,
add $b_{ij}$ arrows from $i$ to $j$. If $i$ is a frozen vertex and $b_{ij}<0$,
then add also $-b_{ij}$ arrows from $j$ to $i.$ If $B$ has skew-symmetric
principal part, then $B(\Gamma(B))=B.$

###  1.2. Seeds, mutations and cluster algebras

A cluster is a pair $(\mathbf{x},\mathbf{y})$ with
$\mathbf{x}=(x_{1}$,$\ldots$, $x_{n})$ and $\mathbf{y}=(x_{n+1}$,$\ldots$,
$x_{n+m})$ such that $(x_{1}$,$\ldots$, $x_{n}$,$\ldots$, $x_{n+m})$ are $n+m$
algebraically independent indeterminates over $K$. We refer to the elements of
$\mathbf{x}$ as exchangeable variables and to the elements of $\mathbf{y}$ as
frozen variables. Given a cluster, the field
$\mathcal{F}=\mathbf{q}(K)(x_{1}$,$\ldots$, $x_{n+m})$ is called the ambient
field.

######  Definition 1.4 (Seeds).

A seed is a triple $\Sigma=(\mathbf{x},\mathbf{y},B)$ such that
$(\mathbf{x},\mathbf{y})$ is a cluster and $B$ is a $(n+m)\times n$ exchange
matrix. We always tacitly assume $\mathbf{x}=(x_{1},\ldots,x_{n})$ and
$\mathbf{y}=(x_{n+1},\ldots,x_{n+m})$. A seed is called acyclic if the ice
quiver $\Gamma(B)$ is acyclic.

We identify two seeds $\Sigma=(\mathbf{x},\mathbf{y},B)$ and
$\Sigma^{\prime}=(\mathbf{x}^{\prime},\mathbf{y}^{\prime},B^{\prime})$ if
there exists a permutation $\sigma\in S_{n+m}$ such that $\sigma(i)\in[1,n]$
for all $i\in[1,n]$ and

  * •

$b_{ij}=b^{\prime}_{\sigma(i),\sigma(j)}$ for every $i,j\in[1,n+m]$;

  * •

$x_{i}=x^{\prime}_{\sigma(i)},\,y_{j}=y^{\prime}_{\sigma(j)}$ for every
$i\in[1,n]$ and $j\in[n+1,n+m]$.

######  Definition 1.5 (Mutation of seeds).

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed with ambient field
$\mathcal{F}$. Fix an exchangeable index $i\in[1,n].$ The mutation of $\Sigma$
in direction $i$ is the triple
$\mu_{i}(\Sigma)=(\mathbf{x}_{i},\mathbf{y}_{i},B_{i})$ defined as follows

  * (a)

$\mathbf{x}_{i}=(x_{1},\dots,x_{i}^{\prime},\dots,x_{n})$ with

$x_{i}^{\prime}=\frac{1}{x_{i}}\left(\prod_{b_{ki}>0}x_{k}^{b_{ki}}+\prod_{b_{ki}<0}x_{k}^{-b_{ki}}\right)\in\mathcal{F};$

  * (b)

$\mathbf{y}_{i}=\mathbf{y}$;

  * (c)

$B_{i}=(b^{\prime}_{jk})\in\mathcal{M}_{(n+m)\times n}(\mathbb{Z})$ with

$b_{jk}^{\prime}=\begin{cases}-b_{jk}&\text{if $j=i$ or $k=i$;}\\\
b_{jk}+\frac{1}{2}(|b_{ji}|b_{ik}+b_{ji}|b_{ik}|)&\text{otherwise.}\end{cases}$

One can prove that $\mu_{i}(\Sigma)$ is a seed with the same number of
exchangeable and frozen variables and the same ambient field as $\Sigma$, and
$B_{i}$ has the same rank as $B$. Notice that
$(\mu_{i}\circ\mu_{i})(\Sigma)=\Sigma.$

######  Definition 1.6 (Exchange polynomials).

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. Suppose that $i\in[1,n]$ is
an exchangeable index. The polynomial

$f_{i}:=x_{i}x_{i}^{\prime}\in K[\mathbf{x},\mathbf{y}]$

is called the exchange polynomial associated to $x_{i}$ (with respect to the
seed $\Sigma$).

Mutations induce an equivalence relation on seeds. Indeed we say that two
seeds $\Sigma=(\mathbf{x},\mathbf{y},B)$ and
$\Sigma^{\prime}=(\mathbf{z},\mathbf{y},C)$ are mutation-equivalent if there
exist $i_{1},\dots,i_{k}\in[1,n]$ such that
$\Sigma^{\prime}=(\mu_{i_{1}}\circ\cdots\circ\mu_{i_{k}})(\Sigma)$. In this
case, we write $\Sigma\sim\Sigma^{\prime},$ or, if no confusion can arise,
$\mathbf{x}\sim\mathbf{z}$. Denote by $\mathcal{M}(\Sigma)$ the mutation
equivalence class of $\Sigma$ and by $\mathcal{X}=\mathcal{X}(\Sigma)$ the set
of all exchangeable variables appearing in $\mathcal{M}(\Sigma).$

######  Definition 1.7 (Cluster algebras).

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. The cluster algebra
associated to $\Sigma$ is the $K$-algebra

$\mathcal{A}=\mathcal{A}(\Sigma)=K[x,y^{\pm 1}\mid
x\in\mathcal{X},y\in\mathbf{y}].$

The elements $x\in\mathcal{X}$ are called cluster variables of
$\mathcal{A}(\Sigma)$; the cluster variables in the initial seed $\Sigma$ are
called initial cluster variables; the elements $y\in\mathbf{y}$ are called
frozen variables.

In the literature one can also find the definition of a cluster algebra
associated to a seed $\Sigma=(\mathbf{x},\mathbf{y},B)$ as the $K$-algebra
$K[x,y\mid x\in\mathcal{X},y\in\mathbf{y}]$. However we only deal with the
case of invertible frozen variables.

For any seed $\Sigma=(\mathbf{x},\mathbf{y},B)$, denote by

$\mathcal{L}_{\mathbf{x}}=K[u^{\pm 1}\mid u\in\mathbf{x}\cup\mathbf{y}]$

the localization of $K[u\mid u\in\mathbf{x}\cup\mathbf{y}]$ at
$S_{\mathbf{x}}:=\\{\,{x_{1}^{a_{1}}\cdots x_{n+m}^{a_{n+m}}\mid
a_{i}\in\mathbb{N}_{0}}\,\\}$ and by

$\mathcal{L}_{\mathbf{x},\mathbb{Z}}=\mathbb{Z}[u^{\pm 1}\mid
u\in\mathbf{x}\cup\mathbf{y}]$

the localization of $\mathbb{Z}[u\mid u\in\mathbf{x}\cup\mathbf{y}]$ at
$S_{\mathbf{x}}.$

######  Definition 1.8 (Upper cluster algebras).

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. The upper cluster algebra
associated to $\Sigma$ is the $K$-algebra

$\mathcal{U}=\mathcal{U}(\Sigma)=\bigcap_{\mathbf{z}\sim\mathbf{x}}\mathcal{L}_{\mathbf{z}}.$

An upper cluster algebra $\mathcal{U}(\Sigma)$ is called a full rank upper
cluster algebra if its initial exchange matrix has full rank. For a full rank
upper cluster algebra $\mathcal{U}$, every exchange matrix of $\mathcal{U}$
has full rank, since, as mentioned above, mutations preserve the rank ([BFZ05,
Lemma 3.2]).

The following result, known as Laurent phenomenon, is due to Fomin and
Zelevinsky and explains the relation between cluster algebras and upper
cluster algebras.

######  Theorem 1.9 (Laurent phenomenon, [FZ02]).

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. Let $\mathcal{X}(\Sigma)$ be
the set of cluster variables associated to $\Sigma$ and let
$\mathcal{A}(\Sigma),\,\mathcal{U}(\Sigma)$ be the cluster algebra and the
upper cluster algebra associated to $\Sigma,$ respectively. Then

$\mathcal{X}(\Sigma)\subseteq\bigcap_{\mathbf{z}\sim\mathbf{x}}\mathcal{L}_{\mathbf{z},\mathbb{Z}}\,,\qquad\text{in
particular,}\qquad\mathcal{A}(\Sigma)\subseteq\mathcal{U}(\Sigma).$

Given a seed $\Sigma=(\mathbf{x},\mathbf{y},B)$, the following inclusions hold

$\mathcal{A}(\Sigma)\subseteq\mathcal{U}(\Sigma)\subseteq\bigcap_{i=0}^{n}\mathcal{L}_{\mathbf{x}_{i}}.$
(1.1)

In certain instances, the inclusions (1.1) turn into equalities, see Examples
1.12.

Locally acyclic cluster algebras were introduced by Muller in [Mul13] to
generalize acyclic cluster algebras maintaining some of their properties. They
are a much large class than the acyclic cluster algebras, in particular they
include cluster algebras arising from marked surfaces. Locally acyclic cluster
algebras are finitely generated, noetherian, and integrally closed ([Mul13,
Theorem 4.2]). Moreover, the following theorem holds.

######  Theorem 1.10 ([Mul14, Theorem 2]).

If $\mathcal{A}$ is locally acyclic, then $\mathcal{A}=\mathcal{U}$.

Another class for which one of the inclusions becomes an equality is the one
of full rank upper cluster algebra. The following result is known as Starfish
lemma.

######  Theorem 1.11 ([CKQ23, Theorem 2.16]).

Let $\mathcal{U}$ be a full rank upper cluster algebra. Then

$\mathcal{U}=\bigcap_{i=0}^{n}\mathcal{L}_{\mathbf{x}_{i}},$

for all seeds $(\mathbf{x},\mathbf{y},B)$.

