# The minimal monomial lifting of cluster algebras I: branching problems

Luca Francone
francone at math.univ-lyon1.fr

###### Abstract

Let $Y$ be a complex scheme with cluster structure, $T$ be a complex torus and
${\mathfrak{X}}$ be a suitable partial compactification of $T\times Y$. We
produce, through a technique called minimal monomial lifting, a canonically
graded upper cluster algebra $\overline{\mathcal{A}}$ inside
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$, which is, in a precise sense,
the best candidate to give a cluster structure on ${\mathfrak{X}}$ compatible
with the one on $Y$. We develop some geometric criteria to prove the equality
between $\overline{\mathcal{A}}$ and
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$, which doesn’t always hold and
has some remarkable consequences. As a special case, through this
construction, we recover in a purely geometric and combinatorial method some
cluster algebras already appearing in the literature.

If ${\widehat{G}}$ is a complex reductive subgroup of the complex, semisimple,
simply connected algebraic group $G$, we set up a geometric framework to study
the branching problem in representation theory, for this pair of groups,
through the minimal monomial lifting. We study in detail the case of a Levi
subgroup and the tensor product case, for which we have the best possible
outcome. Finally, we identify a cluster structure on the base affine space
associated to $G$ and discuss some consequences of these results.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on cluster algebras
    1. 2.1 Cluster algebras
    2. 2.2 Disjoint union of seeds
    3. 2.3 Cluster valuations induced by frozen variables
    4. 2.4 Highly-freezing and semi-freezing
    5. 2.5 Graded seeds
  3. 3 Monomial liftings
    1. 3.1 Pole filtration and lifting graduation
  4. 4 The minimal monomial lifting
    1. 4.1 Equality conditions
  5. 5 Preliminaries on algebraic groups
    1. 5.1 Some classical facts and notation
    2. 5.2 Generalised minors
    3. 5.3 Cluster structure on U(w)
    4. 5.4 Technical properties of minors I: on vanishing and twist
    5. 5.5 Technical properties of minors II: on algebraic independence
  6. 6 Application to G
  7. 7 Monomial lifting for branching problems
    1. 7.1 The branching scheme
    2. 7.2 Monomial lifting for the branching scheme
    3. 7.3 Lifting graduation for the Branching algebra
    4. 7.4 On question 7.2.3
      1. 7.4.1 Levi subgroups
      2. 7.4.2 Products
  8. 8 Study of equality
    1. 8.1 The Levi case
    2. 8.2 The product case: tensor product
    3. 8.3 The base affine space, or branching to a maximal Torus

##  1 Introduction

This is the first step in a project which aims to study branching problems in
representation theory, through the use of cluster algebras. The aims for this
project are multiple, among them we have:

  1. 1.

Provide a possibly uniform framework to study these problems.

  2. 2.

Reveal hidden features of this classical question through the lens of cluster
theory.

  3. 3.

Try to build polyhedral models for multiplicities using cluster algebra
theory.

#### The branching problem

Let ${\widehat{G}}$ be a complex, connected, reductive algebraic subgroup of
the connected reductive group $G$. The branching problem in representation
theory asks to understand how, irreducible representations of $G$, decompose
under the natural ${\widehat{G}}$-action.

Fix maximal tori ${\widehat{T}}\subseteq T$ and Borel subgroups $B\supseteq T$
and ${\widehat{B}}\supseteq{\widehat{T}}$ of $G$ and ${\widehat{G}}$
respectively. Let $X(T)$ denote the group of characters of $T$ and let
$X(T)^{+}$ denote the set of dominant characters. For $\lambda\in X(T)^{+}$,
$V(\lambda)$ denotes the irreducible representation of highest weight
$\lambda$. Similarly, we use the notation $X(\widehat{T})$,
$X(\widehat{T})^{+}$, $V(\widehat{\lambda})$ relatively to $\widehat{G}$. For
any $\lambda\in X(T)^{+}$, there is a natural ${\widehat{G}}$-equivariant
isomorphism:

$\begin{array}[]{rcl}\displaystyle\bigoplus_{\widehat{\lambda}\in
X({\widehat{T}})^{+}}{\operatorname{Hom}}(V(\widehat{\lambda}),V(\lambda))^{\widehat{G}}\otimes
V(\widehat{\lambda})&{\,\longrightarrow}&V(\lambda)\\\ f\otimes
v&\longmapsto&f(v).\end{array}$

Hence the multiplicity of $V(\widehat{\lambda})$ in $V(\lambda)$ is
$c_{\lambda}^{\widehat{\lambda}}=\dim{\operatorname{Hom}}(V(\widehat{\lambda}),V(\lambda))^{G}$.
A result one would like to achieve, in solving the branching problem is the
following.

Objective: for the pair $(G,{\widehat{G}})$, construct a positive
combinatorial model for the multiplicities.

Exemple of such models are given by the Littlewood-Ricardson rule, Gelfand-
Tseltlin’s patterns, Littelmann’s paths [Lit95] or Sundaram’s dominos [Sun90].

Though, for some specific pairs $(G,{\widehat{G}})$ such a model exists, no
general technique for constructing those models, uniformly for some families
of pairs, is known. To the author best knowledge, the only remarkable
exceptions, for which a positive model for multiplicities can be constructed
in families, are the following.

  1. 1.

Weight spaces: ${\widehat{G}}$ is a maximal torus of $G$.

  2. 2.

The Levi case: ${\widehat{G}}$ is a Levi subgroup of $G$.

  3. 3.

The tensor product case: ${\widehat{G}}$ is diagonally embedded in
$G={\widehat{G}}\times{\widehat{G}}$, for which the branching problem amounts
to decompose the tensor product of irreducible representations of
${\widehat{G}}$.

Note that the first family is a subfamily of the second one. A breakthrough
has been realised by Berenstein and Zelevisnky [BZ01], who constructed, under
the assumption that $G$ is semisimple and simply connected, (many) polyhedral
models for multiplicities, for any branching problem belonging to the previous
list. Their proof is actually quite involved and uses deep results and
objects, such as: the canonical basis and the Lusztig’s parametrisations
[Lus10], its relation to total positivity and double Bruhat cells [Lus94]
[FZ99] and tropicalisation. We refer to [Zel02] for a beautiful survey on the
proof. One key ingredient to link all these objects, are the detereminantal
identientities [FZ99][Theorem 1.16,1.17], which also stand at the very base of
all the known cluster algebra structures on the ring of varieties related to
$G$, such as double Bruhat cells [BFZ05] and $T$-stable subgroups of the
unipotent radical of $B$ [GLS11], [GY21]. From this perspective, it’s no
surprise that cluster algebras have been defined by Fomin and Zelevinsky
[FZ02] to study the combinatorial properties of the dual canonical basis.

A more evident link between cluster algebras and branching problems has been
recently realised by Gross, Hacking, Keel, Kontsevich [GHKK18][Corollary 0.20,
0.21]: they constructed (many) polyhedral models for the weight space
decomposition and for the tensor product decomposition of
$\operatorname{SL}_{n}$, using cluster algebras, and recovered some well known
models. Fei obtained the same results [Fei21][Theorem 8.1, 9.2] for any
simple, simply laced and simply connected algebraic group.

#### Why cluster algebras?

Starting from a combinatorial data called seed, usually denoted by $t$, and an
iterative procedure called mutation, one can define the cluster algebra
$\mathcal{A}(t)$ and the upper cluster algebra $\overline{\mathcal{A}}(t)$. In
general one has $\mathcal{A}(t)\subseteq\overline{\mathcal{A}}(t)$. The
cluster algebra is more combinatorial in nature, while the upper cluster
algebra is more geometric. In particular, though $\overline{\mathcal{A}}(t)$
may not be noetherian [GHK15], [Spe13], it may be interpreted as the ring of
global sections of a locally of finite type smooth scheme. Thanks to a huge
effort due to many authors, there’s nowadays a quite well developed structural
theory of cluster algebras.

To explain some features of this theory, we recall that part of the defining
data of a seed $t$ is the vertex set $I$, which has a partition
$I=I_{uf}\sqcup I_{f}$ into un-frozen (or mutable) and frozen vertices. Each
seed also posses a distinguished set of variables $x=(x_{i})_{i\in I}$,
indexed by the vertex set, which are elements of the cluster algebra
$\mathcal{A}(t).$ Defining the degree of the cluster variables, under some
constraints arising from the mutation process, defines a global graduation on
both $\mathcal{A}(t)$ and $\overline{\mathcal{A}}(t)$ to which we refer as a
cluster graduation. There’s a well developed theory for constructing special
bases, called good bases, of $\overline{\mathcal{A}}(t)$ [Dup11], [QIN17],
[GHKK18], [Qin22]. If we allow some frozen variables not to be invertible in
$\overline{\mathcal{A}}(t)$, some results are still conjectural. Nevertheless,
under some technical assumption on the seed $t$ good bases exist and have many
parametrisations by rational points into polyhedral cones.

We say that a scheme ${\mathfrak{X}}$ has cluster structure if
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})=\overline{\mathcal{A}}(t)$ for a
certain seed $t$. Our strategy to study branching problems through cluster
algebras is the following.

#### The strategy

Step 1: Define a $T\times{\widehat{T}}$-scheme
${\mathfrak{X}}={\mathfrak{X}}(G,{\widehat{G}})$ such that for any
$(\lambda,\widehat{\lambda})\in X(T)\times X({\widehat{T}})$

${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})_{\lambda,\widehat{\lambda}}=\begin{cases}{\operatorname{Hom}}(V(\widehat{\lambda}),V(\lambda))^{G}&\text{if}\quad(\lambda,\widehat{\lambda})\in
X(T)^{+}\times X({\widehat{T}})^{+}\\\ 0&\text{otherwise}\end{cases}$

Here
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})_{\lambda,\widehat{\lambda}}$
denotes the homogeneous component, of weight $(\lambda,\widehat{\lambda})$,
for the $T\times{\widehat{T}}$ action.

Step 2: Identify a cluster structure on
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$.

Step 3: Interpret the $X(T)\times X({\widehat{T}})$ graduation on
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ as a cluster graduation.

Step 4: Prove the existence of good bases, for the upper cluster algebra
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$, and derive polyhedral models
for multiplicities.

Step 1 is rather easy and exploits the algebraic Peter-Weyl theorem. To
achieve Step 2 and Step 3, we want to use a new construction called minimal
monomial lifting and the results of [GLS11], [GY21]. For step 4, we want to
use the results of [Qin22] and some ideas of [GHKK18].

In the present paper, we want to describe the minimal monomial lifting
technique, perform step 1 and explain how one could perform step 2 for a given
pair $(G,{\widehat{G}})$. Moreover, we perform step 2 and half of step 3 for
the three families of branching problems described above.

For some of these families, the second half of step 3 and step 4 are achieved.
This is part of an ongoing work.

#### Outline of the paper and main results

Section 2 contains the preliminaries on cluster algebras. We formalise two
notions already existing in the literature: the difference between highly-
frozen and semi-frozen vertices and define the cluster valuation associated to
a frozen vertex. The cluster variable associated to a highly-frozen vertex is
not invertible, while the one associated to a semi-frozen one is. Then, we
recall the notion of graded seed and degree configuration, which is the
combinatorial datum defining a cluster graduation.

In Section 3, we introduce the fundamental notion of lifting configuration.
Let $t$ be a seed, with vertex set $I$, and $D$ be a finite set. By little
abuse, a lifting configuration $\nu$, on $t$, is a matrix
$\nu\in\mathbb{Z}^{D\times I}$. Given a lifting configuration $\nu$, on $t$,
we introduce a seed called monomial lifting of $t$ associated to $\nu$, which
is denoted by $\upharpoonleft\kern-3.50006ptt$. The set $D$ is a subset of the
semi-frozen vertex set of $\upharpoonleft\kern-3.50006ptt$. For every vertex
$d\in D$, the associated cluster variable is denoted by
$\upharpoonleft\kern-3.50006ptx_{d}$. We introduce the notion of mutation of
lifting configurations and we prove the following properties.

######  Theorem 1.0.1.

Let $\nu\in\mathbb{Z}^{D\times I}$ be a lifting configuration on $t$.

  1. 1.

Lifting a seed commutes with mutation.

  2. 2.

For every seed $t^{\prime}$ mutation equivalent to $t$, through the mutation
process, we can associate a well defined lifting configuration $\nu^{\prime}$
on $t^{\prime}.$

  3. 3.

$\mathcal{A}(\upharpoonleft\kern-3.50006ptt)=\mathcal{A}(t)[\upharpoonleft\kern-3.50006ptx_{d}^{\pm
1}]_{d}$.

  4. 4.

If $t$ is of maximal rank,
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)=\overline{\mathcal{A}}(t)[\upharpoonleft\kern-3.50006ptx_{d}^{\pm
1}]_{d}$.

Moreover, if the seed $t$ is graded by a degree configuration $\sigma$, we
produce a degree configuration $\upharpoonleft\kern-3.50006pt\sigma$, on
$\upharpoonleft\kern-3.50006ptt$, called the lifting of the degree
configuration $\sigma$. Since every seed $t$ is trivially graded, the cluster
algebra $\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)$ is
canonically $\mathbb{Z}^{D}$-graded by the lifting of the trivial degree
configuration on $t$, which is denoted by $\upharpoonleft\kern-3.50006pt0$. By
Theorem 1.0.1, if $Y$ is a scheme with cluster structure, under mild
assumptions on $Y$, any lifting configuration on any seed of the cluster
structure of $Y$ allows to produce a cluster structure of
$(\mathbb{C}^{*})^{D}\times Y$. Section 3.1 is devoted to understand
geometrically the canonical $\mathbb{Z}^{D}$-graduation on
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)$ and the kind of
graduations we could hope to understand, using the minimal monomial lifting
technique.

About the minimal monomial lifting. In Section 4 we give the definition of
schemes ${\mathfrak{X}}$ suitable for lifting. Such a scheme contains an open
subset of the form $T\times Y$ and functions on this open subset can be
"homogenised" to global regular functions on ${\mathfrak{X}}$. If
${\mathcal{O}}_{Y}(Y)=\overline{\mathcal{A}}(t)$, then, we can identify
geometrically a special lifting configuration $\nu$, on $t$, called the
minimal lifting configuration. The seed $\upharpoonleft\kern-3.50006ptt^{D}$,
obtained from $\upharpoonleft\kern-3.50006ptt$ by changing the nature of the
vertices of $D$ from semi-frozen to highly-frozen, is called minimal monomial
lifting of $t$, with respect to ${\mathfrak{X}}$. The seed
$\upharpoonleft\kern-3.50006ptt^{D}$ is the best possible candidate, to give
${\mathfrak{X}}$ a cluster structure compatible with the one on $Y$, in the
sense of Theorem 4.0.10.

######  Theorem 1.0.2.

We have that

$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})\subseteq{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})\quad\text{and}\quad\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})_{\prod_{d}\upharpoonleft\kern-2.45004pt\kern
1.39998ptx_{d}}=\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)={\mathcal{O}}_{\mathfrak{X}}(T\times
Y).$

Moreover, if a seed $\widetilde{t}$ is compatible with $t$ in the sense of
Theorem 4.0.10, and
$\overline{\mathcal{A}}(\widetilde{t})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$,
then $\widetilde{t}=\upharpoonleft\kern-3.50006ptt^{D}$.

In the last part of Section 4, we develop some elementary geometric criteria
to study whether $\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$
equals ${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$, since the inclusion
between these two algebras may be strict. Ultimately, the difference between
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ and
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$, depends on the behaviour of
$\upharpoonleft\kern-3.50006ptt^{D}$ along the divisors of ${\mathfrak{X}}$ in
the complement of $T\times Y$. We deduce some consequences of such an
equality, that suggest that it is reasonable to construct a cluster structure
on a suitable for lifting scheme ${\mathfrak{X}}$, through the minimal
monomial lifting, if the $T$-action on $T\times Y$ extends to
${\mathfrak{X}}.$ This is a necessary condition, for the equality between
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ and
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ to hold, if ${\mathfrak{X}}$ is
affine. Moreover, under the existence of such an action, the algebra
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$, which is
canonically graded via the lifting of the trivial degree configuration
$\upharpoonleft\kern-3.50006pt0$, is a graded subalgebra of
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$, which is $X(T)$-graded by the
$T$-action.

Section 5 is devoted to some preliminaries on algebraic groups. We study in
detail some technical properties of generalised minors, that are crucial for
the proof of Theorem 1.0.4. The reader can skip this part at first. In Section
6 we consider an example of minimal monomial lifting where the inclusion
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})\subseteq{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$
is strict. We report this toy example since we consider it instructive.

