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\title{\Large Explicit non-inducible rooted cluster morphisms}
\author{Qiyue Tang}
\date{}

\begin{document}
\maketitle
\begingroup
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\begin{NoHyper}
\footnotetext{\textit{2020 Mathematics Subject Classification.} Primary 13F60; Secondary 13A02.\newline
\textit{Keywords.} Rooted cluster morphism, Markov cluster algebra.}
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\begin{abstract}
The augmentation of the Markov cluster algebra is explicit but not inducible as a rooted cluster morphism. This answers a question of Chang and Zhu negatively.
\end{abstract}

\section{Rooted morphisms}

Rooted cluster morphisms were introduced by Assem, Dupont, and
Schiffler~\cite{ADS}. Chang and Zhu~\cite[Definition~2.8]{CZ} call such
a morphism \emph{explicit} if its values on the initial cluster determine
it uniquely, and \emph{inducible} if it extends to the ambient Laurent
rings. They ask whether every explicit rooted cluster morphism is
inducible~\cite[Problem~1]{CZ}. We give a counterexample using the
Markov cluster algebra.

The construction uses its positive grading, recorded by
Muller~\cite[Section~11.2]{Muller}, and the propagation of zero values
under mutation. The latter argument appears in
Castronovo--Gorsky--Simental--Speyer~\cite[arXiv version, Section~3.4.1]{CGSS} and
Beyer--Muller~\cite[Section~5.2]{BM} in their study of deep points.
Here we use these facts to obtain uniqueness of the augmentation among
homomorphisms to reduced rings, and hence the required rooted cluster
morphism.

All rings are commutative, and all ring homomorphisms preserve the
identity. We work with coefficient-free cluster algebras over $\Z$.
A seed is a pair $\Sigma=(\mathbf x,B)$, where
$\mathbf x=(x_1,\ldots,x_n)$ consists of algebraically independent
variables over $\Q$, $n\geq1$, and $B=(b_{ij})_{1\leq i,j\leq n}$
is a skew-symmetrizable integer matrix. Thus there are positive integers
$d_i$ such that $d_i b_{ij}=-d_j b_{ji}$ for all $i,j$.
Every variable in $\mathbf x$ is exchangeable; there are no frozen
variables. Mutation in direction $k\in\{1,\ldots,n\}$ replaces $x_k$
by
\[
 x_k'=\frac{\prod_{b_{ik}>0}x_i^{b_{ik}}
                  +\prod_{b_{ik}<0}x_i^{-b_{ik}}}{x_k},
\]
where an empty product is $1$, and replaces $B$ by the matrix with
entries
\[
 b_{ij}'=
 \begin{cases}
 -b_{ij},& i=k\text{ or }j=k,\\[2pt]
 b_{ij}+\dfrac{|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2},& i,j\ne k.
 \end{cases}
\]
The cluster algebra $\mathcal A(\Sigma)$ is the $\Z$-subalgebra of
$\Q(x_1,\ldots,x_n)$ generated by all variables obtained by finite
sequences of mutations. A rooted cluster algebra retains the choice
of $\Sigma$. By the Laurent phenomenon~\cite[Theorem~3.1]{FZ},
\[
 \mathcal A(\Sigma)\subseteq
 \Z[x_1^{\pm1},\ldots,x_n^{\pm1}].
\]
Following~\cite{CZ}, its ambient ring is
$L_{\Sigma,\Q}=\Q[x_1^{\pm1},\ldots,x_n^{\pm1}]$.

For another seed $\Sigma'=(\mathbf y,B')$, a rooted cluster morphism
$f:\mathcal A(\Sigma)\to\mathcal A(\Sigma')$ is a ring homomorphism
with the following properties. Each initial variable maps to an initial
variable of $\Sigma'$ or to an integer, and $f$ commutes with every
biadmissible mutation sequence. Write $\mu_u$ for mutation at the
cluster variable $u$, acting on a seed and its labelled variable positions.
A finite sequence $(u_1,\ldots,u_\ell)$, with $\ell\geq0$,
is \emph{admissible} if each $u_j$ is exchangeable after the preceding
mutations; it is \emph{biadmissible} if its image sequence under $f$ is
also admissible in the target. More precisely, for every such sequence
and every initial variable $v$,
\[
 f\bigl(\mu_{u_\ell}\cdots\mu_{u_1}(v)\bigr)
 =\mu_{f(u_\ell)}\cdots\mu_{f(u_1)}\bigl(f(v)\bigr).
\]
Each mutation replaces the variable at its chosen position and leaves
the other positions unchanged; on the right, integers are left fixed.
The empty sequence acts as the identity.
These are the coefficient-free instances of the axioms in~\cite{CZ}.