Assume that $K$ is a field and that $\Sigma$ is an isolated seed, i.e.
$\Sigma=\left(\left(x_{1},\dots,x_{n}\right),\emptyset,0\right)$. Then all the
inclusions (1.1) are equalities. Indeed

$\mathcal{A}(\Sigma)=\mathcal{U}(\Sigma)=\bigcap_{i=0}^{n}\mathcal{L}_{\mathbf{x}_{i}}=K[x_{1}^{\pm
1},\dots,x_{n+m}^{\pm 1}].$

######  Examples 1.12.

  1. (1)

The first example of a cluster algebra for which $\mathcal{A}\neq\mathcal{U}$
is due to Berenstein, Fomin, and Zelevinsky [BFZ05] and it is the Markov
cluster algebra. It is the cluster algebra with base ring $K=\mathbb{Z}$
associated to the following quiver $\mathcal{Q}$

${1}$${3}$${2.}$

$\mathcal{A}(\mathcal{Q})$ is a $\mathbb{N}$-graded algebra [Mul13], with the
degree of all cluster variables equal to $1$. For any cluster
$(x_{1},x_{2},x_{3})$, the element
$M=\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{x_{1}x_{2}x_{3}}$ is in
$\mathcal{U}(\mathcal{Q})$, but it has graded degree $-1$, so it is not in
$\mathcal{A}(\mathcal{Q})$. Moreover, the upper cluster algebra is factorial
and it is given by $\mathbb{Z}[x_{1},x_{2},x_{3},M]$ ([MM15, Proposition
6.7]).

  2. (2)

Consider the quiver $A_{3}=\leavevmode\hbox to64.83pt{\vbox
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$s=\frac{1+x_{2}}{x_{1}x_{3}}\in K(x_{1},x_{2},x_{3}).$

It is easy to see that

$s=\frac{x_{1}^{\prime}}{x_{3}}=\frac{x_{3}^{\prime}}{x_{1}}\in\bigcap_{i=0}^{3}\mathcal{L}_{\mathbf{x}_{i}}.$

However, $(x_{1}^{\prime},x_{2},x_{3}^{\prime})$ is a seed, obtained by
mutating $\mathbf{x}$ in direction 3 and then 1, and $s\notin
K[{x_{1}^{\prime}}^{\pm 1},x_{2}^{\pm 1},{x_{3}^{\prime}}^{\pm 1}].$ Indeed,
if we express $s$ in terms of the new seed we obtain

$s=\frac{x_{1}^{\prime}x_{3}^{\prime}}{1+x_{2}}$

which is not a Laurent polynomial in $(x_{1}^{\prime},x_{2},x_{3}^{\prime})$.
Hence
$\mathcal{U}(A_{3})\subsetneq\bigcap_{i=0}^{3}\mathcal{L}_{\mathbf{x}_{i}}$.

However, there are non-full rank upper cluster algebras for which the Starfish
lemma holds. For example, any factorial upper cluster algebra satisfies the
Starfish lemma as shown in [GLS13, Corollary 1.5].

######  Definition 1.13.

Let $\mathcal{U}$ be an upper cluster algebra. We say that $\mathcal{U}$
satisfies the starfish condition at the seed $(\mathbf{x},\mathbf{y}\,B)$ if

$\mathcal{U}=\bigcap_{i=0}^{n}\mathcal{L}_{\mathbf{x}_{i}}.$

If $\mathcal{U}$ has full rank, then Theorem 1.11 implies that $\mathcal{U}$
satisfies the starfish condition at all seeds. We introduced this definition
because for our results we don’t need the starfish condition at all seeds but
just at one of them (see Section 3).

######  Remark 1.14.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. Suppose that $i\in[1,n]$ is
isolated, i.e., $x_{i}x_{i}^{\prime}=2$. If $K$ is a field, the index $i$ is
isolated if and only if $x_{i}\in K^{\times}$. Therefore, if we freeze $i$, we
obtain an algebra isomorphic to the original one. Hence, from now on, if $K$
is a field we assume without restriction that $[1,n]$ has no isolated indices.

###  1.3. Factorization Theory

In this section, we recall some basic notions of the theory of factorization.
For more details see for instance [GHK06]. We always assume that our monoids
are commutative and cancellative.

######  Definition 1.15.

Let $H$ be a monoid.

  1. (1)

Let $a,b\in H$. We say that $a|b$ if there exists an element $c\in H$ such
that $b=ca.$

  2. (2)

Two elements $a,b\in H$ are associated if there exists a unit $\varepsilon\in
H^{\times}$ such that $a=\varepsilon b.$ In this case we write $a\simeq_{H}b$.

  3. (3)

A non-unit $u\in H$ is an atom, or an irreducible element, if $u=ab$ with
$a,b\in H$ implies $a\in H^{\times}$ or $b\in H^{\times}$.

  4. (4)

An atom $u\in H$ is a strong atom, or an absolutely irreducible element, if
for all positive integers $n>1$ the only factorization (up to associates) of
$u$ is $u^{n}=u\cdots u.$

  5. (5)

A non-unit $p\in H$ is a prime element if $p|ab$ with $a,b\in H$ implies $p|a$
or $p|b$.

It is well known that

$\text{prime}\implies\text{strong atom}\implies\text{atom}.$

We will denote by $\mathcal{A}(H)$ the set of atoms of the monoid $H$.

######  Definition 1.16.

Let $H$ be a monoid.

  1. (1)

$H$ is atomic if every non-unit of $H$ can be written as a finite product of
atoms.

  2. (2)

$H$ is a bounded factorization monoid, in short BF-monoid, if it is atomic and
for every non-unit $a\in H$ there exist $\lambda(a)\in\mathbb{N}_{0}$ such
that $a=x_{1}\cdots x_{n}$ for $x_{1},\dots,x_{n}$ atoms implies
$n\leq\lambda(a).$

  3. (3)

$H$ is a finite factorization monoid, in short FF-monoid, if it is atomic and
every non-unit factors into atoms in only finitely many ways up to order and
associates.

  4. (4)

$H$ is factorial, or UF-monoid, if it is atomic and every non-unit of $A$
factors in a unique way up to order and associates.

The connection between the notions in Definition 1.16 is described by the
following picture:

$\text{factorial monoid}\implies\text{FF-monoid}\implies\text{BF-
monoid}\implies\text{atomic}.$

Recall these different characterizations of BF-monoids and FF-monoids (see
[GHK06, Proposition 1.3.3 and Proposition 1.5.5]).

  * •

$H$ is a BF-monoid if and only if there exists a length function, i.e., a map
$\lambda\colon H\to\mathbb{N}_{0}$ such that $\lambda(a)<\lambda(b)$ whenever
$a|b$ and $a\not\simeq_{H}b$.

  * •

$H$ is an FF-domain if and only if every $a\in H$ has only finitely many non-
associated divisors.

Let $A$ be a domain. We say that $A$ is an atomic (resp. BF, FF, factorial)
domain if the monoid $A^{\bullet}$ is atomic (resp. BF, FF, factorial).

We conclude this subsection with two well-known properties of localizations of
domains. Denote by $\operatorname{Spec}(A)$ the set of all prime ideals of
$A$. Recall that a multiplicative closed subset of $A$ is a subset $S\subseteq
A$ such that $1\in S,\,S\cdot S\subseteq S$, and $0\notin S$.

######  Proposition 1.17.

Let $A$ be a domain and $S\subseteq A$ a multiplicatively closed set of $A$.
There is an inclusion-preserving bijection

$\displaystyle\\{\,\mathfrak{p}\in\operatorname{Spec}(A)\mid\mathfrak{p}\cap
S=\emptyset\,\\}$
$\displaystyle\longleftrightarrow\operatorname{Spec}(S^{-1}A),$
$\displaystyle\mathfrak{p}$ $\displaystyle\mapsto
S^{-1}\mathfrak{p}=\mathfrak{p}(S^{-1}A),$ $\displaystyle\mathfrak{q}\cap A$
$\displaystyle\mapsfrom\mathfrak{q}.$

In particular, the bijection preserves the height of every prime ideal that
does not meet $S$.

######  Proposition 1.18.

Let $A$ be a factorial domain, and $S$ a multiplicatively closed set of $A$.
Denote by $T$ the set of all the prime elements of $A$ that divide an element
of $S$, and by $M=\mathcal{A}(A)\setminus T.$ Then
$S^{-1}T\subseteq(S^{-1}A)^{\times}$ and
$S^{-1}M\subseteq\mathcal{A}(S^{-1}A)$.

###  1.4. Krull Domains

In this section, we recall some basic notions about Krull domains. For more
details, see [Fos73, GHK06].

Recall that a discrete valuation ring, in short DVR, is a factorial domain $A$
with a unique prime element $p$ (up to associates), that is, every $a\in A$
has a unique factorization of the form $a=up^{n}$ with $u\in A^{\times}$ and
$n\in\mathbb{N}_{0}$. Equivalently, a domain $A$ is a discrete valuation ring
if there exists a discrete valuation
$v\colon\mathbf{q}(A)\to\mathbb{Z}\cup\\{\infty\\}$ such that
$A=\\{\,x\in\mathbf{q}(A)\mid v(x)\geq 0\,\\}.$ With this notation, every
$a\in A$ can be uniquely written as $a=up^{v(a)}$, with $u\in A^{\times}$ (see
[GHK06, Theorem 2.3.8]).

######  Definition 1.19 (Krull domains).

Let $A$ be a domain and let $\mathfrak{X}(A)$ be the set of all height-1 prime
ideals of $A$. Then $A$ is a Krull domain if

  1. (1)

$A_{\mathfrak{p}}$ is a discrete valuation ring for every
$\mathfrak{p}\in\mathfrak{X}(A);$

  2. (2)

$A=\bigcap_{\mathfrak{p}\in\mathfrak{X}(A)}A_{\mathfrak{p}}$;

  3. (3)

every non-zero element $a\in A$ is contained in at most a finite number of
height-1 prime ideals of $A$.