In Section 7, we define the branching scheme
${\mathfrak{X}}(G,{\widehat{G}})$, for the pair ${\widehat{G}}\subseteq G$,
where $G$ is semisimple and ${\widehat{G}}$ is reductive. The scheme
${\mathfrak{X}}(G,{\widehat{G}})$ realizes step 1 of our strategy and we prove
that it is suitable for lifting. As a consequence of the results of Section
3.1, we derive the following geometric realization of the multiplicities
spaces of the branching from $G$ to ${\widehat{G}}$.

######  Proposition 1.0.3.

For any $(\lambda,\widehat{\lambda})\in X(T)^{+}\times X({\widehat{T}})^{+}$,
we have an isomorphism

${\operatorname{Hom}}(V(\lambda),V(\widehat{\lambda}))^{\widehat{G}}{\,\longrightarrow\,}\mathbb{C}[U]^{\widehat{U}}_{\lambda,\widehat{\lambda}-\rho(\lambda)}$

where

$\begin{array}[]{r l l
}\mathbb{C}[U]^{\widehat{U}}_{\lambda,\widehat{\lambda}-\rho(\lambda)}=\\{f\in\mathbb{C}[U]^{\widehat{U}}\,:&f(\widehat{h}^{{}^{-1}}u\widehat{h})=(\widehat{\lambda}-\lambda)(\widehat{h})f(u)&\text{for}\quad\widehat{h}\in{\widehat{T}},u\in
U\ ,\,\text{and}\\\ &\mathcal{V}_{\alpha}(1\otimes
f)\geq-\langle\lambda,\alpha^{\vee}\rangle&\text{for}\quad\alpha\in
D\\}.\end{array}$

Actually, the previous proposition can be interpreted as a generalisation of
[Zel99][Proposition 1.4], which is the starting point for the interest in
perfect bases. Perfect bases, which include the classical limit of the dual
canonical basis, have been arguably one of the most used items to study tensor
product decomposition. See [Kam22] for a beautiful survey on perfect bases.
Finally, we prove the main result of the paper.

######  Theorem 1.0.4.

In the following two cases:

  1. 1.

$G$ is simple, simply connected and ${\widehat{G}}$ is a Levi subgroup of $G$.

  2. 2.

${\widehat{G}}$ is semisimple, simply connected and diagonally embedded in
$G={\widehat{G}}\times{\widehat{G}}$.

Then, if ${\mathfrak{X}}={\mathfrak{X}}(G,{\widehat{G}})$, there exists a seed
$t$ constructed in [GY21], and an isomorphism $\mathbb{Z}^{D}\simeq X(T)$,
such that
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$
as graded algebras.

The construction of cluster structures for branching to a Levi subgroup is, to
my best knowledge, new. For the tensor product decomposition it is mostly new,
except for the following few exceptions. In [GHKK18], the authors construct a
cluster structure on a variation of the branching scheme for tensor product
decomposition of $\operatorname{SL}_{n}(\mathbb{C})$. In [Fei21], the author
constructs a cluster structure on the branching scheme for tensor product
decomposition assuming that the group is simple, simply laced and simply
connected. In Section 8.2 we prove that Fei’s structures are obtained through
minimal monomial lifting. It’s probable that Fei’s structures agree with the
ones of the present text, but I’m not capable of doing this comparison in
detail. Finally, Example 8.2.14 and Theorem 1.0.4 partially answer to
[Fei21][Conjecture 8.2] in the $G_{2}$-case.

In Theorem 1.0.4, $\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$
is $\mathbb{Z}^{D}$-graded via the lifting of the trivial degree configuration
on $t$, while ${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ is $X(T)$-graded
by restriction of the $X(T)\times X({\widehat{T}})$-graduation. That is, if
$\lambda\in X(T)$,
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})_{\lambda,\bullet}=\bigoplus_{\widehat{\lambda}\in
X({\widehat{T}})}{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})_{\lambda,\widehat{\lambda}}.$
Roughly speaking, the canonical cluster graduation on
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ detects, for a
single irreducible representation of $G$, into how many
${\widehat{G}}$-irreducible representations it decomposes. Still, it doesn’t
distinguish different ${\widehat{G}}$-modules. Incorporating this information
into a cluster graduation becomes a purely combinatorial problem, that could
be achieved lifting a certain conjectural degree configuration on the seeds
constructed in [GY21]. See Question 5.3.4. As a final remark, we are also
interested in Questions 7.4.1 .

Acknowledgements. I would like to thank Bernard Leclerc for the invitation to
Caen and the fruitful discussions and insights on cluster algebras, and
Nicolas Ressayre, for introducing me to the branching problem, for the support
and the many discussions that made this work possible.

##  2 Preliminaries on cluster algebras

###  2.1 Cluster algebras

We give a definition of cluster algebras and upper cluster algebras of
geometric type in the spirit of [GLS13]. The only differences with the
standard setting are encoded in the notions of semi-frozen (invertible) and
highly-frozen (not invertible) variables. The type of frozen variables affects
the definition of the coefficient ring of the cluster algebra. This choice
provides a natural framework for studying functions over some partial
compactifications of cluster varieties. Note that our cluster algebras are
very special cases of the ones defined in [BMS19] and of generalized cluster
algebras [GSV18].

######  Definition 2.1.1 (seed).

Let $\mathbb{K}$ be a field extension of $\mathbb{C}$. A seed $t$ of
$\mathbb{K}$ is a collection

$(I_{uf},I_{sf},I_{hf},B,x)$

consisting of:

  * –

Three disjoint finite sets $I_{uf},I_{sf},I_{hf}$. An element of
$I:=I_{uf}\sqcup I_{sf}\sqcup I_{hf}$ (resp. $I_{f}:=I_{hf}\sqcup I_{sf}$) is
called vertex (resp. frozen vertex). A vertex is respectively called unfrozen
(or mutable), semi-frozen, highly frozen if it belongs to
$I_{uf},I_{sf},I_{hf}.$

  * –

The extended exchange matrix $B\in\mathbb{Z}^{I\times I_{uf}}$, which is an
integer matrix, such that it’s principal part $B^{\circ}:=B_{|I_{uf}\times
I_{uf}}$ is skew-symmetrizable, meaning that there exists
$d_{i}\in\mathbb{N}_{>0}$, for $i\in I_{uf}$, such that for any $i,j\in
I_{uf}$, $d_{i}b_{i,j}=-d_{j}b_{j,i}.$

  * –

An extended cluster $x\in(\mathbb{K}^{*})^{I}$, consisting of a transcendence
basis of $\mathbb{K}$ over $\mathbb{C}$, whose elements are called cluster
variables. A variable is said to be unfrozen (or mutable) (resp. semi-frozen,
resp. highly-frozen, resp. frozen) if its corresponding vertex is unfrozen
(resp. semi-frozen, resp. highly frozen, resp. frozen).

If a seed is denoted by $t$, we implicitly assume that its defining data are
denoted as in the previous definition. If the seed is called $t^{\bullet}$
(resp. $t_{\bullet}$), where $\bullet$ is any superscript (resp. subscript),
then we add a $\bullet$ superscript (resp. subscript) to all the notation. For
example, we write
$t^{\prime}=(I^{\prime}_{uf},I^{\prime}_{sf},I^{\prime}_{hf},B^{\prime},x^{\prime})$
or $t_{\mathbf{i}}=(I_{\mathbf{i},uf},\dots,x_{\mathbf{i}})$. If needed, we
add the dependence on $t$ writing $B(t)$, $x(t)$ and so on.

Graphical notation for seeds: given a seed $t$, the only data required to
identify the isomorphism class of the associated cluster and upper cluster
algebra is the vertex set and the extended exchange matrix. This is encoded
graphically in a valued quiver $Q$ (or $Q(t)$ is the dependence on $t$ is
needed) as follows.

  * -

The vertex set of $Q$ is $I$. For $i\in I$, the corresponding vertex of $Q$ is
pictured by a symbol $\bigcirc$ (resp. $\square$, resp. $\blacksquare$) if it
is unfrozen (resp. semi-frozen, resp. highly frozen) and labelled by $i$.

Given two vertices $i$ and $j$ of $Q$, we use the convention that $b_{i,j}=0$
if $i,j$ are both frozen and that $b_{i,j}=-b_{j,i}$ if exactly one between
$i$ and $j$ is frozen. Note that, in the last case, exactly one between
$b_{i,j}$ and $b_{j,i}$ is defined as a coefficient of the generalised
exchange matrix, accordingly to which vertex is unfrozen. Then

  * -

There is an arrow between $i$ and $j$, pointing towards $j$, if and only if
$b_{i,j}>0$. In this case, it is labelled by: "$b_{i,j},-b_{j,i}$". Moreover,
if $b_{i,j}=-b_{j,i}$, for low values of $b_{i,j}$, we may write $b_{i,j}$
distinct arrows from $i$ to $j$ instead of a labelled arrow.

######  Example 2.1.2.

Let $t=(\\{1,2\\},\\{3\\},\\{4\\},B,x)$ be a seed whose generalised exchange
matrix is

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

where the rows and columns of $B$ are labelled in the obvious way. The
following two pictures are both a graphical representation of the seed $t$

${{\square 3}}$${{\square 3}}$${{\bigcirc 1}}$${{\bigcirc 2}}$${{\bigcirc
1}}$${{\bigcirc 2}}$${{\blacksquare 4}}$${{\blacksquare
4}}$$\scriptstyle{2,2}$$\scriptstyle{3,1}$$\scriptstyle{3,1}$$\scriptstyle{1,1}$

General notation: If $b\in\mathbb{R}$ then

$b^{+}:=\max\\{b,0\\}\quad b^{-}:=\max\\{-b,0\\}\quad\text{so that}\quad
b=b^{+}-b^{-}.$

If $J$, $K$ are finite sets and $M\in\mathbb{R}^{J\times K}$, then
$M^{\pm}\in\mathbb{R}_{\geq 0}^{J\times K}$ is the matrix obtained applying
component-wise to $M$ the corresponding operation. If $k\in K$, the $k$-th
column of $M$ is $M_{\bullet,k}=M_{|J\times\\{k\\}}\in\mathbb{R}^{J}.$
Similarly, if $M\in\mathbb{R}^{J\times K}$ and $J_{1}\subseteq J$,
$K_{1}\subseteq K$, then $M_{J_{1}\times K_{1}}=M_{|J_{1}\times K_{1}}$ and
$M_{\bullet,K_{1}}=M_{J\times K_{1}}.$
If $S$ is any set and we consider $S$-valued matrices of size $J\times K$,
that is elements of $S^{J\times K}$, then we use analogue notations for
restrictions.
Let $H$ be an abelian group and $h\in H^{J}$. If $m\in\mathbb{Z}^{J}$ and
$M\in\mathbb{Z}^{J\times K}$, then

$h^{m}:=\sum m_{j}h_{j}\quad\text{while}\quad h^{M}=(h^{M_{\bullet},k})_{k\in
K}\in(H)^{K}.$

If $H=\mathbb{K}^{*}$, then $h^{m}$ is a monomial in the $h_{j}$. Note that,
if $I$ is a third finite set and $N\in\mathbb{Z}^{K\times I}$ then

$h^{MN}=(h^{M})^{N}$

where the product $\mathbb{Z}^{J\times K}\times\mathbb{Z}^{K\times
I}{\,\longrightarrow\,}\mathbb{Z}^{J\times I}$ corresponds to composition of
morphisms in the canonical identification between $\mathbb{Z}^{J\times K}$ and
${\operatorname{Hom}}_{\mathbb{Z}}(\mathbb{Z}^{K},\mathbb{Z}^{J}).$ Finally,
the elements of the canonical basis of $\mathbb{Z}^{K}$ are denoted by
$e_{k}$, for $k\in K$.

Given a seed $t$ and a mutable vertex $k\in I_{uf}$, we have an operation
called seed mutation at k, denoted by $\mu_{k}$, that produces a new seed
$\mu_{k}(t)=t^{\prime}$, of $\mathbb{K}$, defined as follows.

  * –

The vertex sets are the same of $t$, that is
$I_{uf}^{\prime}=I_{uf},I_{sf}^{\prime}=I_{sf}$ and $I_{hf}^{\prime}=I_{hf}$.

  * –

The extended exchange matrix $B^{\prime}$ satisfies:

$b^{\prime}_{i,j}=\begin{aligned} \begin{cases}-b_{i,j}&\text{if}\quad
i=k\quad\text{or}\quad j=k\\\
b_{i,j}+b_{i,k}^{+}b_{k,j}^{+}-b_{i,k}^{-}b_{k,j}^{-}&\text{otherwise}.\end{cases}\end{aligned}$
(1)

  * –

The extended cluster $x^{\prime}$ is defined by

$x_{i}^{\prime}=\begin{aligned} \begin{cases}x_{i}&\text{if}\quad i\neq k\\\
x^{B_{\bullet,k}^{+}-e_{k}}+x^{B_{\bullet,k}^{-}-e_{k}}&\text{if}\quad
i=k.\end{cases}\end{aligned}$ (2)

The identity

$x_{k}x_{k}^{\prime}=x^{B_{\bullet,k}^{+}}+x^{B_{\bullet,k}^{-}}$ (3)

is called exchange relation. We will often refer to the two monomials on the
right hand side of the exchange relation as $M^{+}_{k}$ and $M_{k}^{-}$, in
the obvious way.
Since the mutation is an involution, that is $\mu_{k}\circ\mu_{k}(t)=t$, we
can can define an equivalence relation $\sim$ on the set of seeds of
$\mathbb{K}$ with the same set of vertices: we say that $t\sim t^{\prime}$ if
there exists a sequence of mutations that transforms $t$ in $t^{\prime}$. That
is, if there exists $i_{1},\dots,i_{l}\in I_{uf}$ such that

$\mu_{i_{l}}\circ\dots\circ\mu_{i_{1}}(t)=t^{\prime}.$

We call $\Delta$ the set of seeds of $\mathbb{K}$ equivalent to $t$.

From now on, we fix an equivalence class of seeds $\Delta$ of $\mathbb{K}$.
Since any seed in $\Delta$ has the same set of frozen variables, say $x_{i}$
for $i\in I_{f}$, we can define the coefficient ring

${\operatorname{Cf}}[\Delta]:=\mathbb{C}[x_{i}]_{i\in I_{hf}}[x_{i}^{\pm
1}]_{i\in I_{sf}}.$ (4)

######  Definition 2.1.3 (cluster algebra).

The cluster algebra $\mathcal{A}(\Delta)$ is the
${\operatorname{Cf}}[\Delta]$-algebra generated by all the cluster variables
of all the seeds in $\Delta$.

If $t\in\Delta$, we define the ring of Laurent Polynomials

${\mathcal{L}}(t):={\operatorname{Cf}}[\Delta][x_{i}^{\pm 1}]_{i\in I_{uf}}$
(5)

Note that the unfrozen and semi-frozen variable of $t$ are invertible in
${\mathcal{L}}(t)$, while the highly frozen are not.

######  Definition 2.1.4 (upper cluster algebra).

The upper cluster algebra $\overline{\mathcal{A}}(\Delta)$ is

$\overline{\mathcal{A}}(\Delta):=\bigcap_{t\in\Delta}{\mathcal{L}}(t).$

Both $\mathcal{A}(\Delta)$ and $\overline{\mathcal{A}}(\Delta)$ are domains,
moreover $\overline{\mathcal{A}}(\Delta)$ is normal since it is defined as the
intersection of normal rings. Still, both the cluster algebra and the upper
cluster algebra are not noetherian in general. Note that, the choice of the
ambient field $\mathbb{K}$ (as the choice of the variables $x$) is immaterial
for the isomorphism class of the cluster algebra and the upper cluster
algebra. It becomes important when we try to identify cluster algebra
structures on a given ring. So, when we only deal with (upper) cluster
algebras, we often omit to specify the ambient field $\mathbb{K}$. The
following theorem can be found in [GHKK18][Corollary 0.4] or in [LS15] for
skew-symmetric seeds.

######  Theorem 2.1.5 (Positivity of the Laurent phenomenon).

For any $t,t^{\prime}\in\Delta$ and $i\in I$,

$x_{i}^{\prime}\in\mathbb{N}[x_{j}]_{j\in I_{f}}[x_{k}^{\pm 1}]_{k\in I_{uf}}$

In particular $\mathcal{A}(\Delta)\subseteq\overline{\mathcal{A}}(\Delta).$

The weaker statement that $x_{i}^{\prime}\in\mathbb{Z}[x_{j}]_{j\in
I_{f}}[x_{k}^{\pm 1}]_{k\in I_{uf}}$, usually called Laurent phenomenon, is
much more elementary and also sufficient for most applications. It can be
found in[FZ03, Proposition 11.2]. (See also [FZ02, Theorem 3.1]).