Precisely, $f$ is explicit if every rooted cluster morphism
$g:\mathcal A(\Sigma)\to\mathcal A(\Sigma')$ satisfying
$g(x_i)=f(x_i)$ for $1\leq i\leq n$ equals $f$.
It is inducible if there is a ring homomorphism
$L_{\Sigma,\Q}\to L_{\Sigma',\Q}$ whose restriction is $f$.

\section{The Markov example}

Let $x,y,z$ be algebraically independent over $\Q$, and let $A$ be the
cluster algebra with initial seed
\[
 \Sigma=\left((x,y,z),
 B=\begin{pmatrix}0&2&-2\\-2&0&2\\2&-2&0\end{pmatrix}\right).
\]
This is the coefficient-free Markov cluster algebra. Matrix mutation
gives $\mu_k(B)=-B$ and $\mu_k(-B)=B$ for $k=1,2,3$.
Consequently, in every cluster $(a,b,c)$, mutation at $a$ has exchange
relation
\begin{equation}\label{exchange}
 aa'=b^2+c^2,
\end{equation}
with the analogous relations in the other two directions.

\begin{theorem}\label{main}
There is a ring homomorphism $\epsilon:A\to\Z$ sending every cluster
variable to zero. For every coefficient-free rooted cluster algebra
$C$, the composite
\[
 f:A\xrightarrow{\epsilon}\Z\hookrightarrow C
\]
is an explicit rooted cluster morphism which is not inducible.
\end{theorem}

\begin{proof}
Give a Laurent monomial $x^iy^jz^k$, with $i,j,k\in\Z$, degree $i+j+k$.
A Laurent polynomial is homogeneous of degree $d$ if all its nonzero
monomials have degree $d$. The initial variables have degree $1$,
and~\eqref{exchange} shows inductively that every cluster variable is
homogeneous of degree $1$. The Laurent phenomenon ensures that these
are homogeneous elements of the Laurent polynomial ring.

For each integer $d\geq0$, let $A_d$ be the subgroup of $A$ consisting of its homogeneous
elements of degree $d$, together with zero. Since cluster variables
generate $A$ over $\Z$, the Laurent grading restricts to
\[
 A=\bigoplus_{d\geq0}A_d,\qquad A_0=\Z,
 \qquad A_d A_e\subseteq A_{d+e}\quad(d,e\geq0).
\]
Projection onto $A_0$ therefore defines a ring homomorphism
\[
 \epsilon\left(\sum_{d\geq0}a_d\right)=a_0
 \qquad(a_d\in A_d),
\]
where each sum has finitely many nonzero terms. In particular,
$\epsilon(1)=1$, and $\epsilon$ kills every cluster variable.

We next prove uniqueness with a more general target. A ring is
\emph{reduced} if its only nilpotent element is zero. Let $R$ be a
reduced ring, and let $g:A\to R$ be a ring homomorphism satisfying
$g(x)=g(y)=g(z)=0$. Suppose that $g$ kills all variables in a cluster
$(a,b,c)$. Mutate at $a$ to obtain $(a',b,c)$, and then at $b$ to
obtain a variable $b^\star$. The second exchange relation is
\[
 bb^\star=(a')^2+c^2.
\]
Applying $g$ gives $g(a')^2=0$, hence $g(a')=0$. The same argument
works in each direction. Induction on the length of a mutation
sequence shows that $g$ kills every cluster variable. Since these
variables generate $A$, the map $g$ is $\epsilon$ followed by the
canonical map $\Z\to R$.

For the composite $f$ in the statement, all initial variables map to
the integer zero. There is no nonempty biadmissible sequence, since
its first image would have to be an exchangeable variable of $C$.
Thus $f$ is a rooted cluster morphism. The algebra $C$ is a subring
of a rational function field, so it is reduced. The preceding
uniqueness statement applies to every rooted cluster morphism with
the same initial values as $f$, proving that $f$ is explicit.