Equivalently, $A$ is a Krull domain if and only if there exists a family
{$v_{i}\\}_{i\in I}$ of discrete valuations on the quotient field
$\mathbf{q}(A)$ such that, for all $x\in\mathbf{q}(A)\setminus\\{0\\}$,

  1. (i)

$v_{i}(x)=0$ for all but finitely many $i\in I;$

  2. (ii)

$x\in A$ if and only $v_{i}(x)\geq 0$ for all $i\in I.$

Let $A$ be a domain. For any non-empty subsets $X$, $Y\subset\mathbf{q}(A)$,
we define

$(Y:X)=\\{\,a\in\mathbf{q}(A)\mid aX\subset Y\,\\},\quad
X^{-1}=(A:X),\quad\text{and}\quad X_{v}=(X^{-1})^{-1}.$

A subset $\mathfrak{c}\subseteq A$ is called a divisorial ideal of $A$ if
$\mathfrak{c}_{v}=\mathfrak{c}$ and a subset
$\mathfrak{c}\subseteq\mathbf{q}(A)$ is called a fractional divisorial ideal
of $A$ if there exists some $x\in A^{\bullet}$ such that $x\mathfrak{c}$ is a
divisorial ideal of $A$. We denote by $\mathcal{I}_{v}(A)$ the set of all
divisorial ideals of $A$ and by $\mathcal{F}_{v}(A)$ the set of all fractional
divisorial ideal of $A$. For fractional ideals $\mathfrak{a}$,
$\mathfrak{b}\in\mathcal{F}_{v}(A)$, define their $v$-product by
$\mathfrak{a}\cdot_{v}\mathfrak{b}=(\mathfrak{a}\mathfrak{b})_{v}$, and for
$k$ fractional divisorial ideals $\mathfrak{a}_{1},\dots,\mathfrak{a}_{k}$
define their $v$-product to be

$\sideset{}{\mkern-2.0mu\raisebox{1.99997pt}{$\vphantom{\prod}{}^{v}$}}{\prod}_{i=1}^{k}\mathfrak{a}_{i}:=\Bigl{(}\prod_{i=1}^{n}\mathfrak{a}_{i}\Bigr{)}_{v}.$

By convention an empty product is equal to the trivial ideal $A$. A fractional
divisorial ideal $\mathfrak{a}\in\mathcal{F}_{v}(A)$ is v-invertible if
$\mathfrak{a}\cdot_{v}\mathfrak{a}^{-1}=A.$ Every non-zero principal
fractional ideal $xA$ is invertible with inverse $x^{-1}A$. The group
$\mathcal{F}_{v}(A)^{\times}$ is the group of $v$-invertible fractional
ideals. We have that $(xA)\cdot_{v}(yA)=xyA,$ hence the subset
$\mathcal{H}(A)$ of all non-zero principal fractional ideals is a subgroup of
$\mathcal{F}_{v}(A)^{\times}.$ So the following definition makes sense.

######  Definition 1.20 (Class groups).

Let $A$ be a Krull domain. The (divisor) class group of $A$ is the quotient
group $\mathcal{C}(A)=\mathcal{F}_{v}(A)^{\times}/\mathcal{H}(A).$ We write
the group additively. For $\mathfrak{a}\in\mathcal{F}_{v}(A)^{\times}$ we
denote by $[\mathfrak{a}]\in C(A)$ the class containing $\mathfrak{a}$.

A domain $A$ is $v$-noetherian if it satisfies the ascending chain condition
on divisorial ideals. An element $x\in\mathbf{q}(A)$ is almost integral if
there exists $c\in\mathbf{q}(A)^{\bullet}$ such that $cx^{n}\in A$ for all
$n\geq 0$. A domain $A$ is completely integrally closed if
$A=\\{\,x\in\mathbf{q}(A)\mid x\,\text{almost integral}\,\\}$. Every
noetherian domain is $v$-noetherian and a noetherian domain is integrally
closed if and only if it is completely integrally closed. The following
theorem is a characterization of Krull domains in terms of multiplicative
ideal theory.

######  Theorem 1.21 ([Fos73]).

Let $A$ be a domain. Then $A$ is a Krull domain if and only if $A$ is
completely integrally closed and $v$-noetherian.

Dedekind domains are exactly the one-dimension Krull domains [GHK06, Theorem
2.10.6]. In a Dedekind domain every non-zero proper ideal factors into a
product of prime ideals. Krull domains generalize this property of Dedekind
domains, as shown in the next theorem.

######  Theorem 1.22 ([Fos73, Corollary 3.14]).

Let $A$ be a Krull domain. Then every invertible fractional divisorial ideal
$\mathfrak{a}\in\mathcal{F}_{v}(A)^{\times}$ has a representation as
divisorial product

$\mathfrak{a}=\sideset{}{\mkern-2.0mu\raisebox{1.99997pt}{$\vphantom{\prod}{}^{v}$}}{\prod}_{\mathfrak{p}\in\mathfrak{X}(A)}\mathfrak{p}^{n_{\mathfrak{p}}},$

with uniquely determined $n_{\mathfrak{p}}\in\mathbb{Z},$ almost all of which
are 0. We have $a\in\mathcal{I}_{v}(A)^{\bullet}$ if and only if
$n_{\mathfrak{p}}\geq 0$ for all $\mathfrak{p}\in\mathfrak{X}(A)$.

For $\mathfrak{a}\in\mathcal{F}_{v}(A)^{\times}$ we define the
$\mathfrak{p}$-adic valuation of $\mathfrak{a}$ as
$v_{\mathfrak{p}}(\mathfrak{a})=n_{\mathfrak{p}}$ with $n_{\mathfrak{p}}$ as
in the previous theorem. For $x\in\mathbf{q}(A)$ one has $x\in A$ if and only
if $v_{\mathfrak{p}}(x)=v_{\mathfrak{p}}(xA)\geq 0$ for every
$\mathfrak{p}\in\mathfrak{X}(A)$.

Krull domains theory and factorization theory are intimately related by the
following theorem.

######  Theorem 1.23 ([GHK06]).

Let $A$ be a domain. Then the following are equivalent:

  1. (a)

$A$ is factorial;

  2. (b)

$A$ is atomic and every atom is a prime element;

  3. (c)

$A$ is a Krull domain and $\mathcal{C}(A)=0$.

Krull domains have the following important property, which we will use later
to prove our main theorem (cfr. Section 3).

######  Theorem 1.24 (Approximation property, [Fos73]).

Let $A$ be a Krull domain. For all $n\in\mathbb{N}$, pairwise distinct
$\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}\in\mathfrak{X}(A)$ and integers
$e_{1},\dots,e_{n}\in\mathbb{Z}$, there exists an element $x\in\mathbf{q}(A)$
such that $v_{\mathfrak{p}_{i}}(x)=e_{i}$ and $v_{\mathfrak{p}}(x)\geq 0$ for
every
$\mathfrak{p}\in\mathfrak{X}(A)\setminus\\{\mathfrak{p}_{1},\dots,\mathfrak{p}_{n}\\}.$

##  2\. Factorization theory in upper cluster algebras

In this section, we prove that every cluster variable is a strong atom and
that every (upper) cluster algebra is an FF-domain. Recall the following.

######  Theorem 2.1 ([CKQ23]).

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. Let
$\mathcal{A}=\mathcal{A}(\Sigma)$, $\mathcal{U}=\mathcal{U}(\Sigma)$ be the
cluster algebra and upper cluster algebra associated to $\Sigma$,
respectively. Then

  1. (1)

    1. (i)

$\mathcal{A}^{\times}=\\{\,\lambda x_{n+1}^{a_{n+1}}\dots
x_{n+m}^{a_{n+m}}\,\mid\,\lambda\in K^{\times},\,a_{i}\in\mathbb{Z}\,\\}$;

    2. (ii)

every cluster variable is an atom of $\mathcal{A}$ (pairwise non-associated).

  2. (2)

    1. (i)

$\mathcal{U}^{\times}=\\{\,\lambda x_{n+1}^{a_{n+1}}\dots
x_{n+m}^{a_{n+m}}\,\mid\,\lambda\in K^{\times},\,a_{i}\in\mathbb{Z}\,\\}$;

    2. (ii)

every cluster variable is an atom in $\mathcal{U}$ (pairwise non-associated).

From the existing literature ([GLS13, Corollary 4.2], [GELS19, Corollary
1.23], and [CKQ23, Theorem 4.9]), we obtain the following characterization of
factorial (upper) cluster algebras. We remark the directions
(c)$\,\Rightarrow$(a) (or (b)$\,\Rightarrow\,$(a)) actually make use of the
fact that (upper) cluster algebras are atomic, see Remark 2.8 below.

######  Proposition 2.2.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. Let $H$ be either
$\mathcal{A}(\Sigma)$ or $\mathcal{U}(\Sigma)$. Then the following are
equivalent:

  1. (a)

$H$ is factorial,

  2. (b)

every cluster variable is a prime element of $H$,

  3. (c)

the variables $x_{1},\dots,x_{n}$ are prime elements of $H$,

If $H=\mathcal{U}$ and $H$ is factorial, then
$H=\bigcap_{i=0}^{n}\mathcal{L}_{\mathbf{x}_{i}}$.

To show that cluster variables are not just atoms, but even strong atoms, we
need the following.

######  Lemma 2.3 ([CKQ23, Lemma 5.2]).

Let $\mathcal{U}$ be an upper cluster algebra and $(\mathbf{x},\mathbf{y},B)$
be a seed. If $x_{1}^{a_{1}}\ldots x_{n}^{a_{n}}\in\mathcal{U}$ for some
$a_{1},\dots,a_{n}\in\mathbb{Z}$, then $a_{1},\dots,a_{n}\in\mathbb{N}_{0}$.

For every subset $S$ of a domain $A$ we denote by $\llbracket S\rrbracket_{A}$
the divisor-closed submonoid generated by $S$, that is the set of all elements
that divide an element of the submonoid generated by $S$. In particular if
$u\in A$ then

$\llbracket u\rrbracket_{A}=\\{\,a\in A\,\colon a|u^{n},\,\text{for
some}\,n\in\mathbb{N}_{0}\,\\}.$

######  Proposition 2.4.

Let $\mathcal{A}$ be a cluster algebra and $\mathcal{U}$ be the corresponding
upper cluster algebra. Let $(\mathbf{x},\mathbf{y},B)$ be an arbitrary seed.
Then the divisor-closed submonoid generated by a cluster monomial in
$\mathbf{x}$ is

$\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{U}}=\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{A}}=\\{\,\varepsilon x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\mid\varepsilon\in\mathcal{U}^{\times},\,a_{i}\in\mathbb{N}_{0},\,\text{with}\,\phantom{,}a_{i}=0\,\phantom{,}\text{if}\,\phantom{,}e_{i}=0\,\\},$

for all $e_{1},\dots,e_{n}\in\mathbb{N}_{0}$. In particular, the monoid
$\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{U}}=\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{A}}$ is free abelian.