Clearly, if $t\in\Delta$, we also use the notation $\mathcal{A}(t)$ (resp.
$\overline{\mathcal{A}}(t)$) to denote $\mathcal{A}(\Delta)$ (resp.
$\overline{\mathcal{A}}(\Delta)$).

######  Definition 2.1.6.

Let $t\in\Delta$. For $i\in I_{uf}$, let $t_{i}:=\mu_{i}(t)$ and $t_{0}:=t$.
The upper bound $\mathcal{U}(t)$ at $t$ is

$\mathcal{U}(t):=\bigcap_{i\in I_{uf}\cup\\{0\\}}{\mathcal{L}}(t_{i}).$

If the matrix $B$ is of maximal rank, we say that $t$ is of maximal rank. Note
that the rank of the generalised exchange matrix is invariant under mutation
by [BFZ05, Lemma 3.2]. The following theorem is a very special case of [GSV18,
Theorem 3.11]. The case with no highly frozen vertices had already been proved
in [BFZ05, Corollary 1.7].

######  Theorem 2.1.7.

If $t$ is of maximal rank, $\overline{\mathcal{A}}(\Delta)=\mathcal{U}(t)$.

We say that no mutable vertex of $t$ is completely disconnected if, for any
$k\in I_{uf}$, there exists $i\in I$ such that $b_{i,k}\neq 0.$ It’s easy to
see that this property is invariant under mutation, hence if $\Delta$ is an
equivalence class of seeds, we say that no mutable vertex of $\Delta$ is
completely disconnected if the same property holds for one, hence for any, of
its seeds. Note that completely disconnected mutable vertices are the ones
that produce trivial exchange relations.

######  Theorem 2.1.8 ([GLS13]).

Suppose that no mutable vertex of $\Delta$ is completely disconnected. Let
$t,t^{\prime}\in\Delta$ and $i,j\in I_{uf}\sqcup I_{hf}$, then

  1. 1.

The variable $x_{i}$ is irreducible in $\overline{\mathcal{A}}(\Delta)$.

  2. 2.

The ideals $(x_{i})$ and $(x_{j}^{\prime})$ are equal if and only if
$x_{i}=x_{j}^{\prime}$.

  3. 3.

The invertible elements of $\overline{\mathcal{A}}(\Delta)$, up to scalar, are
the monomials in the semi-frozen variables.

The above theorem is proved in [GLS13, Theorem 1.3, Corollary 2.3] for the
ordinary cluster algebra under the assumption that the seed $t$ is connected
(see [GLS13, Section 1.2]). The exact same proof also works for the upper
cluster algebra and the upper bound under the weaker assumption that no
mutable vertex is completely disconnected.

###  2.2 Disjoint union of seeds

This section is only needed for an application in Section 6.

We introduce a notation. If $M\in\mathbb{R}^{(J_{1}\sqcup
J_{2})\times(K_{1}\times K_{2})}$, then

$M=\begin{pmatrix}M_{1,1}&M_{1,2}\\\ M_{2,1}&M_{2,2}\end{pmatrix}$ (6)

means that $M_{J_{a}\times K_{b}}=M_{a,b}$, for $a,b\in\\{1,2\\}$.

######  Definition 2.2.1.

Let $t$ and $t^{\prime}$ be two seeds. The disjoint union of $t$ and
$t^{\prime}$ is the seed $t|t^{\prime}=(J_{uf},J_{sf},J_{hf},C,z)$ of
$\mathbb{C}(x_{i},x^{\prime}_{i^{\prime}})_{i\in I,i^{\prime}\in I^{\prime}}$
defined by

  * -

The set of vertices

$J_{uf}=I_{uf}\sqcup I^{\prime}_{uf}\quad J_{sf}=I_{sf}\sqcup
I^{\prime}_{sf}\quad J_{hf}=I_{hf}\sqcup I^{\prime}_{hf}.$

  * -

The generalised exchange matrix

$C=\begin{pmatrix}B&0\\\ 0&B^{\prime}\end{pmatrix}.$

  * -

The cluster $z$, which is defined by

$z_{I}=x\quad z_{I^{\prime}}=x^{\prime}.$

We identify $\mathbb{C}(x_{i},x^{\prime}_{i^{\prime}})_{i\in I,i^{\prime}\in
I^{\prime}}$ with the fraction field of
${\mathcal{L}}(t)\otimes{\mathcal{L}}(t^{\prime})$ in the natural way. It’s
clear that
${\mathcal{L}}(t|t^{\prime})={\mathcal{L}}(t)\otimes{\mathcal{L}}(t^{\prime})$.
The following property is immediate to verify:

######  Lemma 2.2.2.

Let $t,t^{\prime}$ as above, $i\in I_{uf}$, $i^{\prime}\in I^{\prime}_{uf}$,
then

$\mu_{i}(t|t^{\prime})=\mu_{i}(t)|t^{\prime}\quad\text{and}\quad\mu_{i^{\prime}}(t|t^{\prime})=t|\mu_{i}(t^{\prime}).$

######  Lemma 2.2.3.

If $t$ and $t^{\prime}$ are of maximal rank, then
$\overline{\mathcal{A}}(t|t^{\prime})=\overline{\mathcal{A}}(t)\otimes\overline{\mathcal{A}}(t^{\prime}).$

###### Proof.

Clearly $t|t^{\prime}$ is of maximal rank. The lemma follows easily from
repeated application of Theorem 2.1.7, the previous lemma and the fact that
the tensor product commutes with finite intersections. ∎

###  2.3 Cluster valuations induced by frozen variables

We introduce the notion of cluster valuation at a frozen vertex. This is
something which implicitly appears in the existing literature, but for which
we didn’t find an appropriate reference, or at least not at the level of
detail we need.

If $t\in\Delta$, we denote by $\mathbb{C}[t]$ (resp. $\mathbb{C}(t)$) the
polynomial ring (resp. the field of fractions) in the cluster variables
$x_{i}$. By the Laurent phenomenon (Theorem 2.1.5) we have that
$\mathbb{C}[t]\subseteq\overline{\mathcal{A}}(\Delta)$ and the fraction field
of $\overline{\mathcal{A}}(\Delta)$ is $\mathbb{C}(t)$, which from now on will
be denoted by $\mathbb{C}(\Delta)$.

Any cluster variable $x_{i}$ generates a prime ideal in $\mathbb{C}[t]$, so we
can consider the induced discrete valuation

$\mathcal{V}_{i}^{t}:\mathbb{C}(\Delta){\,\longrightarrow\,}\mathbb{Z}\cup\\{\infty\\}.$

For an element $p\in\mathbb{C}[t]\subseteq\mathbb{C}(\Delta)$, the exponent of
$x_{i}$ in the factorisation of $p$ into irreducible factors, in the ring
$\mathbb{C}[t]$, is $\mathcal{V}_{i}^{t}(p).$

######  Lemma 2.3.1.

If $i\in I_{f}$ and $t\sim t^{\prime}$, then
$\mathcal{V}^{t}_{i}=\mathcal{V}^{t^{\prime}}_{i}$. In particular, we refer to
any such valuation as the cluster valuation at the frozen vertex $i$ and
denote it by $\mathcal{C}\mathcal{V}_{i}$.

###### Proof.

It’s sufficient to prove the case of $t^{\prime}=\mu_{k}(t)$, where $k\in
I_{uf}.$ Let

$\mathbb{C}[I\setminus\\{k\\}]:=\mathbb{C}[x_{j}]_{j\in
I\setminus\\{k\\}}=\mathbb{C}[x_{j}^{\prime}]_{j\in I\setminus\\{k\\}}$

and $\mathbb{C}(I\setminus\\{k\\})$ the fraction field of
$\mathbb{C}[I\setminus\\{k\\}]$. Since the fraction field of
$\mathbb{C}(I\setminus\\{k\\})[x_{k}^{\pm 1}]$ is equal to
$\mathbb{C}(\Delta)$, it’s also sufficient to prove that
$\mathcal{V}_{i}^{t}(f)=\mathcal{V}_{i}^{t^{\prime}}(f)$ for any
$f\in\mathbb{C}(I\setminus\\{k\\})[x_{k}^{\pm 1}].$
It’s easy to see that, if
$\mathcal{V}^{I\setminus\\{k\\}}_{i}:\mathbb{C}(I\setminus\\{k\\}){\,\longrightarrow\,}\mathbb{Z}\cup\\{\infty\\}$
is the valuation induced by the prime ideal $(x_{i})$ of
$\mathbb{C}[I\setminus\\{k\\}]$, and if

$f=\sum_{n=-N}^{N}f_{n}x_{k}^{n}$ (7)

with $f_{n}\in\mathbb{C}(I\setminus\\{k\\})$, then

$\mathcal{V}^{t}_{i}(f)=\min\\{\mathcal{V}_{i}^{I\setminus\\{k\\}}(f_{n})\,:\,-N\leq
n\leq N\\}.$ (8)

To prove (8), it’s sufficient to notice that if $f_{n}\neq 0$, then

$\mathcal{V}_{i}^{I\setminus\\{k\\}}(f_{n})=v_{n}{\,\Longleftrightarrow\,}f_{n}=x_{i}^{v_{n}}\frac{p_{n}}{q_{n}}$
(9)

with $p_{n},q_{n}\in\mathbb{C}[I\setminus\\{k\\}]$ not divisible by $x_{i}$.
Then, we can compute $\mathcal{V}_{i}^{t}(f)$ plugging the right hand side of
(9) into (7) and we easily obtain(8).

Using the exchange relation (3), we compute that

$f=\sum_{n=-N}^{N}f_{n}(M_{k}^{+}+M_{k}^{-})^{n}(x_{k}^{\prime})^{-n}\in\mathbb{C}(I\setminus\\{k\\})[(x_{k}^{\prime})^{\pm
1}]$

But $M_{k}^{+}+M_{k}^{-}$ is not divisible by $x_{i}$, hence

$\mathcal{V}^{I\setminus\\{k\\}}(f_{n})=\mathcal{V}^{I\setminus\\{k\\}}(f_{n}(M_{k}^{+}+M_{k}^{-})^{n}).$

The statement follows using (8) and the similar formula relative to the seed
$t^{\prime}$.

∎

At the level of cluster varieties, the previous lemma says that the gluing
maps between cluster tori preserve the valuation defined by the frozen
variables. We can strengthening Theorem 2.1.8 in the case of highly frozen
variables as follows.

######  Corollary 2.3.2.

If $i\in I_{hf}$, then $x_{i}$ generates a prime ideal of
$\overline{\mathcal{A}}(\Delta)$. Moreover,
$\overline{\mathcal{A}}(\Delta)_{(x_{i})}$ is a discrete valuation ring whose
induced valuation on $\mathbb{C}(\Delta)$ equals the cluster valuation
$\mathcal{C}\mathcal{V}_{i}.$

###### Proof.

Fix $t\in\Delta$. Take $a,b\in\overline{\mathcal{A}}(\Delta)$ such that
$ab\in(x_{i}).$ Since $x_{i}$ generates a prime ideal in ${\mathcal{L}}(t)$,
we can assume that $\frac{a}{x_{i}}\in{\mathcal{L}}(t)$. Note that
${\mathcal{L}}(t)$ is a localisation of $\mathbb{C}[t]$, hence the restriction
of the valuation $\mathcal{V}^{t}_{i}$ on ${\mathcal{L}}(t)$ agrees with the
valuation defined by the prime ideal of ${\mathcal{L}}(t)$ generated by
$x_{i}$. In particular, $\mathcal{V}^{t}_{i}(\frac{a}{x_{i}})\geq 0$. By the
definition of $\overline{\mathcal{A}}(\Delta)$, it’s clear that for any
$t^{\prime}\in\Delta$,
$\frac{a}{x_{i}}\in\frac{1}{x_{i}}{\mathcal{L}}(t^{\prime})\subseteq{\mathcal{L}}(t^{\prime})_{x_{i}}$.
Lemma 2.3.1 implies that $\mathcal{V}^{t^{\prime}}_{i}(\frac{a}{x_{i}})\geq
0$, hence $\frac{a}{x_{i}}\in{\mathcal{L}}(t^{\prime})$. It follows that
$\frac{a}{x_{i}}\in\overline{\mathcal{A}}(\Delta)$ and the first part of the
statement follows. For the second statement, notice that we have inclusions

$\mathbb{C}[t]\subseteq\overline{\mathcal{A}}(\Delta)\subseteq{\mathcal{L}}(t).$

In particular, by the definition of ${\mathcal{L}}(t)$, localizing the above
inclusions at $\prod_{j\in I_{uf}\sqcup I_{sf}}x_{j}$ gives a sequence of
isomorphisms. Thus
$\mathbb{C}[t]_{(x_{i})}=\overline{\mathcal{A}}(\Delta)_{(x_{i})}={\mathcal{L}}(t)_{(x_{i})}$.
∎

######  Remark 2.3.3.

One can easily prove that the cluster valuation is a tropical valuation on
$\overline{\mathcal{A}}(\Delta)$ in the sense of [BFZ05, Definition 7.1], in
particular, it agrees with one of the valuation constructed in [BFZ05, Lemma
7.3].

###  2.4 Highly-freezing and semi-freezing

We introduce two operations on seeds, that change the nature of some frozen
vertices from highly-frozen to semi-frozen, or the other way around. The
notation is designed to be coherent with the fact that, passing from $t$ to
$t_{F}$ corresponds to a localisation at the level of upper cluster algebras,
while passing from $t$ to $t^{F}$ is the inverse operation, as described in
Lemma 2.4.2.

######  Definition 2.4.1.

Let $t$ be a seed.

  * -

If $F\subseteq I_{hf}$, we define the seed
$t_{F}=(I_{F,uf},I_{F,sf},I_{F,hf},B_{F},x_{F})$ by

$I_{F,uf}=I_{uf}\quad I_{F,sf}=I_{sf}\sqcup F\quad I_{F,hf}=I_{hf}\setminus
F\quad B_{F}=B\quad x_{F}=x.$

We say that $t_{F}$ is obtained from $t$ by semi-freezing the vertices in $F$.
If $\Delta=\Delta(t)$, then we denote $\Delta_{F}=\Delta(t_{F})$.

  * –

If $F\subseteq I_{sf}$, we define the seed
$t^{F}=(I_{uf}^{F},I_{sf}^{F},I_{hf}^{F},B^{F},x^{F})$ by

$I_{uf}^{F}=I_{uf}\quad I_{sf}^{F}=I_{sf}\setminus F\quad
I_{hf}^{F}=I_{hf}\sqcup F\quad B^{F}=B\quad x^{F}=x.$

We say that $t^{F}$ is obtained from $t$ by higly-freezing the vertices in
$F$. If $\Delta=\Delta(t)$, then we denote $\Delta^{F}=\Delta(t^{F})$.

######  Lemma 2.4.2.

Let $t$ be a seed, $\Delta=\Delta(t)$, $F\subseteq I_{hf}$, $G\subseteq
I_{sf}$, $k$ (resp. $j$) a mutable (resp. frozen) vertex of $t$.

  1. 1.

$(t_{F})^{F}=t$.

  2. 2.

$(t^{G})_{G}=t$.

  3. 3.

$\mathbb{C}(\Delta)=\mathbb{C}(\Delta_{F})=\mathbb{C}(\Delta^{G})$.

  4. 4.

$\mathcal{V}^{t^{G}}_{j}=\mathcal{V}^{t}_{j}=\mathcal{V}^{t_{F}}_{j}.$

  5. 5.

$\mu_{k}(t_{F})=\mu_{k}(t)_{F}$ and $\mu_{k}(t^{G})=\mu_{k}(t)^{G}$.

  6. 6.

$\Delta_{F}=\\{t^{\prime}_{F}\,:\,t^{\prime}\in\Delta\\}$ and
$\Delta^{G}=\\{(t^{\prime})^{G}\,:\,t^{\prime}\in\Delta\\}.$

  7. 7.

If $z:=\prod_{i\in F}(x_{i})$, the natural inclusion
$\overline{\mathcal{A}}(\Delta)\subseteq\overline{\mathcal{A}}(\Delta_{F})$
induces an equality

$\overline{\mathcal{A}}(\Delta)_{z}=\overline{\mathcal{A}}(\Delta_{F}).$

  8. 8.

$\overline{\mathcal{A}}(\Delta^{G})=(\bigcap_{i\in
G}\mathcal{C}\mathcal{V}_{i}{{}^{-1}}(\mathbb{Z}_{\geq
0}\cup\\{\infty\\}))\bigcap\overline{\mathcal{A}}(\Delta)$.