Finally, an extension $\widetilde f$ to the ambient rings would send
the invertible element $x$ to zero. This would give
\[
 1=\widetilde f(xx^{-1})
   =\widetilde f(x)\widetilde f(x^{-1})=0,
\]
which is impossible in the ambient ring of $C$.
\end{proof}

Taking $C=A$ gives an endomorphism answering Chang--Zhu
Problem~1. Its image is $\Z$, and it satisfies $f^2=f$.

The role of reduced targets can also be seen in a quotient of $A$.
Let $I=(x,y,z)A$ be the ideal generated by the initial variables, and
let $A_{>0}=\bigoplus_{d>0}A_d=\ker\epsilon$.
For an ideal $J$, its radical is
$\sqrt J=\{a:a^m\in J\text{ for some integer }m\geq1\}$;
the reduction of a ring $S$ is $S_{\mathrm{red}}=S/\sqrt{(0)}$.

\begin{proposition}
One has $\sqrt I=A_{>0}$ and $(A/I)_{\mathrm{red}}\cong\Z$,
but $A/I$ is not reduced.
\end{proposition}

\begin{proof}
Apply the uniqueness argument in the proof of Theorem~\ref{main}
to the quotient map $A\to A/\sqrt I$. Every cluster variable lies in
$\sqrt I$, so $A_{>0}\subseteq\sqrt I$. Conversely,
$I\subseteq\ker\epsilon$ and $A/\ker\epsilon\cong\Z$ is reduced,
which gives $\sqrt I\subseteq\ker\epsilon=A_{>0}$.
The stated isomorphism follows.

Let $u=(y^2+z^2)/x$ be the mutation of $x$, and let $v$ be the
mutation of $y$ in the cluster $(u,y,z)$. Then
$u^2=yv-z^2\in I$. If $u\in I$, taking the degree $1$ part of an
expression $u=xh_1+yh_2+zh_3$, with $h_i\in A$, would give
$u\in\Z x+\Z y+\Z z$. This contradicts its Laurent expansion
$u=x^{-1}y^2+x^{-1}z^2$. Thus the class of $u$ in $A/I$ is nonzero
and has square zero.
\end{proof}

\medskip
\noindent\textit{AI use.}
ChatGPT was used to develop the counterexample and its proof.
Codex was used to check the arguments, search the literature, and
prepare the manuscript.

\begin{thebibliography}{9}

\bibitem{ADS}
I.~Assem, G.~Dupont, and R.~Schiffler,
\emph{On a category of cluster algebras},
J. Pure Appl. Algebra \textbf{218} (2014), no.~3, 553--582.

\bibitem{BM}
J.~Beyer and G.~Muller,
\emph{Deep points of cluster algebras},
Int. Math. Res. Not. IMRN \textbf{2025} (2025), no.~4,
rnaf027, 39~pp.

\bibitem{CGSS}
M.~Castronovo, M.~Gorsky, J.~Simental, and D.~E.~Speyer,
\emph{Cluster deep loci and mirror symmetry},
Commun. Amer. Math. Soc. \textbf{6} (2026), 1021--1086;
\href{https://arxiv.org/abs/2402.16970v2}{arXiv:2402.16970v2}.

\bibitem{CZ}
W.~Chang and B.~Zhu,
\emph{On rooted cluster morphisms and cluster structures in
$2$-Calabi--Yau triangulated categories},
J. Algebra \textbf{458} (2016), 387--421;
\href{https://arxiv.org/abs/1410.5702v2}{arXiv:1410.5702v2}.

\bibitem{FZ}
S.~Fomin and A.~Zelevinsky,
\emph{Cluster algebras I: Foundations},
J. Amer. Math. Soc. \textbf{15} (2002), no.~2, 497--529.

\bibitem{Muller}
G.~Muller,
\emph{Locally acyclic cluster algebras},
Adv. Math. \textbf{233} (2013), 207--247.

\end{thebibliography}

\medskip
\noindent Zhili College, Tsinghua University, Beijing 100084, China\par
\noindent\textit{Email:} \href{mailto:tangqy24@mails.tsinghua.edu.cn}{\texttt{tangqy24@mails.tsinghua.edu.cn}}
\end{document}