###### Proof.

Denote by $M$ the set $M=\\{\,\varepsilon x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}\mid\varepsilon\in\mathcal{U}^{\times},\,a_{i}\in\mathbb{N}_{0},\,\text{with}\,\phantom{,}a_{i}=0\,\phantom{,}\text{if}\,\phantom{,}e_{i}=0\,\\}.$

First, we prove that $M\subseteq\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{A}}$. Let $a\in M$, say $a=\varepsilon
x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ for some $a_{i}\in\mathbb{N}_{0}$,
$\varepsilon\in\mathcal{U}^{\times}$. Then $a$ divides the
$N^{\text{th}}$-power of $x_{1}^{e_{1}}\cdots x_{n}^{e_{n}}$ with
$N=\mathrm{max}\\{\lceil\frac{a_{i}}{e_{i}}\rceil,\,i\in[1,n],\,e_{i}\neq
0\\}$, since trivially

$(x_{1}^{e_{1}}\cdots x_{n}^{e_{n}})^{N}=(\varepsilon x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}})(\varepsilon^{-1}x_{1}^{e_{1}N-a_{1}}\cdots
x_{n}^{e_{n}N-a_{n}})$

holds. Hence $a\in\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{A}}$.

Now, we show that $\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{U}}\subseteq M$. Let $a\in\llbracket
x_{1}^{e_{1}}\cdots x_{n}^{e_{n}}\rrbracket_{\mathcal{U}}$, so there exist a
non-negative integer $N\in\mathbb{N}_{0}$ and an element $b\in\mathcal{U}$
such that

$x_{1}^{e_{1}N}\cdots x_{n}^{e_{n}N}=ab.$ (2.1)

Consider the Laurent expansion of $a$ and $b$ with respect to $\mathbf{x}$,
say

$a=\frac{P(x_{1},\dots,x_{n+m})}{x_{1}^{\alpha_{1}}\cdots
x_{n+m}^{\alpha_{n+m}}},\qquad
b=\frac{Q(x_{1},\dots,x_{n+m})}{x_{1}^{\beta_{1}}\cdots
x_{n+m}^{\beta_{n+m}}},$

where $P$, $Q\in K[\mathbf{x},\mathbf{y}]$ and $\alpha_{i}$,
$\beta_{i}\in\mathbb{N}_{0}$ for every $i\in[1,n+m]$.

So Equation (2.1) can be rewritten as

$x_{1}^{\alpha_{1}+\beta_{1}+e_{1}N}\cdots
x_{n}^{\alpha_{n}+\beta_{n}+e_{n}N}x_{n+1}^{\alpha_{n+1}+\beta_{n+1}}\cdots
x_{n+m}^{\alpha_{n+m}+\beta_{n+m}}=PQ,$

hence $P$ and $Q$ must be monomials in $x_{1},\dots,x_{n+m}$, and a fortiori
$a$, $b$ are associated to Laurent monomials in $x_{1},\dots,x_{n+m}.$ By
Lemma 2.3, we get $a=\varepsilon x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ with
$a_{1},\dots,a_{n}\in\mathbb{N}_{0}$ and $\varepsilon\in\mathcal{U}^{\times}.$
Observe that, if there is $i\in[1,n]$ such that $e_{i}=0,$ then $x_{i}\nmid
a,$ whence $a_{i}=0.$

Therefore, we have proved the following inclusions

$M\subseteq\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{A}}\subseteq\llbracket x_{1}^{e_{1}}\cdots
x_{n}^{e_{n}}\rrbracket_{\mathcal{U}}\subseteq M,$

and this concludes the proof. ∎

######  Corollary 2.5.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. Let
$\mathcal{A}=\mathcal{A}(\Sigma)$and $\mathcal{U}=\mathcal{U}(\Sigma)$ be the
cluster algebra and upper cluster algebra associated to $\Sigma$,
respectively. Then every cluster variable is a strong atom of $\mathcal{A}$
and $\mathcal{U}$.

###### Proof.

The statement is a direct consequence of Proposition 2.4 applied to the
divisor-closed submonoid generated by one cluster variable. ∎

The Laurent phenomenon implies that
$A\subseteq\mathcal{U}\subseteq\bigcap_{k=0}^{n}\mathcal{L}_{\mathbf{x}_{k}}$,
for every seed $(\mathbf{x},\mathbf{y},B)$. The ring
$\bigcap_{k=0}^{n}\mathcal{L}_{\mathbf{x}_{k}}$ is a finite intersection of
Laurent polynomial rings, which, being factorial, are in turn intersections of
DVRs. So we can write
$\bigcap_{k=0}^{n}\mathcal{L}_{\mathbf{x}_{k}}=\bigcap_{i\in I}D_{i}$, with
$D_{i}$ a DVR for every $i\in I$. Hence we have a monoid homomorphism

$\mathbf{v}\colon\mathcal{U}\to\mathbb{N}_{0}^{(I)},\qquad
u\mapsto(v_{i}(u))_{i\in I}\,,$

where $v_{i}\colon\mathbf{q}(D_{i})\to\mathbb{Z}$ is the discrete valuation of
$D_{i}$ and $\mathbb{N}_{0}^{(I)}=\\{\,(a_{i})\in\mathbb{N}_{0}^{I}\mid
a_{i}=0\,\,\text{for all but}$ $\text{fi\\-ni\\-te\\-ly many}\,i\in I\,\\}$.
Observe that, by Theorem 2.1,

$\mathcal{A}^{\times}=\mathcal{U}^{\times}=\Bigl{(}\bigcap_{k=0}^{n}\mathcal{L}_{\mathbf{x}_{k}}\Bigr{)}^{\times}=\Bigl{(}\bigcap_{i\in
I}D_{i}\Bigr{)}^{\times}=\bigcap_{i\in I}D_{i}^{\times}.$ (2.2)

######  Lemma 2.6.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed and
$\mathcal{A}=\mathcal{A}(\Sigma)$ and $\mathcal{U}=\mathcal{U}(\Sigma)$ be the
cluster algebra and the upper cluster algebra associated to $\Sigma$,
respectively. Let $\mathbf{v}\colon\mathcal{U}\to\mathbb{N}_{0}^{(I)}$ be the
monoid homomorphism defined above. Then

$\mathbf{v}(a)=\mathbf{v}(b)\iff a\simeq_{\,\mathcal{U}}b\iff
a\simeq_{\mathcal{A}}b.$

###### Proof.

First assume that $a$, $b$ are two elements of $\mathcal{U}$ such that
$\mathbf{v}(a)=\mathbf{v}(b)$, say $v_{i}(a)=v_{i}(b)=n_{i}$ for every $i\in
I$. Let $p_{i}$ denote the unique (up to associates) prime element of $D_{i}$
with $i\in I$. By definition of a DVR (see Section 1.4), for every $i\in I$,
we have that $a=u_{i}p_{i}^{n_{i}}$ and $b=u_{i}^{\prime}p_{i}^{n_{i}}$ with
$u_{i}$, $u^{\prime}_{i}\in D_{i}^{\times}$, that is $a=w_{i}b$ for some
$w_{i}\in D_{i}^{\times}$. Hence $w_{i}b=w_{j}b$ for all $i$, $j\in I$ implies
$w_{i}=w_{j}$ and so $a=wb$ for some $w\in\bigcap_{i\in I}D_{i}^{\times}$.
Thus Equation (2.2) implies that $a\simeq_{\,\mathcal{U}}b$, and this is
equivalent to $a\simeq_{\mathcal{A}}b$. The other direction is
straightforward. ∎

######  Theorem 2.7.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed and
$\mathcal{A}=\mathcal{A}(\Sigma)$ and $\,\mathcal{U}=\mathcal{U}(\Sigma)$ be
the cluster algebra and the upper cluster algebra associated to $\Sigma$,
respectively. Then $\mathcal{A}$ and $\mathcal{U}$ are FF-domains. In
particular, they are BF-domains and atomic.

###### Proof.

Let $H$ denote either $\mathcal{A}^{\bullet}$ or $\mathcal{U}^{\bullet}$, and
let $a\in H$. We want to show that $a$ has only finitely many non-associated
divisors. Consider the set $\Omega=\\{\,e_{i}\mid v_{i}(a)>0\,\\}$ of all the
atoms of $\mathbb{N}_{0}^{(I)}$ that divide $\mathbf{v}(a)$. Here $e_{i}$
denotes the tuple with all components $0$ except the $i$-th entry that is $1$.
Notice that $\Omega$ is finite.

Let $v\in H$ be such that $v|a$. Then, since $\mathbf{v}(v)|\mathbf{v}(a)$,
there exist $\omega_{1},\dots,\omega_{k}\in\Omega$ such that
$\mathbf{v}(v)=\omega_{1}+\cdots+\omega_{k},$ hence, since $\Omega$ is finite,
there are only finitely many possibilities for $\mathbf{v}(v)$ for each
divisor $v$ of $a$. So let $u$ be another divisor of $a$ such that
$\mathbf{v}(u)=\omega_{1}+\cdots+\omega_{k}=\mathbf{v}(v).$ Then Lemma 2.6
implies that $u\simeq_{H}v$, thus $a$ has only finitely many non-associated
divisors. ∎

######  Remark 2.8.

In [GELS19] the cluster algebras under investigation are usually Krull
domains, and hence FF-domains (in particular, atomic). However, [GELS19,
Corollary 1.23] (which is subsumed in our Proposition 2.2), concerns arbitrary
cluster algebras. To deduce that the cluster variables being prime is
sufficient for an (upper) cluster algebra to be factorial, one needs to know a
priori that the cluster algebra is atomic. This was taken for granted in
[GELS19, Corollary 1.23] and in [CKQ23, Lemma 4.1]. However, it is somewhat
non-trivial and our Theorem 2.7 provides a proof of this fact. More generally,
in the statement [GELS19, Corollary 1.20] (respectively [CKQ23, Lemma 4.1]) of
a corollary of Nagata’s Theorem, an assumption that $A$ be atomic is missing.
See [AAZ92, Section 1] for details.

Recall that a domain is a Krull domain if and only if it is completely
integrally closed and $v$-noetherian. Moreover, a Krull domain is always an
FF-domain.