###### Proof.

The first five statements are obvious. Statement 6 follows from 5 and the fact
that semi-freezing or highly-freezing a seed doesn’t change the set of mutable
vertices. Then it’s clear that
${\mathcal{L}}(t^{G})\subseteq{\mathcal{L}}(t)\subseteq{\mathcal{L}}(t_{F})$.
Using 6, we get that

$\overline{\mathcal{A}}(\Delta^{G})\subseteq\overline{\mathcal{A}}(\Delta)\subseteq\overline{\mathcal{A}}(\Delta_{F}).$

Moreover, $z$ is invertible in $\overline{\mathcal{A}}(\Delta_{F})$, hence we
have an inclusion
$\overline{\mathcal{A}}(\Delta)_{z}\subseteq\overline{\mathcal{A}}(\Delta_{F})$.
If $f\in\overline{\mathcal{A}}(\Delta_{F})$, by Lemma 2.3.1 and the definition
of upper cluster algebra, we get that

$(\prod_{i\in
F}x_{i}^{-\mathcal{V}^{t}_{i}(f)})f\in\overline{\mathcal{A}}(\Delta).$

This proves 7. Statement 8 also follows immediately from Lemma 2.3.1 and the
definition of upper cluster algebra.

∎

###  2.5 Graded seeds

Cluster algebras and upper cluster algebras can be graded almost as if they
where polynomial rings. In particular, defining the degree of the cluster
variables of an initial seed and using the mutation process induces, under
some hypothesis, global graduations. This is formalised by the notion of
graded seed. We refer to [Gra15] for more details.

Let $t$ be a seed.

######  Definition 2.5.1.

Let $H$ be an abelian group and $\sigma\in H^{I}$. We say that $\sigma$ is an
$H$-degree configuration on $t$ if

$\sigma^{B^{+}}=\sigma^{B^{-}}.$ (10)

A seed with an $H$-degree configuration is called $H$-graded seed.

We drop the dependence on $H$, in the notation, if the group is clear from the
context or meaningless. If we denote a graded seed by $t$ (resp.
$t^{\bullet}$, resp. $t_{\bullet}$ for a certain symbol $\bullet$), we
implicitly assume that its degree configuration is $\sigma$ (resp.
$\sigma^{\bullet}$, resp. $\sigma_{\bullet}$) and that the grading group is
$H$.

From now on, $t$ is an $H$-graded seed. We have a notion of mutation of degree
configuration: if $k\in I_{uf}$ and $t^{\prime}=\mu_{k}(t)$, then
$\sigma^{\prime}=\mu_{k}(\sigma)$ is the $H$-degree configuration, on
$t^{\prime}$, defined by the following formula:

$\sigma^{\prime}_{i}=\begin{aligned} \begin{cases}\sigma_{i}&\text{if}\quad
i\neq k\\\ \sigma^{B_{\bullet,k}^{+}}-\sigma_{k}&\text{if}\quad
i=k.\end{cases}\end{aligned}$ (11)

We say that $\sigma^{\prime}$ is the mutation at $k$ of the degree
configuration $\sigma.$ Any sequence of mutations defines a degree
configuration on the resulting seed, which actually only depends on the seed
and not on the sequence of mutations. Hence, any $t^{\prime}\in\Delta(t)$ is
in a canonical way a graded seed.

Note that the algebra ${\mathcal{L}}(t)$ is canonically $H$-graded by setting
$\deg(x_{i})=\sigma_{i}$. The set of Laurent monomials is an homogeneous base
of ${\mathcal{L}}(t)$.

######  Proposition 2.5.2.

Let $t^{*}\in\Delta(t)$. Then $\mathcal{A}(t)$ and $\overline{\mathcal{A}}(t)$
are graded subalgebras of ${\mathcal{L}}(t^{*})$. The graduation defined on
$\mathcal{A}(t)$ and $\overline{\mathcal{A}}(t)$, by this inclusion, is
independent on $t^{*}$. In particular, $\mathcal{A}(t)$ and
$\overline{\mathcal{A}}(t)$ are canonically graded. Moreover, cluster
variables are homogeneous.

######  Remark 2.5.3.

Since any cluster variable is homogeneous, the construction above is
compatible with the operation of highly freezing or semi-freezing a set of
vertices.

Note that all the basis constructed in [Qin22] consist of homogeneous
elements. Finally, if $t$ is a not necessarily graded seed, and we identify
$B$ as an element of
${\operatorname{Hom}}_{\mathbb{Z}}(\mathbb{Z}^{I_{uf}},\mathbb{Z}^{I})$, then
we have an universal ${\operatorname{coker}}(B)$-graduation
$\overline{\sigma}$ on $t$ defined by
$\overline{\sigma}_{i}=\overline{e}_{i}$, where $\overline{e}_{i}$ is the
class of $e_{i}$ in ${\operatorname{coker}}(B)$. In particular we see that,
roughly speaking, having many frozen vertices is good for constructing fine
graduation on $\mathcal{A}(t)$ and $\overline{\mathcal{A}}(t)$.

##  3 Monomial liftings

From now on, $D$ denotes a finite set. We define the notion of $D$-lifting
configuration on a seed $t$. For any such configuration, we construct a seed
$\upharpoonleft\kern-3.50006ptt$, to which we refer as a monomial lifting of
$t$. The name is explained by the fact that, under mild assumptions, if $Y$ is
a variety carrying a cluster structure, then any monomial lifting of $t$ gives
to $(\mathbb{C}^{*})^{D}\times Y$ a cluster structure, compatible with the one
on $Y$. Conversely, any cluster structure on $(\mathbb{C}^{*})^{D}\times Y$,
which is compatible with the one on $Y$ in the sense of Proposition 3.0.9, is
defined by a monomial lifting. We start with some definitions.

From now on, $t$ is a seed of the field $\mathbb{K}$ and $\Delta=\Delta(t).$
We assume that $I\cap D=\emptyset.$

######  Definition 3.0.1.

A $D$-pointed field extension of $\mathbb{K}$ is a field extension
$\mathbb{L}/\mathbb{K}$, with a $D$-uple $(x_{d})_{d\in
D}\in(\mathbb{L}^{*})^{D}$ of $\mathbb{K}$-algebraically independent elements,
such that $\mathbb{L}=\mathbb{K}(x_{d})_{d\in D}.$

If we denote a $D$-pointed field extension of $\mathbb{K}$ by $\mathbb{L}$, we
implicitly assume that it’s defining $D$-uple of elements is denoted by
$(x_{d})_{d}.$ Whenever a seed $t$ of $\mathbb{K}$ and a $D$-pointed field
extension $\mathbb{L}$ of $\mathbb{K}$ are given, we denote by

${\widehat{x}}=(x_{j})_{j\in I\sqcup D}\in(\mathbb{L}^{*})^{I\sqcup D}.$ (12)

######  Definition 3.0.2.

A $D$-seed extension of $t$ is a seed $t^{*}$, of a field extension
$\mathbb{L}/\mathbb{K}$, such that

  * -

$I_{uf}=I^{*}_{uf}\quad I_{sf}\subseteq I^{*}_{sf}\quad I_{hf}\subseteq
I^{*}_{hf}$ and $I^{*}_{f}\setminus I_{f}=D$.

  * -

$B^{*}_{I\times I_{uf}}=B.$

If moreover $\mathbb{L}$ is $D$-pointed and for any $d\in D$ we have
$x^{*}_{d}=x_{d}$, then we say that $t^{*}$ is a $D$-pointed seed extension of
$t$.

In the next definition, we repeatedly use the notation introduced in (6).

######  Definition 3.0.3 (monomial lifting).

A $D$-lifting configuration on $t$ is a pair $(\mathbb{L},\nu)$ consisting of
a $D$-pointed extension $\mathbb{L}$ of $\mathbb{K}$ and a lifting matrix
$\nu\in\mathbb{Z}^{D\times I}$. The monomial lifting of $t$, defined by
$(\mathbb{L},\nu)$, is the seed
$\upharpoonleft\kern-3.50006ptt=(\upharpoonleft\kern-3.50006ptI_{uf},\upharpoonleft\kern-3.50006ptI_{sf},\upharpoonleft\kern-3.50006ptI_{uf},\upharpoonleft\kern-3.50006ptB,\upharpoonleft\kern-3.50006ptx)$
defined by:

  * –

The set of vertices
$\upharpoonleft\kern-3.50006ptI_{uf}=I_{uf},\quad\upharpoonleft\kern-3.50006ptI_{sf}=I_{sf}\sqcup
D,\quad\upharpoonleft\kern-3.50006ptI_{hf}=I_{hf}.$

  * –

The matrix

$\upharpoonleft\kern-3.50006ptB=\begin{pmatrix}B\\\ -\nu
B\end{pmatrix}\in\mathbb{Z}^{(I\sqcup D)\times I_{uf}}.$ (13)

The coefficients of $\upharpoonleft\kern-3.50006ptB$ are denoted by
$\upharpoonleft\kern-3.50006ptb_{i,j}$ for
$i\in\,\upharpoonleft\kern-3.50006ptI,j\in\,\upharpoonleft\kern-3.50006ptI_{uf}$.

  * –

The variables are

$\upharpoonleft\kern-3.50006ptx={\widehat{x}}^{{\widehat{\nu}}}\quad\text{where}\quad{\widehat{\nu}}=\begin{pmatrix}Id&0\\\
\nu&Id\end{pmatrix}\in\mathbb{Z}^{(I\sqcup D)\times(I\sqcup D)}.$ (14)

In this case we say that $\nu$ lifts $t$ to $\upharpoonleft\kern-3.50006ptt$
and write: $\upharpoonleft\kern-3.50006ptt\xlongleftarrow{\nu}t.$

Any monomial lifting $\upharpoonleft\kern-3.50006ptt$ is a $D$-pointed
extension of $t$. Note that, according to our notation,

${\widehat{x}}_{D}=(x_{d})_{d}\in(\mathbb{L}^{*})^{D}.$

Formula (14) implies that for any $i\in I$

$\upharpoonleft\kern-3.50006ptx_{i}=x_{i}\,{\widehat{x}}^{\nu_{\bullet,i}}_{D}.$

Hence, we can think about $\nu$ as a collection of monomials, indexed by $I$,
in the variables ${\widehat{x}}_{D}$. The cluster variables
$\upharpoonleft\kern-3.50006ptx_{i}$, for $i\in I$, are the product of $x_{i}$
and the corresponding monomial.

######  Example 3.0.4.

Let $t$ be a seed of the field $\mathbb{K}$, which graphically is described by
the following quiver

$\leavevmode\hbox to121.06pt{\vbox
to16.31pt{\pgfpicture\makeatletter\hbox{\hskip 60.5277pt\lower-8.204pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-60.5277pt}{-8.10416pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
12.91664pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.6111pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\bigcirc 1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
12.91664pt\hfil&\hfil\hskip 36.91661pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.6111pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\bigcirc 2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
12.91664pt\hfil&\hfil\hskip 34.69441pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.3889pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\blacksquare 3}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
10.69444pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{38.93881pt}{-5.60416pt}\pgfsys@lineto{15.7388pt}{-5.60416pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{15.53882pt}{-5.60416pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-10.89444pt}{-5.60416pt}\pgfsys@lineto{-34.09445pt}{-5.60416pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-34.29443pt}{-5.60416pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
That is: $I_{uf}=\\{1,2\\}$, $I_{sf}=\emptyset$, $I_{hf}=\\{3\\}$ and

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

Let $D=\\{d\\}$, here $d$ is considered as a symbol,
$\mathbb{L}=\mathbb{K}(x_{d})$ where $x_{d}$ is a variable and $\nu=(1,2,3).$
Then, a simple matrix multiplication implies that
$\upharpoonleft\kern-3.50006ptt$ corresponds graphically to the quiver

$\leavevmode\hbox to121.06pt{\vbox
to48.57pt{\pgfpicture\makeatletter\hbox{\hskip 60.5277pt\lower-24.33594pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-60.5277pt}{-24.2361pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 30.90794pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.60243pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${d}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.90797pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
12.91664pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.6111pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\bigcirc 1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
12.91664pt\hfil&\hfil\hskip 36.91661pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.6111pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\bigcirc 2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
12.91664pt\hfil&\hfil\hskip 34.69441pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.3889pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\blacksquare 3}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
10.69444pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{38.93881pt}{-21.7361pt}\pgfsys@lineto{15.7388pt}{-21.7361pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{15.53882pt}{-21.7361pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-10.89444pt}{-21.7361pt}\pgfsys@lineto{-34.09445pt}{-21.7361pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-34.29443pt}{-21.7361pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-36.0962pt}{-11.40799pt}\pgfsys@lineto{-6.53188pt}{7.73283pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.83943}{0.54347}{-0.54347}{0.83943}{-6.364pt}{7.8415pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-0.1889pt}{-13.43195pt}\pgfsys@lineto{-0.1889pt}{3.76814pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-0.1889pt}{3.96812pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-34.78584pt}{-13.43195pt}\pgfsys@lineto{-5.22151pt}{5.70888pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.83943}{0.54347}{-0.54347}{0.83943}{-5.05363pt}{5.81755pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{2.2222pt}{-13.43195pt}\pgfsys@lineto{2.2222pt}{3.76814pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{2.2222pt}{3.96812pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>1&0\\\ 0&1\\\ -2&-2\end{pmatrix}.$

Moreover, from the definition, we have that

$\upharpoonleft\kern-3.50006ptx_{1}=x_{1}x_{d}\quad\upharpoonleft\kern-3.50006ptx_{2}=x_{2}x_{d}^{2}\quad\upharpoonleft\kern-3.50006ptx_{3}=x_{3}x_{d}^{3}\quad\upharpoonleft\kern-3.50006ptx_{d}=x_{d}.$

Monomial liftings can be performed in steps. In fact, suppose that
$D=D_{1}\sqcup D_{2}$ and $\mathbb{L}$ is a $D$-field extension of
$\mathbb{K}$. Then the field $\mathbb{L}_{1}=\mathbb{K}(x_{d_{1}})_{d_{1}\in
D_{1}}$ is a $D_{1}$-pointed extension of $\mathbb{K}$ and $\mathbb{L}$ is a
$D_{2}$-pointed extension of $\mathbb{L}_{1}$. The following lemma is obvious
from the definitions.

######  Lemma 3.0.5.

Let $(\mathbb{L},\nu)$ be a $D$-lifting data on $t$. If $D=D_{1}\sqcup D_{2}$,
and $\nu_{j}=\nu_{D_{j},\bullet}$, for $j=1,2$, then the following diagram is
commutative.

${\bullet}$${\bullet}$${t}$$\scriptstyle{\nu_{1}}$$\scriptstyle{(\nu_{2},0)}$$\scriptstyle{\nu}$

We introduce the following notion of mutation of lifting configuration.

######  Definition 3.0.6 (mutation of lifting configuration).

Let $(\mathbb{L},\nu)$ be a $D$-lifting configuration on $t$, $k\in I_{uf}$
and $t^{\prime}:=\mu_{k}(t)$. Consider the $D$-lifting matrix $\nu^{\prime}$
on $t^{\prime}$ defined by

$\nu^{\prime}_{\bullet,i}=\begin{aligned}
\begin{cases}\nu_{\bullet,i}&\text{if}\quad i\neq k\\\ \max\\{\nu
B_{\bullet,k}^{+}\,,\,\nu B_{\bullet,k}^{-}\\}-\nu_{\bullet,k}&\text{if}\quad
i=k\end{cases}\end{aligned}$ (15)

where the $\max$ in (15) is taken component-wise. We say that
$\nu^{\prime}=\mu_{k}(\nu)$ is the mutation at $k$ of the lifting matrix $\nu$
and that the lifting configuration
$(\mathbb{L},\nu^{\prime})=\mu_{k}(\mathbb{L},\nu)$, on $t^{\prime}$, is the
mutation at $k$ of the lifting configuration $(\mathbb{L},\nu).$

######  Remark 3.0.7.

For any $a,b\in\mathbb{R}$ we have that $\max\\{a,b\\}=(a-b)^{+}+b$. Using
this formula and the fact that $b=b^{+}-b^{-}$, we get two alternative forms
of formula (15), namely:

$\nu^{\prime}_{\bullet,k}=(\nu B_{\bullet,k})^{+}+\nu
B^{-}_{\bullet,k}-\nu_{\bullet,k}$ (16)

and

$\nu^{\prime}_{\bullet,k}=(-\nu B_{\bullet,k})^{+}+\nu
B^{+}_{\bullet,k}-\nu_{\bullet,k}.$ (17)

If the $D$-pointed field $\mathbb{L}$ is clear, we identify a lifting
configuration with its lifting matrix. From now on, we fix $\mathbb{L}$ and
suppose that any lifting configuration has $\mathbb{L}$ as defining field. The
following lemma is crucial.

######  Lemma 3.0.8.

Lifting seeds commutes with mutation. That is, for any lifting configuration
$\nu$ on $t$ and any $k\in I_{uf}$, if we denote by $t^{\prime}=\mu_{k}(t)$
and $\nu^{\prime}=\mu_{k}(\nu)$, then the following diagram is commutative

${\upharpoonleft\kern-3.50006ptt}$${\upharpoonleft\kern-3.50006ptt^{\prime}}$${t}$${{t^{\prime}}}$$\scriptstyle{\nu^{\prime}}$$\scriptstyle{\mu_{k}}$$\scriptstyle{\nu}$$\scriptstyle{\mu_{k}}$

###### Proof.

Call $t^{*}=\mu_{k}(\upharpoonleft\kern-3.50006ptt)$. We have to prove that
$t^{*}=\upharpoonleft\kern-3.50006ptt^{\prime}$. It’s clear that the two seeds
have the same vertices sets. To avoid possible confusion, we stress that
$\upharpoonleft\kern-3.50006ptB^{\pm}$ denotes
$(\upharpoonleft\kern-3.50006ptB)^{\pm}$ and that
$\upharpoonleft\kern-3.50006ptb_{i,j}^{\pm}$ are the coefficients of
$\upharpoonleft\kern-3.50006ptB^{\pm}$.

First we prove that $x^{*}=\upharpoonleft\kern-3.50006ptx^{\prime}$. Clearly,
if $d\in D$, then
$x^{*}_{d}=\upharpoonleft\kern-3.50006ptx^{\prime}_{d}=x_{d}$. If $i\in I$ and
$i\neq k$, then we have
$x_{i}^{*}=\upharpoonleft\kern-3.50006ptx_{i}=\upharpoonleft\kern-3.50006ptx_{i}^{\prime}$
because of (15) and (2). Finally, we can compute that:

$\displaystyle x_{k}^{*}=$ $\displaystyle\kern
1.99997pt\displaystyle\upharpoonleft\kern-3.50006ptx^{\upharpoonleft\kern-2.45004pt\kern
1.39998ptB_{\bullet,k}^{+}-e_{k}}+\upharpoonleft\kern-3.50006ptx^{\upharpoonleft\kern-2.45004pt\kern
1.39998ptB_{\bullet,k}^{-}-e_{k}}$ $\displaystyle=$ $\displaystyle\kern
1.99997pt{\widehat{x}}^{{\widehat{\nu}}\cdot\upharpoonleft\kern-2.45004pt\kern
1.39998ptB^{+}_{\bullet,k}-{\widehat{\nu}}_{\bullet,k}}+{\widehat{x}}^{{\widehat{\nu}}\cdot\upharpoonleft\kern-2.45004pt\kern
1.39998ptB^{-}_{\bullet,k}-{\widehat{\nu}}_{\bullet,k}}$ $\displaystyle=$
$\displaystyle\kern
1.99997ptx^{B^{+}_{\bullet,k}-e_{k}}{\widehat{x}}_{D}^{\nu\cdot
B^{+}_{\bullet,k}+(-\nu\cdot
B_{\bullet,k})^{+}-\nu_{\bullet,k}}+x^{B^{-}_{\bullet,k}-e_{k}}{\widehat{x}}_{D}^{\nu\cdot
B^{-}_{\bullet,k}+(-\nu\cdot B_{\bullet,k})^{-}-\nu_{\bullet,k}}$ (18)

The first equality is the mutation rule (2), the second is obtained from the
definition of $\upharpoonleft\kern-3.50006ptx$ (14) and the third follows from
the following expressions:

$\begin{split}{\widehat{\nu}}\cdot\upharpoonleft\kern-3.50006ptB^{+}=\begin{pmatrix}Id&0\\\
\nu&Id\end{pmatrix}\begin{pmatrix}B^{+}\\\ (-\nu
B)^{+}\end{pmatrix}=\begin{pmatrix}B^{+}\\\ \nu B^{+}+(-\nu
B)^{+}\end{pmatrix}\\\
{\widehat{\nu}}\cdot\upharpoonleft\kern-3.50006ptB^{-}=\begin{pmatrix}Id&0\\\
\nu&Id\end{pmatrix}\begin{pmatrix}B^{-}\\\ (-\nu
B)^{-}\end{pmatrix}=\begin{pmatrix}B^{-}\\\ \nu B^{-}+(-\nu
B)^{-}\end{pmatrix}\end{split}$ (19)

Recall that
$\upharpoonleft\kern-3.50006ptB^{+}-\upharpoonleft\kern-3.50006ptB^{-}=\upharpoonleft\kern-3.50006ptB$,
in particular

${\widehat{\nu}}\cdot\upharpoonleft\kern-3.50006ptB^{+}-{\widehat{\nu}}\cdot\upharpoonleft\kern-3.50006ptB^{-}=\begin{pmatrix}Id&0\\\
\nu&Id\end{pmatrix}\begin{pmatrix}B\\\ -\nu B\end{pmatrix}=\begin{pmatrix}B\\\
0\end{pmatrix}.$ (20)

From (19) and (20) we deduce that the exponents of ${\widehat{x}}_{D}$ in the
two monomials of expression (18) are the same. Hence, using (2), we deduce
from (18) that

$x_{k}^{*}=x_{k}^{\prime}{\widehat{x}}_{D}^{\nu\cdot
B^{+}_{\bullet,k}+(-\nu\cdot B_{\bullet,k})^{+}-\nu_{\bullet,k}}.$

Comparing this formula to the expression of
$\upharpoonleft\kern-3.50006ptx_{k}^{\prime}$, we see that it’s sufficient to
prove that

$\nu B^{+}_{\bullet,k}+(-\nu
B_{\bullet,k})^{+}-\nu_{\bullet,k}=\nu^{\prime}_{\bullet,k}.$

But this is Formula (17).

Next, we consider the exchange matrices, that is we prove that
$B^{*}=\upharpoonleft\kern-3.50006ptB^{\prime}$. It’s clear from (1) and (13)
that for any $j\in I$ and $i\in I_{uf}$,
$b^{*}_{j,i}=b^{\prime}_{j,i}=\upharpoonleft\kern-3.50006ptb^{\prime}_{j,i}$.
For $d\in D$, then

$b^{*}_{d,k}=-\upharpoonleft\kern-3.50006ptb_{d,k}=\nu_{d,\bullet}B_{\bullet,k}.$

and

$\upharpoonleft\kern-3.50006ptb_{d,k}^{\prime}=-\nu^{\prime}_{d,\bullet}B^{\prime}_{\bullet,k}=\nu^{\prime}_{d,\bullet}B_{\bullet,k}.$

The equality $b^{*}_{d,k}=\upharpoonleft\kern-3.50006ptb^{\prime}_{d,k}$
follows from the fact that $b_{k,k}=0$ and the definition of $\nu^{\prime}$
(15). Next, if $i\in I_{uf}$, $i\neq k$, using (1) and (13) we get that

$\displaystyle b^{*}_{d,i}=$
$\displaystyle\upharpoonleft\kern-3.50006ptb_{d,i}+\upharpoonleft\kern-3.50006ptb_{d,k}^{+}b_{k,i}^{+}-\upharpoonleft\kern-3.50006ptb_{d,k}^{-}b_{k,i}^{-}$
$\displaystyle=$
$\displaystyle\upharpoonleft\kern-3.50006ptb_{d,i}+(-\nu_{d,\bullet}B_{\bullet,k})^{+}b_{k,i}^{+}-(-\nu_{d,\bullet}B_{\bullet,k})^{-}b_{k,i}^{-}.$

Similarly we have that

$\displaystyle\upharpoonleft\kern-3.50006ptb^{\prime}_{d,i}=$
$\displaystyle-\nu^{\prime}_{d,\bullet}B^{\prime}_{\bullet,i}$
$\displaystyle=$ $\displaystyle-\nu_{d,k}^{\prime}b_{k,i}^{\prime}+\sum_{j\neq
k}-\nu_{d,j}b_{j,i}^{\prime}$ $\displaystyle=$
$\displaystyle-\nu_{d,k}b_{k,i}+\max\\{\nu_{d,\bullet}B^{+}_{\bullet,k}\,,\,\nu_{d,\bullet}B^{-}_{\bullet,k}\\}b_{k,i}-\sum_{j\in
I\setminus\\{k\\}}\nu_{d,j}(b_{j,i}+b^{+}_{j,k}b^{+}_{k,i}-b^{-}_{j,k}b^{-}_{k,i})$
$\displaystyle=$
$\displaystyle-\nu_{d,\bullet}B_{\bullet,i}+\max\\{\nu_{d,\bullet}B_{\bullet,k}^{+}\,,\,\nu_{d,\bullet}B_{\bullet,k}^{-}\\}b_{k,i}-\nu_{d,\bullet}B_{\bullet,k}^{+}b_{k,i}^{+}+\nu_{d,\bullet}B^{-}_{\bullet,k}b_{k,i}^{-}$
$\displaystyle=$
$\displaystyle\upharpoonleft\kern-3.50006ptb_{d,i}+(-\nu_{d,\bullet}B_{\bullet,k})^{+}b_{k,i}+\nu_{d,\bullet}B^{+}_{\bullet,k}b_{k,i}-\nu_{d,\bullet}B^{+}_{\bullet,k}b_{k,i}^{+}+\nu_{d,\bullet}B^{-}_{\bullet,k}b_{k,i}^{-}$
(21)

The first equality follows from formula (13), the second, third and fourth are
obtained from formulas (1), (15) and some elementary manipulations. The last
equality is a consequence of formulas (13) and (17). Finally, using that
$b_{k,i}=b_{k,i}^{+}-b_{k,i}^{-}$ in Formula (21), we obtain immediately that
$\upharpoonleft\kern-3.50006ptb^{\prime}_{d,i}=b_{d,i}^{*}$. ∎

The following characterisation of monomial liftings turns out to be useful.

######  Proposition 3.0.9.

Let $\widetilde{t}$ be a seed of $\mathbb{L}$. Suppose that there exist
$\nu\in\mathbb{Z}^{D\times I}$, and $\lambda\in\mathbb{Z}^{D\times I_{uf}}$
such that

  1. 1.

$\widetilde{t}$ is a $D$-pointed field extension of $t$ with
$D\subset\widetilde{I}_{sf}$.

  2. 2.

For any $i\in I$,
$\widetilde{x}_{i}=x_{i}\,{\widehat{x}}^{\nu_{\bullet,i}}_{D}.$

  3. 3.

For any $k\in I_{uf}$,
$\mu_{k}(\widetilde{x}_{k})=\mu_{k}(x_{k})\,{\widehat{x}}^{\lambda_{\bullet,k}}_{D}$

Then $\widetilde{t}\xlongleftarrow{\nu}t$ and for any $k\in I_{uf}$,
$\lambda_{\bullet,k}=\mu_{k}(\nu)_{\bullet,k}.$

###### Proof.

Denote by $\upharpoonleft\kern-3.50006ptt$ the monomial lifting of $t$
determined by $(\mathbb{L},\nu)$. We have to prove that
$\widetilde{t}=\upharpoonleft\kern-3.50006ptt$. The first assumption implies
that $\widetilde{t}$ and $\upharpoonleft\kern-3.50006ptt$ have the same
vertices set. Moreover, assumption 1 and 2 imply that
$\widetilde{x}=\upharpoonleft\kern-3.50006ptx$. By assumption 1,
$\widetilde{B}_{I,\bullet}=B=\upharpoonleft\kern-3.50006ptB_{I,\bullet}.$ So
we just have to prove that $\widetilde{B}_{D,\bullet}=-\nu B.$ Fix $k\in
I_{uf}$. Call $t^{\prime}=\mu_{k}(t)$ and $t^{*}=\mu_{k}(\widetilde{t})$. By
assumption 3 and (3), we have that

$\begin{array}[]{rl}x_{k}^{*}\widetilde{x}_{k}=&x_{k}^{\prime}x_{k}{\widehat{x}}_{D}^{\nu_{\bullet,k}+\lambda_{\bullet,k}}\\\\[1.99997pt]
=&x^{B^{+}_{\bullet,k}}{\widehat{x}}_{D}^{\nu_{\bullet,k}+\lambda_{\bullet,k}}+x^{B^{-}_{\bullet,k}}{\widehat{x}}_{D}^{\nu_{\bullet,k}+\lambda_{\bullet,k}}.\end{array}$

Always using (3), we can compute that

$\begin{array}[]{rl}x_{k}^{*}\widetilde{x}_{k}=&\widetilde{x}^{\widetilde{B}^{+}_{\bullet,k}}+\widetilde{x}^{\widetilde{B}^{-}_{\bullet,k}}\\\\[1.99997pt]
=&{\widehat{x}}^{{\widehat{\nu}}\widetilde{B}^{+}_{\bullet,k}}+{\widehat{x}}^{{\widehat{\nu}}\widetilde{B}^{-}_{\bullet,k}}\\\\[1.99997pt]
=&x^{B^{+}_{\bullet,k}}{\widehat{x}}^{\nu
B_{\bullet,k}^{+}+\widetilde{B}^{+}_{D\times
k}}+x^{B^{-}_{\bullet,k}}{\widehat{x}}^{\nu
B_{\bullet,k}^{-}+\widetilde{B}^{-}_{D\times k}}.\end{array}$

We deduce that

$\nu B_{\bullet,k}^{+}+\widetilde{B}^{+}_{D\times
k}=\nu_{\bullet,k}+\lambda_{\bullet,k}=\nu
B_{\bullet,k}^{-}+\widetilde{B}^{-}_{D\times k}.$

In particular

$\widetilde{B}_{D\times k}=\widetilde{B}_{D\times
k}^{+}-\widetilde{B}_{D\times k}^{-}=\nu B_{\bullet,k}^{-}-\nu
B_{\bullet,k}^{+}=-\nu B_{\bullet,k}.$

Finally, the statement about $\lambda$ can be easily deduced from the two
above identities and (17).

∎

We introduce some notation. For any finite sets $J$ and $K$, then
$\mathbb{Z}^{J\times K}$ and $(\mathbb{Z}^{J})^{K}$ are canonically identified
by the map that assigns to a matrix the collection of its columns. From now
on, we make no difference between these two objects. Let
$M\in\mathbb{Z}^{J\times K}=(\mathbb{Z}^{J})^{K}$. Note that if $I$ is a third
finite set and $N\in\mathbb{Z}^{K\times I}$, then $MN=M^{N}$. We write $MN$
when we think about $M$ as a matrix and $M^{N}$ when we think about $M$ as the
collection of its columns.
If $H,K$ are groups and $h\in H^{J},k\in K^{J}$, then the notation $h\times k$
stands for the element of $(H\times K)^{J}$ defined by $(h\times
k)_{j}=(h_{j},k_{j})$.

From now on, fix a lifting matrix $\nu\in\mathbb{Z}^{D\times I}$. In
particular $(\mathbb{L},\nu)$ is a fixed $D$-lifting configuration on $t$.

If the seed $t$ is a graded seed, then the degree configuration of $t$ can be
lifted along with $t$.

######  Definition 3.0.10 (lifting of degree configuration).

Let $\upharpoonleft\kern-3.50006pt0:=(\nu,Id)\in(\mathbb{Z}^{D})^{I\sqcup D}$.
If $\sigma\in H^{I}$ is a degree configuration on $t$, then we define

$\upharpoonleft\kern-3.50006pt\sigma=\upharpoonleft\kern-3.50006pt0\times(\sigma,0)\in(\mathbb{Z}^{D}\times
H)^{I\sqcup D}.$ (22)

In this case we say that $\nu$ lifts $\sigma$ to
$\upharpoonleft\kern-3.50006pt\sigma$ and write:
$\upharpoonleft\kern-3.50006pt\sigma\xlongleftarrow{\nu}\sigma.$

######  Lemma 3.0.11.

For any degree configuration $\sigma$ on $t$,
$\upharpoonleft\kern-3.50006pt\sigma$ is a degree configuration on
$\upharpoonleft\kern-3.50006ptt$.