######  Theorem 2.9.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed and
$\,\mathcal{U}=\mathcal{U}(\Sigma)$ be the upper cluster algebra associated to
$\Sigma$. Then $\mathcal{U}$ is completely integrally closed. Moreover, if
there exists a seed such that $\,\mathcal{U}$ satisfies the starfish condition
at that seed, then $\mathcal{U}$ is a Krull domain.

###### Proof.

The upper cluster algebra $\mathcal{U}$, being an intersection of completely
integrally closed domains, is completely integrally closed. Moreover, if we
assume $\mathcal{U}=\bigcap_{i=0}^{n}\mathcal{L}_{\mathbf{x}_{i}}$ for some
seed $(\mathbf{x},\mathbf{y},B)$, then $\mathcal{U}$, being a finite
intersection of Krull domains, is a Krull domain. ∎

##  3\. Class Groups of Upper Cluster Algebras

In this section we determine the class groups of full rank upper cluster
algebras (more generally, upper cluster algebras, satisfying the starfish
condition). To do so, we need some preliminary results. We prove them in the
more general setting of upper cluster algebras that are Krull domains.

######  Proposition 3.1.

Let $A$ be a domain and let $x_{1},\dots,x_{n}\in A$ be such that
$A_{\mathbf{x}}:=A[x_{1}^{-1},\dots,x_{n}^{-1}]=D[x_{1}^{\pm
1},\dots,x_{n}^{\pm 1}]$ is a factorial Laurent polynomial ring for some
subring $D$ of $A$. If $f\in A$ is such that $f\notin A_{\mathbf{x}}^{\times}$
and $f$ has no repeated factors in $A_{\mathbf{x}}$, then there exist a
height-$1$ prime ideal $\mathfrak{p}$ of $A$ and a discrete valuation
$v_{\mathfrak{p}}\colon\mathbf{q}(A)\to\mathbb{Z}\cup\\{\infty\\}$ such that
$v_{\mathfrak{p}}(f)=1$ and $v_{\mathfrak{p}}(x_{i})=0$ for $i\in[1,n].$

###### Proof.

Let $p\in A_{\mathbf{x}}$ be a prime factor of $f$. Set
$\mathfrak{p}=pA_{\mathbf{x}}\cap A.$ The prime ideal $pA_{\mathbf{x}}$ has
height $1$ by Krull’s Principal Ideal Theorem and hence so does $\mathfrak{p}$
(Proposition 1.17). Since $f$ has no repeated factors,
$v_{\mathfrak{p}}(f)=1$. For $i\in[1,n]$, $x_{i}$ is a unit of
$A_{\mathbf{x}}$, therefore $x_{i}\notin\mathfrak{p}$. Hence
$v_{\mathfrak{p}}(x_{i})=0.$ ∎

Notice that for the proof a weaker assumption is sufficient. It is enough to
assume that there exists $f\notin A_{\mathbf{x}}^{\times}$ that has a prime
factor of multiplicity $1$.

######  Corollary 3.2.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed. Let
$\mathcal{U}=\mathcal{U}(\Sigma)$ be the upper cluster algebra associated to
$\Sigma$. Then, for every $i\in I$ there exists a height-$1$ prime ideal
$\mathfrak{p}$ of $\mathcal{U}$ such that $v_{\mathfrak{p}}(x_{i})=1$ and
$v_{\mathfrak{p}}(x_{j})=0$ for every $j\in[1,n]\setminus\\{i\\}$.

###### Proof.

We mutate $\mathbf{x}=(x_{1},\dots,x_{n})$ in direction $i$ obtaining a new
seed $\mathbf{x}_{i}.$ Let $f_{i}=x_{i}x_{i}^{\prime}\in
K[\mathbf{x}_{i},\mathbf{y}]$ be the exchange polynomial of $x_{i}$ associated
to $\Sigma.$ We have that

$\mathcal{U}[\mathbf{x}_{i}^{-1},\mathbf{y}^{-1}]=K[\mathbf{x}_{i}^{\pm
1},\mathbf{y}^{\pm 1}],$

hence $K[\mathbf{x}_{i}^{\pm 1},\mathbf{y}^{\pm 1}]$ is a Laurent polynomial
ring, and therefore factorial. By [GELS19, Proposition 2.3] we know that
$f_{i}$ does not have repeated factors. Moreover, due to our assumption on
isolated seeds (Remark 1.14), one has $f_{i}\notin K[\mathbf{x}_{i}^{\pm
1},\mathbf{y}^{\pm 1}]^{\times}$, so $f_{i}$ satisfies the hypothesis of
Proposition 3.1. Hence we can conclude that there exists a height-$1$ prime
ideal $\mathfrak{p}$ of $\mathcal{U}$ such that $v_{\mathfrak{p}}(f_{i})=1$,
$v_{\mathfrak{p}}(x_{j})=0$ for every $j\in[1,n]\setminus\\{i\\}$, and
$v_{\mathfrak{p}}(x_{i}^{\prime})=0$, hence $v_{\mathfrak{p}}(x_{i})=1$. ∎

Let us recall this very general result on Krull domains.

######  Theorem 3.3 ([GELS19, Theorems 3.1 and 3.2]).

Let $A$ be a Krull domain, and let $x_{1},\dots,x_{n}\in A$ be such that
$A_{\mathbf{x}}:=A[x_{1}^{-1},\dots,x_{n}^{-1}]=D[x_{1}^{\pm
1},\dots,x_{n}^{\pm 1}]$ is a factorial Laurent polynomial ring for some
subring $D$ of $A$. Let $\mathfrak{p}_{1},\dots,\mathfrak{p}_{t}$ be the
pairwise distinct height-1 prime ideals of $A$ containing one of the elements
$x_{1},\dots,x_{n}$. Suppose that

$x_{i}A={\sideset{}{\mkern-2.0mu\raisebox{1.99997pt}{$\vphantom{\prod}{}^{v}$}}{\prod}_{l=1}^{t}}\mathfrak{p}_{j}^{a_{ij}},$

with $\mathbf{a}_{i}=(a_{ij})_{j=1}^{t}\in\mathbb{N}_{0}^{t}.$ Then
$\mathcal{C}(A)\cong\mathbb{Z}^{t}/\langle\mathbf{a}_{i}\mid i\in[1,n]\rangle$
and it is generated by $[\mathfrak{p}_{1}],\dots,[\mathfrak{p}_{t}]$.

Suppose in addition that $D$ is infinite and either $n\geq 2$ or $n=1$ and $D$
has at least $|D|$ height-1 prime ideals. Then every class of $\mathcal{C}(A)$
contains precisely $|D|$ height-1 prime ideals.

We are now ready to prove our main theorem on the class groups of upper
cluster algebras that are Krull domains.

######  Theorem 3.4.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed and let
$\mathcal{U}=\mathcal{U}(\Sigma)$ be the upper cluster algebra associated to
$\Sigma$. Suppose that $\mathcal{U}$ is a Krull domain. Let
$t\in\mathbb{N}_{0}$ denote the number of height-$1$ prime ideals that contain
one of the exchangeable variables $x_{1},\cdots,x_{n}$. Then the class group
$\mathcal{C}(\mathcal{U})$ of $\mathcal{U}$ is a free abelian group of rank
$t-n$.

In particular, each class contains exactly $|K|$ height-1 prime ideals.

###### Proof.

The proof follows the strategy of [GELS19, Theorem A] where they proved the
statement for cluster algebras that are Krull domains.

In order to apply Theorem 3.3, notice that the Laurent phenomenon implies that

$\mathcal{U}[\mathbf{x}^{-1},\mathbf{y}^{-1}]=K[\mathbf{x}^{\pm
1},\mathbf{y}^{\pm 1}]$

and $K[\mathbf{x}^{\pm 1},\mathbf{y}^{\pm 1}]$ is a factorial domain.
$\mathcal{U}$ is a Krull domain by assumption, hence any principal ideal is a
divisorial product of height-$1$ prime ideals, in particular, for every
$i\in[1,n]$ there exist $a_{i1},\dots,a_{it}\in\mathbb{N}_{0}$ such that

$x_{i}\mathcal{U}={\sideset{}{\mkern-2.0mu\raisebox{1.99997pt}{$\vphantom{\prod}{}^{v}$}}{\prod}_{l=1}^{t}}\mathfrak{p}_{j}^{a_{ij}},$
(3.1)

with $\mathfrak{p}_{1},\dots,\mathfrak{p_{t}}$ pairwise distinct height-$1$
prime ideals of $\mathcal{U}$ that contain one of the elements
$x_{1},\dots,x_{n}$ and $a_{ij}=v_{\mathfrak{p}_{j}}(x_{i})$. Thus Theorem 3.3
implies that $C(\mathcal{U})$ is generated by
$[\mathfrak{p}_{1}],\dots,[\mathfrak{p}_{t}]$.

By Corollary 3.2, for every $i\in[1,n]$ there exists
$\mathfrak{p}\in\mathfrak{X}(\mathcal{U})$ such that
$v_{\mathfrak{p}}(x_{i})=1$ and $v_{\mathfrak{p}}(x_{j})=0$ for every $j\neq
i$. Thus, since $\mathfrak{p}_{1},\dots,\mathfrak{p_{t}}$ are all the height-1
prime ideals that contain $x_{i}$, there exists $k_{i}\in[1,t]$ such that
$v_{\mathfrak{p}_{k_{i}}}(x_{i})=a_{ik_{i}}=1$ and
$v_{\mathfrak{p}_{k_{i}}}(x_{j})=a_{jk_{i}}=0$ for every
$j\in[1,n]\setminus\\{i\\}$. Hence, by (3.1),

$0=a_{i1}[\mathfrak{p}_{1}]+\cdots+[\mathfrak{p}_{k_{i}}]+\cdots+a_{it}[\mathfrak{p}_{t}],$

that is $[\mathfrak{p}_{k_{i}}]$ is a linear combination of
$[\mathfrak{p}_{1}],\dots,[\mathfrak{p}_{t}]$. If $i,j\in[1,n]$ with $j\neq
i$, then $\mathfrak{p}_{k_{i}}\neq\mathfrak{p}_{k_{j}}.$ Indeed, if they were
equal, Corollary 3.2 would imply $1=a_{ik_{i}}=a_{ik_{j}}=0$ and this would be
a contradiction. Thus $\mathfrak{p}_{k_{1}},\dots,\mathfrak{p}_{k_{n}}$ are
$n$ superfluous generators and $C(\mathcal{U})$ is a free abelian group
generated by $t-n$ elements.