###### Proof.

Note that
$\upharpoonleft\kern-3.50006pt\sigma^{\upharpoonleft\kern-2.45004pt\kern
1.39998ptB}=\upharpoonleft\kern-3.50006pt0^{\upharpoonleft\kern-2.45004pt\kern
1.39998ptB}\times(\sigma,0)^{\upharpoonleft\kern-2.45004pt\kern 1.39998ptB}.$
Moreover $(\sigma,0)^{\upharpoonleft\kern-2.45004pt\kern
1.39998ptB}=\sigma^{B}=0$ and

$\upharpoonleft\kern-3.50006pt0^{\upharpoonleft\kern-2.45004pt\kern
1.39998ptB}=(\nu,{\operatorname{Id}})\begin{pmatrix}B\\\ -\nu
B\end{pmatrix}=0.$

∎

######  Remark 3.0.12.

Note that $\upharpoonleft\kern-3.50006ptt$ has a canonical
$\mathbb{Z}^{D}$-degree configuration given by
$\upharpoonleft\kern-3.50006pt0=(\nu,{\operatorname{Id}})$, which is the
lifting of the trivial degree configuration on $t$. The data of this degree
configuration is equivalent to the one of $\nu$. As a warning, the degree
configuration $\upharpoonleft\kern-3.50006pt0$ depends both on $t$ and on
$\nu$. To avoid ambiguity, we use the convention that
$\upharpoonleft\kern-3.50006pt0$ is always considered as a degree
configuration on a seed which is denoted by $\upharpoonleft\kern-3.50006ptt$
and whose corresponding lifting matrix $\nu$ is clear from the context. If a
seed is called $t^{\bullet}$ (resp. $t_{\bullet}$), for a certain symbol
$\bullet$, and a lifting $\upharpoonleft\kern-3.50006ptt^{\bullet}$ (resp.
$\upharpoonleft\kern-3.50006ptt_{\bullet}$) of $t^{\bullet}$ (resp.
$t_{\bullet}$) is defined, then we use the notation
$\upharpoonleft\kern-3.50006pt0^{\bullet}$ (resp.
$\upharpoonleft\kern-3.50006pt0_{\bullet}$).

######  Lemma 3.0.13.

Lifting a degree configuration commutes with mutation. In particular, for any
degree configuration $\sigma$ on $t$ and any $k\in I_{uf}$, if
$\sigma^{\prime}=\mu_{k}(\sigma)$, then we have a commutative diagram

${\upharpoonleft\kern-3.50006pt\sigma}$${\upharpoonleft\kern-3.50006pt\sigma^{\prime}}$${\sigma}$${{\sigma^{\prime}}}$$\scriptstyle{\nu^{\prime}}$$\scriptstyle{\mu_{k}}$$\scriptstyle{\nu}$$\scriptstyle{\mu_{k}}$

###### Proof.

Let $t^{\prime}=\mu_{k}(t)$. From the mutation rule (11) it’s clear that
$\mu_{k}(\upharpoonleft\kern-3.50006pt\sigma)=\mu_{k}(\upharpoonleft\kern-3.50006pt0)\times(\sigma^{\prime},0).$
Hence it’s sufficient to prove that
$\mu_{k}(\upharpoonleft\kern-3.50006pt0)=\upharpoonleft\kern-3.50006pt0^{\prime}$.
Call $0^{*}=\mu_{k}(\upharpoonleft\kern-3.50006pt0)$. For $i\in
I\setminus\\{k\\}$ and $d\in D$, the identities

$0^{*}_{i}=\nu_{i}=\nu^{\prime}_{i}=\upharpoonleft\kern-3.50006pt0^{\prime}_{i}\quad\text{and}\quad
0^{*}_{d}=e_{d}=\upharpoonleft\kern-3.50006pt0^{\prime}_{d}$

follows at once from (11) and (15). From the same formulas we compute that

$\begin{array}[]{rl}0^{*}_{k}=&\upharpoonleft\kern-3.50006pt0^{\upharpoonleft\kern-2.45004pt\kern
1.39998ptB^{+}_{\bullet,k}}-\nu_{\bullet,k}\\\\[1.49994pt] =&\nu
B^{+}_{\bullet,k}+(-\nu B_{\bullet,k})^{+}-\nu_{\bullet,k}.\end{array}$

Then the identity $0^{*}_{k}=\upharpoonleft\kern-3.50006pt0^{\prime}_{k}$
follows from (17). ∎

######  Corollary 3.0.14.

For any $k_{1},\dots k_{r}\in I_{uf}$ and $j_{1},\dots j_{s}\in I_{uf}$ such
that
$\mu_{k_{r}}\circ\dots\circ\mu_{k_{1}}(t)=\mu_{j_{s}}\circ\dots\circ\mu_{j_{1}}(t)$,
then
$\mu_{k_{r}}\circ\dots\circ\mu_{k_{1}}(\nu)=\mu_{j_{s}}\circ\dots\circ\mu_{j_{1}}(\nu)$.
In particular, any $t^{\prime}\in\Delta(t)$ has a canonical lifting
configuration $\nu^{\prime}$ defined by the composition of (15) along any
sequence of mutations from $t$ to $t^{\prime}.$ Moreover, the map sending
$t^{\prime}$ to $\upharpoonleft\kern-3.50006ptt^{\prime}$ is a bijection
between $\Delta(t)$ and $\Delta(\upharpoonleft\kern-3.50006ptt).$

###### Proof.

One can recover $\nu$ from $\upharpoonleft\kern-3.50006pt0$. We know that
$\mu_{k_{r}}\circ\dots\circ\mu_{k_{1}}(\upharpoonleft\kern-3.50006pt0)=\mu_{j_{s}}\circ\dots\circ\mu_{j_{1}}(\upharpoonleft\kern-3.50006pt0)$,
then Lemma 3.0.13 implies that
$\mu_{k_{r}}\circ\dots\circ\mu_{k_{1}}(\nu)=\mu_{j_{s}}\circ\dots\circ\mu_{j_{1}}(\nu)$.
Alternatively, the same result can be obtained in the same way as a direct
consequence of Lemma 3.0.8. By Lemma 3.0.8, the map
$\Delta(t){\,\longrightarrow\,}\Delta(\upharpoonleft\kern-3.50006ptt)$ sending
$t^{\prime}$ to $\upharpoonleft\kern-3.50006ptt^{\prime}$ is well defined. It
is surjective since $t$ and $\upharpoonleft\kern-3.50006ptt$ have the same set
of mutable vertices. Moreover, it’s clear from Definition 3.0.3 that if
$\upharpoonleft\kern-3.50006ptt^{\prime}\xlongleftarrow{\nu^{\prime}}t^{\prime}$,
then $\upharpoonleft\kern-3.50006ptt^{\prime}$ completely determines
$\nu^{\prime}$ and $t^{\prime}$. In particular, the map is injective. ∎

Note that, with little abuse of notation, we can consider the $x_{d}$ as
abstract independent variables over $\mathbb{K}$. Any $t^{\prime}\in\Delta$
gives an isomorphism

$\begin{array}[]{r c l }\mathbb{K}[x_{d}^{\pm
1}]_{d}&{\,\longrightarrow}&\mathbb{C}[x_{d}^{\pm
1}]_{d}\otimes\mathbb{K}\\\\[3.99994pt]
\upharpoonleft\kern-3.50006ptx^{\prime}_{i}&\longmapsto&\begin{cases}x_{d}\otimes
1&\text{if}\quad i=d\in D\\\ (x_{d})_{d}^{\nu^{\prime}_{\bullet,i}}\otimes
x^{\prime}_{i}&\text{if}\quad i\in I\end{cases}\end{array}$ (23)

where $\nu^{\prime}$ is the lifting matrix on $t^{\prime}$ determined by
$\nu$. Because of Lemma 3.0.8, the above isomorphism doesn’t depend on
$t^{\prime}$ but only on the chosen initial lifting matrix $\nu$ on $t$. So,
whenever $\nu$ is clear, we identify the above two rings by the previous
isomorphism. In particular, we can think about
$\upharpoonleft\kern-3.50006ptt$ as a seed of the fraction field of
$\mathbb{C}[x_{d}^{\pm 1}]_{d}\otimes\mathbb{K}$. With this identification,
it’s immediate that
${\mathcal{L}}(\upharpoonleft\kern-3.50006ptt)=\mathbb{C}[x_{d}^{\pm
1}]_{d}\otimes{\mathcal{L}}(t).$

######  Theorem 3.0.15.

We have equality

$\mathcal{A}(\upharpoonleft\kern-3.50006ptt)=\mathcal{A}(t)[x_{d}^{\pm
1}]_{d}.$

Moreover, if $t$ is of maximal rank, then

$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)=\overline{\mathcal{A}}(t)[x_{d}^{\pm
1}]_{d}.$

###### Proof.

By Corollary 3.0.14, for any $t^{*}\in\Delta(\upharpoonleft\kern-3.50006ptt)$,
there exists a unique $t^{\prime}\in\Delta(t)$, with a well defined lifting
configuration $\nu^{\prime}$, such that
$t^{*}=\upharpoonleft\kern-3.50006ptt^{\prime}$. The equality involving
$\mathcal{A}(\upharpoonleft\kern-3.50006ptt)$ is then obvious from the
definition of $\upharpoonleft\kern-3.50006ptx^{\prime}$ and the fact that, for
any $d\in D$, then $x_{d}^{\pm
1}\in\mathcal{A}(\upharpoonleft\kern-3.50006ptt)$ since
$D\subseteq\,\,\upharpoonleft\kern-3.50006ptI_{sf}$.

In the following, we think about $\upharpoonleft\kern-3.50006ptt$ as a seed of
the fraction field of $\mathbb{C}[x_{d}^{\pm 1}]_{d}\otimes\mathbb{K}$. If $t$
is of maximal rank, so is $\upharpoonleft\kern-3.50006ptt$. Then using Theorem
2.1.7 and the convention that $\mu_{0}(t)=t$, we deduce that (intersections
run over $I_{uf}\cup\\{0\\}$):

$\begin{array}[]{rl}\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)=&\displaystyle\bigcap\biggl{[}\mathbb{C}[x_{d}^{\pm
1}]_{d}\otimes{\mathcal{L}}(\mu_{k}(t))\biggr{]}\\\\[15.00002pt]
=&\displaystyle\mathbb{C}[x_{d}^{\pm
1}]_{d}\otimes\biggl{[}\bigcap{\mathcal{L}}(\mu_{k}(t)\biggr{]}\\\\[15.00002pt]
=&\overline{\mathcal{A}}(t)[x_{d}^{\pm 1}]_{d}\end{array}$

where central equality follows because $\mathbb{C}[x_{d}^{\pm 1}]_{d}\otimes-$
commutes with finite intersections. ∎

We expect the answer to the following question to be negative in general.

######  Question 3.0.16.

Does the equality between
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)$ and
$\overline{\mathcal{A}}(t)[x_{d}^{\pm 1}]_{d}$ hold without the assumption
that $t$ is of maximal rank?

######  Remark 3.0.17.

In the special case of $t_{prin}$ (the seed obtained from $t$ by adding
principal coefficients), the second statement of the previous theorem is a
consequence of [GHKK18, Proposition 8.27]. We stress the fact that, the
variables $(x_{d})_{d}$ being algebraically independent over $\mathbb{K}$,
$\mathcal{A}(t)[x_{d}^{\pm 1}]_{d}$ and $\overline{\mathcal{A}}(t)[x_{d}^{\pm
1}]_{d}$ are Laurent polynomial rings over $\mathcal{A}(t)$ and
$\overline{\mathcal{A}}(t)$ respectively.

######  Corollary 3.0.18.

The deletion map
$s:\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt){\,\longrightarrow\,}\overline{\mathcal{A}}(t)$,
that is the morphism defined by $s(\upharpoonleft\kern-3.50006ptx_{i})=x_{i}$
for $i\in I$ and $s(\upharpoonleft\kern-3.50006ptx_{d})=1$ for $d\in D$, is
surjective. The kernel is the ideal generated by
$\upharpoonleft\kern-3.50006ptx_{d}-1$, for $d\in D$.

###### Proof.

This is obvious from Theorem 3.0.15. ∎

######  Lemma 3.0.19.

Let $\widetilde{t}$ be a $D$-seed extension of $t$ such that
$D\subseteq\widetilde{I}_{sf}$. If
$B^{t}:\mathbb{Z}^{I}{\,\longrightarrow\,}\mathbb{Z}^{I_{uf}}$ is surjective,
there exist a lifting matrix $\nu\in\mathbb{Z}^{D\times I}$ such that
$\upharpoonleft\kern-3.50006ptB=\widetilde{B}$.

###### Proof.

Since $B^{t}$ is surjective, we can find $\nu$ such that
$\widetilde{B}_{D,\bullet}=-\nu B.$ ∎

To the author best knowledge, the next statement is new.

######  Corollary 3.0.20.

Let $\widetilde{t}$ be a $D$-seed extension of $t$ such that
$D\subseteq\widetilde{I}_{sf}$. If
$B^{t}:\mathbb{Z}^{I}{\,\longrightarrow\,}\mathbb{Z}^{I_{uf}}$ is surjective,
then the deletion map
$s:\overline{\mathcal{A}}(\widetilde{t}){\,\longrightarrow\,}\overline{\mathcal{A}}(t)$
is surjective and its kernel is the ideal generated by $\widetilde{x}_{d}-1$,
for $d\in D.$

###### Proof.

Note that the isomorphism class of an upper cluster algebra only depends on
the $B$-matrix and on the vertices sets of one of its seeds, and so does the
desired property of $s$. The statement follows from the previous lemma and
Corollary 3.0.18. ∎

If $\widetilde{t}$ is a seed extension of $t$, the deletion map
$s:\overline{\mathcal{A}}(\widetilde{t}){\,\longrightarrow\,}\overline{\mathcal{A}}(t)$
is not always surjective,. An example is given in [MM15][Section 7.1,
Corollary 7.1.2]. The deletion map is always surjective at the level of
cluster algebras.

######  Remark 3.0.21.

Let $t$ be a seed and set $D:=\\{i^{\prime}:i\in I\\}$. We identify
$\mathbb{Z}^{D}$ with $\mathbb{Z}^{I}$ in the obvious way. If we consider
$\nu={\operatorname{Id}}\in\mathbb{Z}^{I\times I}$, then

$\upharpoonleft\kern-3.50006ptB=\begin{pmatrix}B\\\ -B\end{pmatrix}.$

Consider the degree transformation $\psi$ defined in [Qin22, Definition
3.3.1]. By [Qin22, Remark 3.3.4], for any $k\in I_{uf}$,

$e_{k}-\psi_{t,\mu_{k}(t)}\psi_{\mu_{k}(t),t}(e_{k})=\upharpoonleft\kern-3.50006ptB_{D\times
k}.$

In particular, the monomial lifting for this special $\nu$ encodes the non-
trivial monodromy of the maps $\psi$ after one mutation. See [Qin22, Remark
3.3.4] for more details. It turns out that, under some hypothesis, the lifting
matrix $\nu$ has an interpretation in terms of the Cox ring of a certain
partial compactification of the cluster manifold. This is part of an ongoing
work.

The results of this section should also work, with the same proofs, for
cluster and upper cluster algebras defined over $\mathbb{Z}$ instead then over
$\mathbb{C}$.

######  Remark 3.0.22.

If $\nu,\mu\in\mathbb{Z}^{D\times I}$ are distinct lifting matrices and
$\upharpoonleft\kern-3.50006ptt_{\nu}\xlongleftarrow{\nu}t$,
$\upharpoonleft\kern-3.50006ptt_{\mu}\xlongleftarrow{\mu}$ are the respective
monomial liftings, clearly $\upharpoonleft\kern-3.50006ptt_{\nu}$ and
$\upharpoonleft\kern-3.50006ptt_{\mu}$ are distinct. Nevertheless, they are
quasi-equivalent in the sense of [Fra16] by [Fra16, Corollary 4.5].

###  3.1 Pole filtration and lifting graduation

It turns out that the graduation induced by $\upharpoonleft\kern-3.50006pt0$
on $\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ has a nice
geometric interpretation. To explain this fact, we switch for a moment to a
more general setting. This allows us to apply the same ideas in the context of
schemes which are suitable for lifting, which are introduced in Section 4.

In this section, $D$ is a finite set and $T$ is a torus of rank $|D|$. We
denote by $X(T)$ the character group of $T$.

######  Definition 3.1.1.

Let $A$ be a $\mathbb{C}$-algebra. An almost polynomial ring over $A$ is a
quadruple $(R,\mathcal{V},X,\phi^{*})$ where

  1. 1.

$R$ is a normal (non necessarily noetherian) $\mathbb{C}$-algebra which is a
domain.

  2. 2.

$\mathcal{V}$ is a collection of discrete valuations
$\mathcal{V}_{d}:R{\,\longrightarrow\,}\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$,
indexed by $d\in D$.

  3. 3.

$X\subseteq R^{D}$ is a collection of elements such that
$\mathcal{V}_{d_{1}}(X_{d_{2}})=\delta_{d_{1},d_{2}}$ and

$R=\bigcap_{d\in D}\mathcal{V}_{d}{{}^{-1}}(\mathbb{Z}_{\geq
0}\cup\\{\infty\\})\bigcap R_{\prod X_{d}}.$

  4. 4.