If $n+m\geq 2$, or $n+m=1$ and $K=\mathbb{Z}$, we can apply Theorem 3.3 to
obtain that every class contains exactly $|K|$ height-1 prime ideals. Suppose
then $n+m=1$ and $K$ is a field. Since we assumed there is no isolated
exchangeable index, necessarily $n=0$ and $m=1$. Then
$\mathcal{U}=K[x_{1}^{\pm 1}]$, $\mathcal{C}(\mathcal{U})=0$ and $\mathcal{U}$
contains $|K|$ pairwise non-associated prime elements. ∎

This theorem leads us to a dichotomy between factorial upper cluster algebras
and non-factorial ones. Let $A$ be a domain and $a\in A^{\bullet}$. We call
$k\geq 0$ a length of $a$ if there exist atoms $u_{1},\dots,u_{k}\in
A^{\bullet}$ such that $a=u_{1}\cdots u_{k}$. The length set of $a$, denoted
by $\mathsf{L}(a)$, is the set of all such lengths; we set
$\mathsf{L}(a)=\\{0\\}$ for $a\in A^{\times}$. Then $\mathsf{L}(a)=\\{0\\}$ if
and only if $a$ is a unit, and $\mathsf{L}(a)=\\{1\\}$ if and only if $a$ is
an atom. In a Krull domain $\mathsf{L}(a)$ is always a finite set.

######  Corollary 3.5.

Let $\mathcal{U}$ be an upper cluster algebra. Assume that $\mathcal{U}$ is a
Krull domain.

  * •

If $\mathcal{U}$ is factorial, then $\mathsf{L}(u)$ is a singleton for each
non-zero element $u\in\mathcal{U}.$

  * •

If $\mathcal{U}$ is not factorial, then for every finite set
$L\subseteq\mathbb{Z}_{\geq 2}$ there exists an element
$u\in\mathcal{U}^{\bullet}$ such that $\mathsf{L}(u)=L$.

###### Proof.

If $\mathcal{U}$ is factorial the claim is trivial. If $\mathcal{U}$ is not
factorial, the claim follows by Theorem 3.4 and a result of Kainrath [Kai99,
Theorem 1]. ∎

Now we compute the rank of the class group of a full rank upper cluster
algebra in terms of the irreducible factors of the exchange polynomials.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed and
$\mathcal{U}=\mathcal{U}(\Sigma)$ be the upper cluster algebra associated to
$\Sigma.$ Recall that, by definition, the upper cluster algebra $\mathcal{U}$
is an intersection of some $\mathcal{L}_{\mathbf{z}}$, where
$\mathcal{L}_{\mathbf{z}}=K[u^{\pm 1}\mid u\in\mathbf{z}\cup\mathbf{y}]$ is
the localization of $\mathcal{U}$ to the set
$S_{\mathbf{z}}=\\{\,z_{1}^{a_{1}}\cdots z_{n}^{a_{n}}x_{n+1}^{a_{n+1}}\cdots
x_{n+m}^{a_{n+m}}\mid a_{i}\in\mathbb{N}_{0}\,\\},$ where
$\mathbf{z}\sim\mathbf{x}$. On the other hand if we suppose that $\mathcal{U}$
is a Krull domain, then
$\mathcal{U}=\bigcap_{\mathfrak{p}\in\mathfrak{X}(\mathcal{U})}\mathcal{U}_{\mathfrak{p}}.$
Moreover, the following equalities holds

$\mathcal{U}=\bigcap_{\mathbf{z}\sim\mathbf{x}}\mathcal{L}_{\mathbf{z}}=\bigcap_{\mathbf{z}\sim\mathbf{x}}\bigcap_{\mathfrak{p}\in\mathfrak{X}(\mathcal{L}_{\mathbf{z}})}(\mathcal{L}_{\mathbf{z}})_{\mathfrak{p}}=\bigcap_{\mathbf{z}\sim\mathbf{x}}\bigcap_{\begin{subarray}{c}\mathfrak{q}\in\mathfrak{X}(\mathcal{U})\\\
\mathfrak{q}\cap
S_{\mathbf{z}}=\emptyset\end{subarray}}\mathcal{U}_{\mathfrak{q}},$

where the last equality follows from Proposition 1.17, since
$\mathcal{L}_{\mathbf{z}}$ is the localization of $\mathcal{U}$ at
$S_{\mathbf{z}}.$

######  Theorem 3.6.

Let $\Sigma=(\mathbf{x},\mathbf{y},B)$ be a seed and
$\mathcal{U}=\mathcal{U}(\Sigma)$ be the upper cluster algebra associated to
$\Sigma.$ Suppose in addition that $\mathcal{U}$ is a Krull domain. Then for
every height-1 prime ideal $\mathfrak{p}$ of $\mathcal{U}$ there exists a seed
$(\mathbf{z},\mathbf{y},C)$ such that
$\mathfrak{p}\mathcal{L}_{\mathbf{z}}\in\mathfrak{X}(\mathcal{L}_{\mathbf{z}}).$

Furthermore, if $\mathcal{U}=\bigcap_{i=0}^{n}\mathcal{L}_{\mathbf{z}_{i}}$
for some seed $(\mathbf{z},\mathbf{y},C)$, then for every height-1 prime ideal
$\mathfrak{p}$ of $\mathcal{U}$ there exists $k\in[0,n]$ such that
$\mathfrak{p}\mathcal{L}_{\mathbf{z}_{k}}\in\mathfrak{X}(\mathcal{L}_{\mathbf{z}_{k}}).$

###### Proof.

We prove the first claim, the second is completely analogous. Proceed by
contradiction, i.e., suppose that there exist
$\mathfrak{p}_{0}\in\mathcal{X}(\mathcal{U})$ such that
$\mathfrak{p}_{0}\mathcal{L}_{\mathbf{z}}\notin\mathfrak{X}(\mathcal{L}_{\mathbf{z}})$
for every seed $(\mathbf{z},\mathbf{y},C).$ We claim that there exists an
element $a\in\mathcal{U}$ such that $v_{\mathfrak{p}_{0}}(a)<0.$

Let $y\in\mathfrak{p}_{0},$ and let
$\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}\in\mathfrak{X}(\mathcal{U})$ be such
that $\\{\mathfrak{p}_{0},\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}\\}$ is the
set of all the distinct height-1 prime ideals of $\mathcal{U}$ that contain
$y$. Notice that $v_{\mathfrak{p}_{i}}(y)>0$ for every $i\in[0,r]$ and
$v_{\mathfrak{q}}(y)=0$ for every $\mathfrak{q}\neq\mathfrak{p}_{i}$. Set
$e_{i}:=v_{\mathfrak{p}_{i}}(y)\in\mathbb{N}$ for every $i\in[1,r]$ and
$e_{0}=0$. By the Approximation property (Theorem 1.24) there exists an
element $x\in\mathcal{U}$ such that $v_{\mathfrak{p}_{i}}(x)=e_{i}$ for every
$i\in[0,r]$ and $v_{\mathfrak{q}}(x)\geq 0$ for every
$\mathfrak{q}\in\mathfrak{X}(\mathcal{U})\setminus\\{\mathfrak{p}_{0},\dots,\mathfrak{p}_{r}\\}.$

Define now $a:=x/y\in\mathbf{q}(\mathcal{U}).$ By construction then

$v_{\mathfrak{p}_{0}}(a)=v_{\mathfrak{p}_{0}}(1/y)<0,\,\,v_{\mathfrak{p}_{i}}(a)=v_{\mathfrak{p}_{i}}(x)-v_{\mathfrak{p}_{i}}(y)=0$

for every $i\in[1,r]$ and $v_{\mathfrak{q}}(a)=v_{\mathfrak{q}}(x)\geq 0$ for
every
$\mathfrak{q}\in\mathfrak{X}(\mathcal{U})\setminus\\{\mathfrak{p}_{0},\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}\\}.$
Hence this implies that $a\in\mathcal{U}_{\mathfrak{q}}$ for every
$\mathfrak{q}\in\mathfrak{X}(\mathcal{U})\setminus\\{\mathfrak{p}_{0}\\}$ and
$a\notin\mathcal{U}_{\mathfrak{p}_{0}}$, hence $a\notin\mathcal{U}$. On the
other hand, by assumption $\mathfrak{p}_{0}\cap S_{\mathbf{z}}\neq\emptyset$
for every seed $(\mathbf{z},\mathbf{y},C)$, hence

$a\in\bigcap_{\mathbf{z}\sim\mathbf{x}}\bigcap_{\begin{subarray}{c}\mathfrak{q}\in\mathfrak{X}(\mathcal{U})\\\
\mathfrak{q}\cap
S_{\mathbf{z}}=\emptyset\end{subarray}}\mathcal{U}_{\mathfrak{q}}=\mathcal{U},$

therefore we found our contradiction. ∎

The theorem has an important consequence if we consider the case of upper
cluster algebras that satisfy the starfish condition at one seed.

######  Corollary 3.7.

Let $\mathcal{U}$ be an upper cluster algebra that satisfies the starfish
condition at a seed $(\mathbf{x},\mathbf{y},B)$. Let $\mathfrak{p}$ be a
height-1 prime ideal of $\mathcal{U}$ that contains $x_{i}$ for some
$i\in[1,n]$. Then $\mathfrak{p}\mathcal{L}_{\mathbf{x}_{i}}$ is a height-1 one
prime ideal of $\mathcal{L}_{\mathbf{x}_{i}}$. In particular, every height-1
prime ideal of $\mathcal{U}$ that contains the cluster variable $x_{i}$ does
not contain any element of the set
$\\{x_{1},\dots,x_{i}^{\prime},\dots,x_{n+m}\\}.$

###### Proof.

By Theorem 3.6 there exists $k\in[0,n]$ such that
$\mathfrak{p}\mathcal{L}_{\mathbf{x}_{k}}\in\mathfrak{X}(\mathcal{L}_{\mathbf{x}_{k}}).$
In particular
$\mathfrak{p}\cap\\{x_{1},\dots,x_{k}^{\prime},\dots,x_{n+m}\\}=\emptyset.$
Therefore $k$ must be $i$ and $x_{j}\notin\mathfrak{p}$ for every
$j\in[1,n+m]\setminus\\{i\\}.$ ∎

We are now ready to determine the rank of the class group of an upper cluster
algebra that satisfies the starfish condition at one seed.

######  Theorem 3.8.

Let $\mathcal{U}$ be an upper cluster algebra that satisfies the starfish
condition at a seed $(\mathbf{x},\mathbf{y},B)$. For every $i\in[1,n]$ let
$l_{i}$ be the number of irreducible factors of the exchange polynomial
$f_{i}$. Then

$\mathcal{C}(\mathcal{U})\cong\mathbb{Z}^{r},\,\,\text{with}\,\,r=\sum_{i=1}^{n}l_{i}-n.$

###### Proof.

Theorem 2.9 implies that $\mathcal{U}$ is a Krull domain, so we can apply
Theorem 3.4 and claim that $\mathcal{C}(\mathcal{U})$ is a free abelian group
of rank $t-n$ with $t$ the number of prime ideals that contain one of the
cluster variables $x_{1},\dots,x_{n}$. By Corollary 3.7, if a height-1 prime
ideal of $\mathcal{U}$ contains a cluster variable $x_{i}$, then it does not
contain $x_{j}$ for every $j\in[1,n]\setminus\\{i\\}$, that is

$t=\sum_{i=1}^{n}|\\{\,\mathfrak{p}\in\mathfrak{X}(\mathcal{U})\mid
x_{i}\in\mathfrak{p}\,\\}|.$

Fix an index $i\in[1,n]$, and let $r_{1},\dots,r_{l_{i}}\in
K[\mathbf{x}_{i},\mathbf{y}]$ be the pairwise non-associated irreducible
factors of the exchangeable polynomial $f_{i}=x_{i}x_{i}^{\prime}\in
K[\mathbf{x}_{i},\mathbf{y}].$ We claim that

$l_{i}=|\\{\,\mathfrak{p}\in\mathfrak{X}(\mathcal{U})\mid
x_{i}\in\mathfrak{p}\,\\}|.$

Let $\mathfrak{p}\in\mathfrak{X}(\mathcal{U})$ be a height-1 prime ideal that
contains the cluster variable $x_{i}$. Corollary 3.7 implies that
$\mathfrak{p}^{\prime}:=\mathfrak{p}\mathcal{L}_{\mathbf{x}_{i}}$ is a
height-1 prime ideal of $\mathcal{L}_{\mathbf{x}_{i}}$ that contains $x_{i}$.
Observe that $f_{i}\in\mathfrak{p}^{\prime}$. Since none of the factors of
$f_{i}$ can be a monomial in $\mathbf{x}$, Lemma 1.18 implies that, for all
$k\in[1,l_{i}]$, $r_{k}$ is also irreducible in
$\mathcal{L}_{\mathbf{x}_{i}}.$ The ideal $\mathfrak{p}^{\prime}$ is prime,
hence necessarily one of the irreducible factors of $f_{i}$, say $r_{k}$, must
be in $\mathfrak{p}^{\prime}.$ Therefore, since $\mathfrak{p}^{\prime}$ has
height $1$ and $\mathcal{L}_{\mathbf{x}_{i}}$ is factorial, we have that
$\mathfrak{p}^{\prime}=r_{k}\mathcal{L}_{\mathbf{x}_{i}},$ and hence
$\mathfrak{p}=\mathcal{U}\cap r_{k}\mathcal{L}_{\mathbf{x}_{i}}$. This proves
that

$|\\{\,\mathfrak{p}\in\mathfrak{X}(\mathcal{U})\mid
x_{i}\in\mathfrak{p}\,\\}|\leq l_{i}.$

A similar argument shows that $\mathfrak{p}_{j}:=\mathcal{U}\cap
r_{j}\mathcal{L}_{\mathbf{x}_{i}}$ is a height-1 prime ideal of $\mathcal{U}$
that contains $x_{i}$, for every $j\in[1,l_{i}]$. Assume
$\mathfrak{p}_{j}=\mathfrak{p}_{k}$ for some $j\neq k$. Then
$r_{j}\mathcal{L}_{\mathbf{x}_{i}}=r_{k}\mathcal{L}_{\mathbf{x}_{i}}$ and
hence $r_{j}\simeq r_{k}$, that is impossible by assumption, whence

$l_{i}=|\\{\,\mathfrak{p}\in\mathfrak{X}(\mathcal{U})\mid
x_{i}\in\mathfrak{p}\,\\}|.\qed$

If $l_{i}=1$ for every $i\in[1,n]$, we get immediately the following
corollary, which is a slight generalization of [CKQ23, Theorem 4.9].

######  Corollary 3.9.

Let $\mathcal{U}$ be an upper cluster algebra that satisfies the starfish
condition at a seed $(\mathbf{x},\mathbf{y},B)$. Then $\mathcal{U}$ is
factorial if and only if the exchange polynomials $f_{i}=x_{i}x_{i}^{\prime}$
are irreducible.

Notice we cannot extend this theorem to upper cluster algebras that are just
Krull domains. Indeed, consider the (upper) cluster algebra of finite type
$A_{3}.$ The cluster algebra is a Krull domain and we know that is not
factorial ($1+x_{2}^{2}=x_{1}x_{1}^{\prime}=x_{3}x_{3}^{\prime}$), hence
$\mathcal{C}(\mathcal{U})\neq 0.$ However, $\sum_{i=1}^{3}l_{i}-3=0,$ so the
theorem does not apply in this case.

######  Remark 3.10.

If the seed is in addition acyclic, the cluster algebra and the upper cluster
algebra coincide and we could get the conclusions of Theorem 3.8 as a
consequence of the main Theorem of [GELS19]. Two indices $i$, $j\in[1,n]$ are
partners if $f_{i}$ and $f_{j}$ have a common non-trivial factor. If
$\,\mathcal{U}$ satisfies the starfish condition at a seed
$(\mathbf{x},\mathbf{y},B)$, then no distinct indices are partners. Indeed,
first notice that, if $i$ and $j$ are partners, then
$(x_{1},\dots,x_{i}^{\prime},\dots,x_{j}^{\prime},\dots,x_{n})$ is a seed,
since $b_{ij}=b_{ji}=0$ (cf. [GELS19, Lemma 2.7]). Suppose now, by
contradiction, that there exist two partners $i,j\in[1,n]$ with $i<j$. Thus
there exists $h\in K[\mathbf{x},\mathbf{y}]\setminus
K[\mathbf{x},\mathbf{y}]^{\times}$ such that $f_{i}=hg_{i}$ and $f_{j}=hg_{j}$
for some $g_{i},g_{j}\in K[\mathbf{x},\mathbf{y}]$. Hence the element
$s=\frac{hg_{i}g_{j}}{x_{i}x_{j}}\in\bigcap_{k=0}^{n}\mathcal{L}_{\mathbf{x}_{k}},$
but $s\notin K[x_{1}^{\pm 1},\dots,{x_{i}^{\prime}}^{\pm
1},\dots,{x_{j}^{\prime}}^{\pm 1},\dots,x_{n}^{\pm 1}]$, and this is a
contradiction. Assume in addition that the (upper) cluster algebra
$\,\mathcal{U}$ is acyclic. Then, since the only partner sets are the sets
$\\{i\\}$, $i\in[1,n]$, Theorem A in [GELS19] implies that the rank of the
class group of $\,\mathcal{U}$ is equal to $t-n$ with
$t=\sum_{i=1}^{n}|\\{\,\mathfrak{p}\in\mathfrak{X}(\mathcal{U})\mid
x_{i}\in\mathfrak{p}\,\\}|$.

Denote by $\mu_{d}^{*}(K)$ the set of primitive $d$-th roots of unit in $K$.

######  Examples 3.11.

Let us compute the class group in some specific examples.

  1. (1)

Consider the seed
$\Sigma=\left(\left(x_{1},x_{2},x_{3},x_{4}\right),\emptyset,B\right)$ where
$B$ is the matrix

$B=\begin{pmatrix}0&-1&0&4\\\ 2&0&3&6\\\ 0&-3&0&0\\\ -4&-3&0&0\end{pmatrix}.$

$B$ is a full rank skew-symmetrizable matrix (consider
$d_{1}=d_{4}=2,\,d_{2}=d_{3}=1$).

The exchange polynomials associated to $\Sigma$ are:

$\displaystyle f_{1}=x_{2}^{2}+x_{4}^{4},\quad
f_{2}=x_{1}x_{3}^{3}x_{4}^{3}+1,$ $\displaystyle f_{3}=x_{2}^{3}+1,\quad
f_{4}=x_{1}^{4}x_{2}^{6}+1.$

The polynomial $f_{1}$ has 2 factors if $\mu_{4}^{*}(K)\neq\emptyset$,
otherwise 1, the polynomial $f_{3}$ has 3 factors if
$\mu_{6}^{*}(K)\neq\emptyset$, otherwise 2, and the polynomial $f_{4}$ has 2
factors if $\mu_{4}^{*}(K)\neq\emptyset$, otherwise 1. The following table
shows the class group of $\mathcal{U}$ in all the possible cases:

| $\mu_{4}^{*}(K)\neq\emptyset$ | $\mu_{4}^{*}(K)=\emptyset$
---|---|---
$\mu_{6}^{*}(K)\neq\emptyset$ | $\mathbb{Z}^{4}$ | $\mathbb{Z}^{2}$
$\mu_{6}^{*}(K)=\emptyset$ | $\mathbb{Z}^{3}$ | $\mathbb{Z}$

Notice that $U(\Sigma)$ is not factorial independently from the choice of the
field.

  2. (2)

Consider the following quiver $\mathcal{Q}$:

${1}$${2}$${3}$${4.}$

The matrix associated to $\mathcal{Q}$ is

$B=\begin{pmatrix}0&0&1&-1\\\ 0&0&-1&-1\\\ -1&1&0&1\\\ 1&1&-1&0\end{pmatrix}$

and the exchange polynomials are:

$\displaystyle f_{1}=x_{3}+x_{4},\quad f_{2}=x_{3}x_{4}+1,$ $\displaystyle
f_{3}=x_{2}x_{4}+x_{1},\quad f_{4}=x_{1}x_{2}+x_{3}.$

$B$ has full rank and the polynomials are irreducible, hence
$\mathcal{U}(\mathcal{Q})$ is factorial.