$\phi^{*}:R_{\prod X_{d}}{\,\longrightarrow\,}\mathbb{C}[T]\otimes A$ is an
isomorphism such that $\phi^{*}(X_{d})=x_{d}\otimes 1$, where the collection
$(x_{d})_{d}$ is a basis of $X(T).$

When $\mathcal{V},X$ and $\phi^{*}$ are clear, we just write that $R$ is an
almost polynomial ring over $A$. If an almost polynomial ring is called $R$,
we implicitly assume that the rest of the data is denoted by
$\mathcal{V},X,\phi^{*}$ and that $R$ is almost polynomial over $A$. Moreover,
$R_{\prod{X_{d}}}$ is canonically identified with $\mathbb{C}[T]\otimes A$ via
$\phi^{*}$. In particular, we write $X_{d}=x_{d}\otimes 1$.

######  Example 3.1.2.

Let $t$ be a seed of maximal rank, $\nu$ a $D$-lifting matrix and
$\upharpoonleft\kern-3.50006ptt\xlongleftarrow{\nu}t$. Then we have an almost
polynomial ring over $\overline{\mathcal{A}}(t)$:

$(\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D}),\mathcal{C}\mathcal{V},\upharpoonleft\kern-3.50006ptx_{D},\phi^{*}).$

Recall that $\upharpoonleft\kern-3.50006ptx_{d}=x_{d}$ and
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})_{\prod_{d}x_{d}}=\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)$
because of Lemma 2.4.2. Moreover, the map
$\phi^{*}:\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt){\,\longrightarrow\,}\mathbb{C}[x_{d}^{\pm
1}]_{d}\otimes\overline{\mathcal{A}}(t)$ defined by formula (23) is an
isomorphism because of Theorem 3.0.15. The third axiom of Definition 3.1.1
holds by statement 8 of Lemma 2.4.2. The other axioms are trivial to check.

We refer to this as the standard quasi polynomial structure of
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$. Of course,
${\operatorname{Spec}}(\mathbb{C}[x_{d}^{\pm 1}])_{d}$ is a torus with
$(x_{d})_{d}$ as base of characters.

Form now on, we fix an almost polynomial ring $R$ over $A$. Let
$s:R{\,\longrightarrow\,}A$ be the morphism induced by the map
$\mathbb{C}[T]\otimes A{\,\longrightarrow\,}A$ sending $p\otimes a$ to
$p(1)a$. Note that, if
$R=\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$, considered
with its standard quasi-polynomial structure, then $s$ is the deletion map of
Corollary 3.0.18. We consider $\mathbb{Z}^{D}$ ordered by $\lambda\leq\mu$ if
the inequality holds component-wise.

######  Definition 3.1.3 (Pole filtration).

The pole filtration on the algebra $A$ is the $(\mathbb{Z}^{D},\leq)$
filtration defined by

$A_{\lambda}:=\\{a\in A\,:\,\forall\,d\in D,\mathcal{V}_{d}(1\otimes
a)\geq-\lambda(d)\\}.$

If $H$ is an abelian group and $A=\bigoplus_{h\in H}A_{h}$ is a graduation, we
set $A_{\lambda,h}=A_{\lambda}\cap A_{h}$. We say that the pole filtration is
$H$-graded, if for any $\lambda\in\mathbb{Z}^{d}$,
$A_{\lambda}=\bigoplus_{h}A_{\lambda,h}.$

It’s easy to verify that, if $\mu\leq\lambda$, then $A_{\mu}\subseteq
A_{\lambda}$, moreover for any $\lambda,\mu$ we have $A_{\lambda}\cdot
A_{\mu}\subseteq A_{\lambda+\mu}$ and finally $A_{\lambda}+A_{\mu}\subseteq
A_{\max\\{\lambda,\mu\\}}$. We express this last inclusion saying that the
pole filtration is a tropical filtration.

We introduce a notation. Let $G$ and $H$ be abelian groups and
$B=\displaystyle\bigoplus_{(g,h)\in G\times H}B_{g,h}$ be a $G\times H$-graded
$\mathbb{C}$-algebra. The $G$-graduation induced by the $G\times H$ one, on
$B$, is defined as follows:

$B=\bigoplus_{g\in G}B_{g,\bullet}\quad\text{where, for}\,\,g\in G\quad
B_{g,\bullet}=\bigoplus_{h\in H}B_{g,h}.$

Similarly, we have an induced $H$-graduation, for which we use an analogue
notation.

######  Definition 3.1.4 (lifting graduation).

A $\mathbb{Z}^{D}$-graduation
$R=\bigoplus_{\lambda\in\mathbb{Z}^{D}}R_{\lambda}$ is called lifting
graduation, of the pole filtration, if the following conditions hold

  1. 1.

For any $d\in D$, $X_{d}\in R_{e_{d}}$.

  2. 2.

The isomorphism $\phi^{*}$ identifies $1\otimes A$ with the $0$-degree
component of $R_{\prod X_{D}}$.

  3. 3.

For any $\lambda\in\mathbb{Z}^{D}$, the restriction of $s$ to $R_{\lambda}$ is
an isomorphism with $A_{\lambda}.$

If the pole filtration is $H$-graded, we say that a $\mathbb{Z}^{D}\times H$
graduation $R=\bigoplus_{\lambda,h}R_{\lambda,h}$ is an $H$-graded lifting
graduation if

  1. 1.

$R=\bigoplus_{\lambda}R_{\lambda,\bullet}$ is a lifting graduation.

  2. 2.

For any $(\lambda,h)\in\mathbb{Z}^{D}\times H$, the restriction of $s$ to
$R_{\lambda,h}$ is an isomorphism with $A_{\lambda,h}.$

It often happens, in representation theory, that we study the components of
the pole filtration on some algebra, because they are the shadows of the
homogeneous components of the lifting graduation, which is the real object of
interest. For example, some classical expressions for the generalised
Littlewood-Richardson coefficients, such as [Zel02, Proposition 1.4], can be
interpreted in this setting. We explain this in section 7.1.

The following lemma is obvious but useful.

######  Lemma 3.1.5.

A $\mathbb{Z}^{D}\times H$ graduation $R=\bigoplus_{\lambda,h}R_{\lambda,h}$
is an $H$-graded lifting graduation if and only if

  1. 1.

$R=\bigoplus_{\lambda}R_{\lambda,\bullet}$ is a lifting graduation.

  2. 2.

The map $s$ is $H$-equivariant.

Recall that if $\widetilde{T}$ is a torus and $\widetilde{R}$ is a
$\mathbb{C}$-algebra, there is a bijection between left actions of
$\widetilde{T}$ on ${\operatorname{Spec}}(\widetilde{R})$ and (multiplicative)
graduations $\widetilde{R}=\bigoplus_{\lambda\in
X(\widetilde{T})}\widetilde{R}_{\lambda}$. Explicitly, if
$\alpha:\widetilde{T}\times{\operatorname{Spec}}(\widetilde{R}){\,\longrightarrow\,}{\operatorname{Spec}}(\widetilde{R})$
is an action, then $\widetilde{R}_{\lambda}$ is the set of $r\in\widetilde{R}$
such that $\alpha^{*}(r)=\lambda\otimes r$. We adopt the convention that, for
any $\mathbb{C}$-scheme $Z$, $\widetilde{T}\times Z$ is canonically acted on
by $\widetilde{T}$ by multiplication on the $\widetilde{T}$ component. In
particular, when $Z={\operatorname{Spec}}(\widetilde{A})$, for a
$\mathbb{C}$-algebra $\widetilde{A}$, then
$(\mathbb{C}[\widetilde{T}]\otimes\widetilde{A})_{\lambda}=\lambda\otimes\widetilde{A}$.
When $\widetilde{T}$ has a fixed base of characters $\varpi\subseteq
X(\widetilde{T})^{J}$, then we identify $\mathbb{Z}^{J}$ with
$X(\widetilde{T})$ through the map $n\longmapsto\varpi^{n}$.

In our context, the torus $T$ has a fixed base of characters determined by
$R$: $(x_{d})_{d}.$

######  Proposition 3.1.6.

Let $R=\bigoplus_{\lambda\in\mathbb{Z}^{D}}R_{\lambda}$ be a graduation and
consider the associated $T$ action on ${\operatorname{Spec}}(R)$. The
graduation on $R$ is a lifting graduation if and only if the morphism
$\phi:T\times{\operatorname{Spec}}(A){\,\longrightarrow\,}{\operatorname{Spec}}(R)$,
induced by $\phi^{*}$, is $T$-equivariant. In particular, if the lifting
graduation exists, it is unique.

###### Proof.

We identify $R$ with a subring of $\mathbb{C}[T]\otimes A$, using $\phi^{*}$.
Suppose that $R=\bigoplus_{\lambda\in\mathbb{Z}^{D}}R_{\lambda}$ is a lifting
graduation. We need to prove that $R_{\lambda}\subseteq x^{\lambda}\otimes A$.
Let $r\in R_{\lambda}$, by condition 3 in Definition 3.1.4, $s(r)\in
A_{\lambda}$. Recall that $X_{d}=x_{d}\otimes 1$. Since $R=\bigcap_{d\in
D}\mathcal{V}_{d}(\mathbb{Z}_{\geq 0}\cup\\{\infty\\})\bigcap R_{\prod X_{d}}$
and $\mathcal{V}_{d_{1}}(X_{d_{2}})=\delta_{d_{1},d_{2}}$, we have that
$x^{\lambda}\otimes s(r)\in R$. Moreover, since $X_{d}\in R_{e_{d}}$ and
$1\otimes s(r)$ is of degree zero, then $x^{\lambda}\otimes s(r)\in
R_{\lambda}$. But the restriction of $s$ to $R_{\lambda}$ is injective. Hence
$r=x^{\lambda}\otimes s(r)$.

Conversely, suppose that $\phi$ is equivariant. Since $X_{d}=x_{d}\otimes 1$,
we clearly have that $X_{d}\in R_{e_{d}}$. Similarly, condition 2 of
Definition 3.1.4 is trivial to check. We want to verify condition 3. Take
$r\in R_{\lambda}$, then $r\in x^{\lambda}\otimes A$, in particular
$r=x^{\lambda}\otimes s(r)$. Again, since $R=\bigcap_{d\in
D}\mathcal{V}_{d}{{}^{-1}}(\mathbb{Z}_{\geq 0}\cup\\{\infty\\})\bigcap
R_{\prod X_{D}}$ and $\mathcal{V}_{d_{1}}(X_{d_{2}})=\delta_{d_{1},d_{2}}$, we
deduce that $s(r)\in A_{\lambda}$. Moreover, if $s(r)=0$, then clearly $r=0.$
This proves that the restriction of $s$ to $R_{\lambda}$ is injective with
image in $A_{\lambda}$. For the surjectivity, notice that if $a\in
A_{\lambda}$, then $x^{\lambda}\otimes a\in R_{\lambda}$ and
$s(x^{\lambda}\otimes a)=a$.

For the unicity part, it’s sufficient to prove that if a torus action as in
the statement of the proposition exists, then it is unique. But this is clear
since $R$ is a domain, hence ${\operatorname{Spec}}(R_{\prod X_{D}})$ is
schematically dense in ${\operatorname{Spec}}(R)$. ∎

The following corollary is a useful reformulation of the above proposition.

######  Corollary 3.1.7.

A graduation $R=\bigoplus_{\lambda\in\mathbb{Z}^{D}}R_{\lambda}$ is the
lifting graduation if and only if, for any $r\in R_{\lambda}$, then
$\phi^{*}(r)=x^{\lambda}\otimes s(r)$.

######  Proposition 3.1.8.

If the lifting graduation $R=\bigoplus_{\lambda\in\mathbb{Z}^{D}}R_{\lambda}$
exists, then $\ker(s)=(X_{d}-1\,:\,d\in D).$

###### Proof.

Let $J=(X_{d}-1\,:\,d\in D)$. A standard calculation allows see that if
$\lambda\geq 0$, then $X^{\lambda}-1$ is in $J$.

We claim that, for any $r\in R$ such that $s(r)\neq 0$ and for any
sufficiently large $\nu$, $r-x^{\nu}\otimes s(r)$ is in $J$. Let’s see that
the claim allows to conclude. Indeed, take $y\in\ker(s)$ and any
$r_{\lambda}\in R_{\lambda}$ such that $s(r_{\lambda})\neq 0$. In particular,
$r_{\lambda}=x^{\lambda}\otimes s(r_{\lambda})$. Then for any $\nu$ we can
write

$y=(y+r_{\lambda})-r_{\lambda}=(y+r_{\lambda}-x^{\nu}\otimes
s(r_{\lambda}))+r_{\lambda}(x^{\nu-\lambda}\otimes 1-1\otimes 1).$

Observe that $s(y+r_{\lambda})=s(r_{\lambda})\neq 0$. In particular, if $\nu$
is sufficiently large, the claim implies that $(y+r_{\lambda}-x^{\nu}\otimes
s(r_{\lambda}))$ is in $J$, but if $\nu\geq\lambda$, we also have that
$r_{\lambda}(x^{\nu-\lambda}\otimes 1-1\otimes
1)=r_{\lambda}(X^{\nu-\lambda}-1)$ is in $J$.

Let’s prove the claim. Write $r=\sum r_{\mu}$, where $r_{\mu}\in R_{\mu}$ and
the sums runs over a finite set $M$, such that $r_{\mu}\neq 0$ if $\mu\in M$.
By the definition of lifting graduation we have that, for any $d\in D$,
$\mathcal{V}_{d}(1\otimes s(r_{\mu}))\geq-\mu(d)$. Then, if
$\nu\geq\max\\{\mu\,:\,\mu\in M\\}$, we have that for any $\mu\in M$,
$x^{\nu}\otimes s(r_{\mu})\in R_{\nu}$, hence $x^{\nu}\otimes s(r)\in
R_{\nu}$. Then, for any such $\nu$,

$r-x^{\nu}\otimes s(r)=\sum_{\mu}(x^{\mu}\otimes s(r_{\mu}))(1\otimes
1-x^{\nu-\mu}\otimes 1)=\sum r_{\mu}(1-X^{\nu-\mu}).$

Since for any $\mu$ in the above sum $\nu-\mu\geq 0$, this proves the claim. ∎

######  Proposition 3.1.9.

In the notation of Example 3.1.2, the $\mathbb{Z}^{D}$ graduation on
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ induced by
$\upharpoonleft\kern-3.50006pt0$ is the lifting graduation. If $t$ is a graded
seed, then the pole filtration is $H$-graded and the graduation induced by
$\upharpoonleft\kern-3.50006pt\sigma$, on
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$, is the
$H$-graded lifting graduation.

###### Proof.

For the first statement, we want to apply Corollary 3.1.7. Recall that
$\upharpoonleft\kern-3.50006pt0$ defines a graduation on
${\mathcal{L}}(\upharpoonleft\kern-3.50006ptt^{D})$, for which the cluster
monomials form a basis of homogeneous elements. Moreover,
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ is a graded
subring of ${\mathcal{L}}(\upharpoonleft\kern-3.50006ptt^{D})$. In particular,
it’s sufficient to prove that for any
$i\in\upharpoonleft\kern-3.50006pt\kern-1.49994ptI$,
$\phi^{*}(\upharpoonleft\kern-3.50006ptx_{i})={\widehat{x}}_{D}^{\upharpoonleft\kern-2.45004pt\kern
1.39998pt\,0_{i}}\otimes s(\upharpoonleft\kern-3.50006ptx_{i})$. But this is
obvious from the definitions.
For the second statement, notice that the same argument as before proves that
$s:\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D}){\,\longrightarrow\,}\overline{\mathcal{A}}(t)$
is $H$-equivariant. Using the first statement of the proposition, we have that
for any $\lambda\in\mathbb{Z}^{D}$,
$s:\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})_{\lambda,\bullet}{\,\longrightarrow\,}\overline{\mathcal{A}}(t)_{\lambda}$
is an isomorphism. It follows at once that the pole filtration is $H$-graded.
Then the second statement follows from Lemma 3.1.5.

∎

######  Corollary 3.1.10.

The deletion map
$s:\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D}){\,\longrightarrow\,}\overline{\mathcal{A}}(t)$
is surjective with kernel the ideal generated by
$\upharpoonleft\kern-3.50006ptx_{d}-1$, for $d\in D$.

###### Proof.

Apply Propositions 3.1.9 and 3.1.8. ∎

To the author it’s not clear if Corollary 3.1.10 can be deduced from Corollary
3.0.18.