  3. (3)

Consider the seed $\Sigma=((x_{1},x_{2},x_{3}),\\{x_{4}\\},B)$ where $B$ is
the matrix

$B=\begin{pmatrix}0&2&-2\\\ -2&0&2\\\ 2&-2&0\\\ 2&0&0\end{pmatrix}.$

$B$ has full rank and the exchange polynomials are

$f_{1}=x_{2}^{2}+x_{3}^{2}x_{4}^{2},\quad f_{2}=x_{1}^{2}+x_{3}^{2},\quad
f_{3}=x_{1}^{2}+x_{2}^{2}.$

Hence, if $\mu_{4}^{*}(K)\neq\emptyset$, then
$\mathcal{C}(\mathcal{U})\cong\mathbb{Z}^{3}$, otherwise $\mathcal{U}$ is
factorial.

Observe that the exchangeable part of the quiver $\Gamma(B)$, being the Markov
quiver, is not acyclic, hence we could not have applied Theorem B in [GELS19].

##  4\. Valuation Pairing

In [CKQ23], the authors introduced the notion of a _valuation pairing_ on an
upper cluster algebra and proved a local unique factorization for full rank
upper cluster algebras. In this section we give an interpretation of the
valuation pairing in terms of the $\mathfrak{p}$-adic valuation in Krull
domains.

######  Definition 4.1 (Valuation pairing, [CKQ23]).

Let $\mathcal{U}$ be an upper cluster algebra and $\mathcal{X}$ be the set of
cluster variables of $\mathcal{U}.$ Define

$(-|-)_{v}\colon\mathcal{X}\times\mathcal{U}\to\mathbb{N}\cup\\{\infty\\}\qquad(x,u)\mapsto(x\mid
u)_{v}:=\text{max}\\{\,s\in\mathbb{N}\mid u/x^{s}\in\mathcal{U}.\,\\}$

Clearly, $(x\mid u)_{v}=0$ if and only if $x\nmid u.$ If $r=(x\mid u)_{v}>0$,
then $u/x^{r}\in\mathcal{U}$ and $u/x^{r+1}\notin\mathcal{U}$, that is there
exists $y\in\mathcal{U}$ such that $u=x^{r}y$ and $x\nmid y$.

######  Definition 4.2 (Local Factorization, [CKQ23]).

Let $\mathcal{U}$ be an upper cluster algebra and $u\in\mathcal{U}$. We say
that $u=ab$ is a local factorization of $u$ with respect to the seed
$(\mathbf{x},\mathbf{y},B)$ if $a$ is a monomial in $\mathbf{x}$ and
$(x_{i}\mid b)_{v}=0$ for every $i\in[1,n].$

Using valuation pairings, Cao, Keller, and Qin proved the following.

######  Theorem 4.3 ([CKQ23, Propisition 3.5 and Theorem 3.7]).

Let $\mathcal{U}$ be an upper cluster algebra and $(\mathbf{x},\mathbf{y},B)$
any seed. Then every $0\neq u\in\mathcal{U}$ admits a local factorization with
respect to $(\mathbf{x},\mathbf{y},B)$. Moreover, if $\mathcal{U}$ has full
rank, then this local factorization is unique (up to associates).

Let $\mathcal{U}$ be an upper cluster algebra. We can interpret the valuation
pairing in terms of discrete valuations on $\mathcal{U}$. By definition,
$\mathcal{U}$ is an intersection of (possibly infinitely many) Laurent
polynomial rings. Each of these Laurent polynomial rings gives rise to a
family of discrete valuations, arising from the height-1 prime ideals of the
Laurent polynomial ring. Let $\\{\,v_{i}:i\in I\,\\}$ be the set of all such
discrete valuations arising from all the Laurent polynomial rings.

Let $x\in\mathcal{X}$ and let $V\subseteq\\{\,v_{i}:i\in I\,\\}$ the subset of
all valuations $v_{i}$ for which $v_{i}(x)>0$. Now consider an element
$u\in\mathcal{U}^{\bullet}$ and set $r=(x\mid u)_{v}.$ Let us write $u=x^{r}y$
with $y\in\mathcal{U}$ such that $x\nmid y$. Clearly, we have that
$v_{i}(x)r\leq v_{i}(u),$ for every $v_{i}\in V$, hence

$r\leq\inf_{v_{i}\in V}\bigg{\lfloor}\frac{v_{i}(u)}{v_{i}(x)}\bigg{\rfloor}.$

Let $s:=\inf_{v_{i}\in V}\Big{\lfloor}\frac{v_{i}(u)}{v_{i}(x)}\Big{\rfloor}.$
Then $u/x^{s}\in\mathcal{U}$ since

$v_{i}(ux^{-s})=v_{i}(u)-sv_{i}(x)\geq
v_{i}(u)-\bigg{\lfloor}\frac{v_{i}(u)}{v_{i}(x)}\bigg{\rfloor}v_{i}(x)\geq
v_{i}(u)-\frac{v_{i}(u)}{v_{i}(x)}v_{i}(x)=0,$

hence

$(x\mid u)_{v}=\inf_{v_{i}\in
V}\frac{v_{\mathfrak{p}_{i}}(u)}{v_{\mathfrak{p}_{i}}(x)}.$

######  Remark 4.4.

If $\mathcal{U}$ is a Krull domain, then the discrete valuations $V$ are just
those arising from the finitely many height-1 prime ideals of $\mathcal{U}$
containing $x$, so in this case

$(x\mid
u)_{v}=\min_{\begin{subarray}{c}\mathfrak{p}\in\mathfrak{X}(\mathcal{U})\\\
x\in\mathfrak{p}\end{subarray}}\bigg{\lfloor}\frac{v_{\mathfrak{p}}(u)}{v_{\mathfrak{p}}(x)}\bigg{\rfloor}.$

## References

  * [AAZ92] D. D. Anderson, David F. Anderson, and Muhammad Zafrullah.  Factorization in integral domains. II.  J. Algebra, 152(1):78–93, 1992.
  * [BFZ05] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky.  Cluster algebras. III. Upper bounds and double Bruhat cells.  Duke Math. J., 126(1):1–52, 2005.  https://doi.org/10.1215/S0012-7094-04-12611-9.
  * [CKQ23] Peigen Cao, Bernhard Keller, and Fan Qin.  The valuation pairing on an upper cluster algebra.  J. Reine Angew. Math, 2023.  online first, https://doi.org/10.1515/crelle-2023-0080.
  * [Fos73] Robert M. Fossum.  The Divisor Class Group of a Krull Domain, volume 74 of Ergebnisse der Mathematik und Ihrer Grenzgebiete.  Springer Berlin, Heidelberg, 1973.
  * [FZ02] Sergey Fomin and Andrei Zelevinsky.  Cluster algebras. I. Foundations.  J. Amer. Math. Soc., 15(2):497–529, 2002.  https://doi.org/10.1090/S0894-0347-01-00385-X.
  * [FZ03] Sergey Fomin and Andrei Zelevinsky.  Cluster algebras: notes for the CDM-03 conference.  In Current developments in mathematics, 2003, pages 1–34. Int. Press, Somerville, MA, 2003.
  * [GELS19] Ana Garcia Elsener, Philipp Lampe, and Daniel Smertnig.  Factoriality and class groups of cluster algebras.  Adv. Math., 358:106858, 48, 2019.  https://doi.org/10.1016/j.aim.2019.106858.
  * [GHK06] Alfred Geroldinger and Franz Halter-Koch.  Non-unique factorizations, volume 278 of Pure and Applied Mathematics (Boca Raton).  Chapman & Hall/CRC, Boca Raton, FL, 2006.  Algebraic, combinatorial and analytic theory.
  * [GLS13] Christof Geiss, Bernard Leclerc, and Jan Schröer.  Factorial cluster algebras.  Doc. Math., 18:249–274, 2013.  https://arxiv.org/abs/1110.1199.
  * [Kai99] Florian Kainrath.  Factorization in Krull monoids with infinite class group.  Colloq. Math., 80(1):23–30, 1999.  https://doi.org/10.4064/cm-80-1-23-30.
  * [Kel12] Bernhard Keller.  Cluster algebras and derived categories.  In Derived categories in algebraic geometry, EMS Ser. Congr. Rep., pages 123–183. Eur. Math. Soc., Zürich, 2012.
  * [Lec10] Bernard Leclerc.  Cluster algebras and representation theory.  In Proceedings of the International Congress of Mathematicians. Volume IV, pages 2471–2488. Hindustan Book Agency, New Delhi, 2010.
  * [LW14] Bernard Leclerc and Lauren K. Williams.  Cluster algebras.  Proc. Natl. Acad. Sci. USA, 111(27):9676–9679, 2014.  https://doi.org/10.1073/pnas.1410635111.
  * [MM15] Jacob P. Matherne and Greg Muller.  Computing upper cluster algebras.  Int. Math. Res. Not. IMRN, (11):3121–3149, 2015.  https://doi.org/10.1093/imrn/rnu027.
  * [Mul13] Greg Muller.  Locally acyclic cluster algebras.  Adv. Math., 233:207–247, 2013.  https://doi.org/10.1016/j.aim.2012.10.002.
  * [Mul14] Greg Muller.  $\mathcal{A}=\mathcal{U}$ for locally acyclic cluster algebras.  SIGMA Symmetry Integrability Geom. Methods Appl., 10:Paper 094, 8, 2014.  https://doi.org/10.3842/SIGMA.2014.094.
  * [Nak22] Tomoki Nakanishi.  Cluster Algebras and Scattering Diagrams, Part I. Basics in Cluster Algebras, 2022.  Preprint, https://arxiv.org/abs/2201.11371.
  * [Zel05] Andrei Zelevinsky.  Cluster algebras: origins, results and conjectures.  In Advances in algebra towards millennium problems, pages 85–105. SAS Int. Publ., Delhi, 2005.