######  Corollary 3.1.11.

Suppose that $D=D_{1}\sqcup D_{2}$. Denote by $\nu_{1}=\nu_{D_{1},\bullet}$,
$\upharpoonleft\kern-3.50006ptt\xlongleftarrow{\nu}t$ and
$\widetilde{\upharpoonleft\kern-3.50006ptt}\xlongleftarrow{\nu_{1}}t$. Then
$\overline{\mathcal{A}}(\widetilde{\upharpoonleft\kern-3.50006ptt}^{D_{1}})$
is the quotient of
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ by the ideal
$(\upharpoonleft\kern-3.50006ptx_{d}-1\,:d\in D_{2}).$

###### Proof.

By Lemma 3.0.5, we have that if $\nu_{2}=\nu_{D_{2},\bullet}$, then
$\upharpoonleft\kern-3.50006ptt^{D_{1}}\xleftarrow{(\nu_{2},0)}\widetilde{\upharpoonleft\kern-3.50006ptt}^{D_{1}}.$
The statement follows from Corollary 3.1.10. ∎

##  4 The minimal monomial lifting

We use the following notation. If $Z$ is a $\mathbb{C}$-scheme and
$f\in{\mathcal{O}}_{Z}(Z)$, we denote with $V(f)$ the zero locus of $f$ in
$Z$, which is a closed subspace of the topological space underlying $Z$. We
write $Z^{(1)}$ for the set of codimension one points of $Z$. If $Z$ is normal
and integral, for $p\in Z^{(1)}$, we denote with
$\mathcal{V}_{p}:\mathbb{C}(Z):{\,\longrightarrow\,}\mathbb{Z}\cup\\{\infty\\}$
the valuation induced, on the fraction field of $Z$, by the discrete valuation
ring ${\mathcal{O}}_{Z,p}$. If $D\subseteq Z$ is closed, irreducible and of
codimension one, it has a unique generic point $p$. Then we set
$\mathcal{V}_{D}=\mathcal{V}_{p}$ and we say that $\mathcal{V}_{p}$ is the
valuation associated to $D$. We say that $f,g\in{\mathcal{O}}_{Z}(Z)$ are
coprime on $Z$ if $V(f)\cap V(g)\cap Z^{(1)}=\emptyset$. Finally, we say that
an open subset $\Omega\subseteq Z$ is big if $\Omega^{(1)}=Z^{(1)}.$

From now on, $D$ is a finite set and $T$ is torus of rank $|D|.$

######  Definition 4.0.1.

A suitable for $D$-lifting scheme is a triple $({\mathfrak{X}},\phi,X)$ such
that:

  1. 1.

${\mathfrak{X}}$ is a noetherian, normal, integral $\mathbb{C}$-scheme.

  2. 2.

$X\subseteq{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})^{D}$ is a collection
of global sections such that, for each $d\in D$, $V(X_{d})$ is irreducible and
if $\mathcal{V}_{d}$ is the associated valuation, then
$\mathcal{V}_{d_{1}}(X_{d_{2}})=\delta_{d_{1},d_{2}}.$

  3. 3.

$\phi:T\times Y{\,\longrightarrow\,}{\mathfrak{X}}\setminus\cup_{d}V(X_{d})$
is an open embedding such that the image is big and
$\phi^{*}(X_{d})=(x_{d}\otimes 1)$, where the collection of $x_{d}$ is a base
of $X(T)$. Here $Y$ is an irreducible $\mathbb{C}$-scheme.

When the data of $\phi$ and $X$ is clear, we just say that ${\mathfrak{X}}$ is
suitable for $D$-lifting. If a scheme suitable for $D$-lifting is called
${\mathfrak{X}}$, we implicitly assume that the rest of the data defining it
is denoted by $\phi$, $Y$ and $X$ and that $\mathcal{V}_{d}$ is the valuation
associated to $V(X_{d}).$

We use the following convention. If ${\mathfrak{X}}$ is suitable for
$D$-lifting, then we identify $T\times Y$ with an open subset of
${\mathfrak{X}}$ using $\phi$. Hence, we write
$\mathbb{C}({\mathfrak{X}})=\mathbb{C}(T\times Y)$ and we identify
$\mathbb{C}(Y)$ with
$1\otimes\mathbb{C}(Y)\subseteq\mathbb{C}({\mathfrak{X}})$. Then,
$\mathbb{C}({\mathfrak{X}})$ is canonically a $D$\- field extension of
$\mathbb{C}(Y)$: its $D$-uple is $X$. We stress that, with these
identifications, if $t$ is a seed of $\mathbb{C}(Y)$ such that
$\overline{\mathcal{A}}(t)={\mathcal{O}}_{Y}(Y)$, then
${\widehat{x}}\subseteq(\mathbb{C}({\mathfrak{X}})^{*})^{I\sqcup D}$ is
defined by

${\widehat{x}}_{i}=\begin{cases}1\otimes x_{i}&\text{if}\quad i\in I\\\
X_{d}&\text{if}\quad i=d\in D\end{cases}$

We can construct affine schemes which are suitable for lifting as follows

######  Example 4.0.2.

Let $R$ be a finite type $\mathbb{C}$-algebra which is a normal domain.
Consider a collection of elements $X\subseteq R^{D}$ such that the ideals
$(X_{d})$ are prime and pairwise different. Suppose that there is an
isomorphism $\psi:R_{\prod_{d}X_{d}}{\,\longrightarrow\,}\mathbb{C}[T]\otimes
A$ such that $\psi(X_{d})=x_{d}\otimes 1$, where the $x_{d}$ form a base of
$X(T)$. Here $A$ is a $\mathbb{C}$-algebra. Then
$({\operatorname{Spec}}(R),\psi^{*},X)$ is suitable for $D$-lifting.

We now state the main theorem of this section.

######  Theorem 4.0.3.

Let ${\mathfrak{X}}$ be a suitable for $D$-lifting scheme and $t$ be a maximal
rank seed of $\mathbb{C}(Y)$ such that
$\overline{\mathcal{A}}(t)={\mathcal{O}}_{Y}(Y).$ Suppose that, for any non-
equal $i,j\in I_{uf}$, $x_{i}$ and $x_{j}$ are coprime on $Y$ and that, for
any $k\in I_{uf}$, $x_{k}$ and $x_{k}^{\prime}=\mu_{k}(x_{k})$ are coprime on
$Y$.

Consider $(\mathbb{C}({\mathfrak{X}}),\nu)$: the $D$-lifting configuration on
$t$ defined by

$\nu_{d,i}:=-\mathcal{V}_{d}(1\otimes x_{i})$

Then

  1. 1.

$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)={\mathcal{O}}_{{\mathfrak{X}}}(T\times
Y)$.

  2. 2.

$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})\subseteq{\mathcal{O}}_{\mathfrak{X}}(X).$

######  Definition 4.0.4.

In the setting of Theorem 4.0.3, $\nu$ and
$\upharpoonleft\kern-3.50006ptt^{D}$ are called respectively the minimal
lifting matrix and the minimal monomial lifting of $t$, associated to
${\mathfrak{X}}.$

The proof is based on the well known algebraic Hartogs’ lemma

######  Lemma 4.0.5 (Hartogs’).

Let $Z$ be a locally noetherian normal integral scheme, then
${\mathcal{O}}_{Z}(Z)=\cap_{p\in Z^{(1)}}{\mathcal{O}}_{Z,p}$, where the
intersection takes place in $\mathbb{C}(Z)$.

Note that ${\mathcal{O}}_{Z,p}$ consists of the rational functions with non-
negative valuation on $p$.

For the proof, we need the following, which is usually called starfish lemma.
Notice that the proof is exactly the same of the affine case [FP16,
Proposition 3.6].

######  Lemma 4.0.6 (starfish lemma).

Let $Z$ be a locally noetherian normal integral scheme over $\mathbb{C}$ and
$t$ be a seed of $\mathbb{C}(Z)$ such that

  1. 1.

For any $i\in I$, $x_{i}\in{\mathcal{O}}_{Z}(Z)$ and if $i\in I_{sf}$,
$x_{i}\in{\mathcal{O}}_{Z}(Z)^{*}.$

  2. 2.

For any $k\in I_{uf}$,
$x_{k}^{\prime}:=\mu_{k}(x_{k})\in{\mathcal{O}}_{Z}(Z)$.

  3. 3.

For any non equal $i,j\in I_{uf}$, $x_{i}$ and $x_{j}$ are coprime on $Z$ and
for any $i\in I_{uf}$, $x_{i}$ and $x_{i}^{\prime}$ are coprime on $Z$.

Then $\overline{\mathcal{A}}(t)\subseteq{\mathcal{O}}_{Z}(Z).$

###### Proof.

Let $f\in\overline{\mathcal{A}}(t)$ and $p\in Z^{(1)}$, since
$f\in{\mathcal{L}}(t)$ and for any $i\in I_{sf}$
$x_{i}\in{\mathcal{O}}_{Z}(Z)^{*}$, if there is no $k\in I_{uf}$ such that
$x_{k}(p)=0$, then $f\in{\mathcal{O}}_{Z,p}$. If it’s not the case, take $k\in
I_{uf}$ such that $x_{k}(p)=0$. Then, for any $j\in I_{uf}\setminus\\{k\\}$,
$x_{j}(p)\neq 0$ and $x_{k}^{\prime}(p)\neq 0$. But
$f\in{\mathcal{L}}(\mu_{k}(t))$, hence we deduce that
$f\in{\mathcal{O}}_{Z,p}$. We conclude using Lemma 4.0.5. Actually, this
proves the stronger statement that the upper bound $\mathcal{U}(t)$ is
included in ${\mathcal{O}}_{Z}(Z).$ ∎

###### Proof of Theorem 4.0.3.

First, if $e\in T(\mathbb{C})$ is the identity, identifying $Y$ with
$\\{e\\}\times Y$ gives an immersion (neither closed nor open)
$\iota:Y{\,\longrightarrow\,}{\mathfrak{X}}$. Since $\iota$ is a monomorphism
and ${\mathfrak{X}}$ is quasi-separated, being noetherian, it follows that $Y$
is quasi-separated. Moreover, ${\mathfrak{X}}$ is locally noetherian and
$\iota$ is an immersion, hence $\iota$ is quasi-compact. In particular, $Y$ is
quasi-compact since ${\mathfrak{X}}$ is. Then, we can cover $Y$ with finitely
many affine open subsets $Y_{i}$. Since $Y$ is quasi-separated, we can cover
each intersection $Y_{i}\cap Y_{j}$ with finitely many affine open subsets
$Y_{i,j,k}$. Consider the natural exact sequence

${0}$${{{\mathcal{O}}_{Y}(Y)}}$${{\displaystyle\prod_{i}{\mathcal{O}}_{Y}(Y_{i})}}$${{\displaystyle\prod_{i,j,k}{\mathcal{O}}_{Y}(Y_{i,j,k})}}$

Note that the two products in the exact sequence are finite. Since
$\mathbb{C}[T]\otimes(-)$ is exact, commutes with finite products and the
$Y_{i},Y_{i,j,k}$ are affine, tensoring with $\mathbb{C}[T]$ yields the exact
sequence

${0}$${{\mathbb{C}[T]\otimes{\mathcal{O}}_{Y}(Y)}}$${{\prod_{i}{\mathcal{O}}_{T\times
Y}(T\times Y_{i})}}$${{\prod_{i,j,k}{\mathcal{O}}_{T\times Y}(T\times
Y_{i,j,k})}}$

But since the $T\times Y_{i}$ cover $T\times Y$, and the $T\times Y_{i,j,k}$
cover $(T\times Y_{i})\cap(T\times Y_{j})$, we deduce from the above exact
sequence that
$\mathbb{C}[T]\otimes{\mathcal{O}}_{Y}(Y)\simeq{\mathcal{O}}_{T\times
Y}(T\times Y)$. Finally, we deduce the first statement from Theorem 3.0.15 and
the fact that $\phi$ is an open embedding.

For the second statement, we want to apply the starfish lemma. Let
$p\in{\mathfrak{X}}^{(1)}$. Since ${\mathfrak{X}}$ is suitable for lifting,
$p\in(T\times Y)^{(1)}$ or $\overline{p}=V(X_{d})$ for a certain $d\in D$. In
the last case, $d$ is unique. Note that for any $m\in\mathbb{Z}^{D}$, then

$\mathcal{V}_{p}({\widehat{x}}_{D}^{m})=\begin{cases}0&\text{if}\quad
p\in(T\times Y)^{(1)}\\\
m_{d}&\text{if}\quad\overline{p}=V(X_{d}).\end{cases}$

Then we easily compute that, for any $i\in I$,

$\mathcal{V}_{p}(\upharpoonleft\kern-3.50006ptx_{i})=\begin{cases}\mathcal{V}_{p}(1\otimes
x_{i})&\text{if}\quad p\in(T\times Y)^{(1)}\\\
0&\text{if}\quad\overline{p}=V(X_{d}).\end{cases}$ (24)

It follows from the Hartogs’ lemma that
$\upharpoonleft\kern-3.50006ptx_{i}\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$.
Moreover, if $i\in\upharpoonleft\kern-3.50006ptI_{sf}=I_{sf}$ then
$\upharpoonleft\kern-3.50006ptx_{i}\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})^{*}$
and clearly
$\upharpoonleft\kern-3.50006ptx_{d}=X_{d}\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}}).$
Now fix $i,j\in I_{uf}=\upharpoonleft\kern-3.50006ptI^{D}_{uf}$ with $i\neq
j$, we want to prove that $\upharpoonleft\kern-3.50006ptx_{i}$ and
$\upharpoonleft\kern-3.50006ptx_{j}$ are coprime on $X$. By the previous
calculation, it’s sufficient to prove that they’re coprime on $T\times Y$,
which is equivalent to the comprimality of $1\otimes x_{i}$ and $1\otimes
x_{j}$ because, for any $d\in D$,
$X_{d}\in{\mathcal{O}}_{{\mathfrak{X}}}(T\times Y)^{*}$. Take a point $p\in
T\times Y$ such that $1\otimes x_{i}(p)=0=1\otimes x_{j}(p).$ Take an affine
open subset $U\simeq{\operatorname{Spec}}A$, of $Y$, such that $p\in T\times
U$. Note that $A$ is a normal domain. Then we can identify $p$ with a prime of
$\mathbb{C}[T]\otimes A$ which contains $1\otimes x_{i}$ and $1\otimes x_{j}$.
If we look at the natural map $A{\,\longrightarrow\,}\mathbb{C}[T]\otimes A$
that sends $a$ to $1\otimes a$, we have that $p^{c}$ is a prime ideal of $A$
that contains $x_{i}$ and $x_{j}$ ($(-)^{c}$ denotes the contraction of
ideals). Since $x_{i}$ and $x_{j}$ are coprime on $Y$, there exists a non-zero
prime $q$ of $A$ such that $q$ is strictly contained in $p^{c}$. Since
$\mathbb{C}$ is algebraically closed and $\mathbb{C}[T]$ is a finitely
generated $\mathbb{C}$-algebra which is a domain, then for any
$\mathbb{C}$-algebra $B$ which is a domain, $\mathbb{C}[T]\otimes B$ is again
a domain. In particular, tensoring with $\mathbb{C}[T]$ sends prime ideals of
$A$ to prime ideals of $\mathbb{C}[T]\otimes A$ and it preserves strict
inclusion because of exactness. It follows that
$0\subsetneq\mathbb{C}[T]\otimes q\subsetneq\mathbb{C}[T]\otimes
p^{c}\subseteq p$. Hence $p$ has height at least 2, so $1\otimes x_{i}$ and
$1\otimes x_{j}$ are comprime on $Y$.

Finally, observe that for any $k\in I_{uf}$,
$x_{k}^{*}:=\mu_{k}(\upharpoonleft\kern-3.50006ptx_{k})$ is regular on
$T\times Y$ because of Lemma 3.0.8. Moreover, from the exchange relation (3)
and the fact that, for any $d\in D$,
$\mathcal{V}_{d}(\upharpoonleft\kern-3.50006ptx_{k})=0$, it follows that
$\mathcal{V}_{d}(x_{k}^{*})\geq 0$. Hence $x_{k}^{*}\in{\mathcal{O}}_{X}(X)$
by Hartogs’ Lemma 4.0.5. Finally, call $t^{\prime}=\mu_{k}(t)$ and
$\nu^{\prime}=\mu_{k}(\nu)$. Since $x_{k}^{*}=(1\otimes
x_{k}^{\prime}){\widehat{x}}^{\nu^{\prime}_{\bullet,k}}$ by Lemma 3.0.8, the
previous argument also proves that $x_{k}^{*}$ and
$\upharpoonleft\kern-3.50006ptx_{k}$ are coprime on $X$. Then,
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})\subseteq{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$
follows from Lemma 4.0.6. ∎

######  Corollary 4.0.7 (Of the proof).

If ${\mathfrak{X}}$ is suitable for $D$-lifting, then
$({\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}}),\mathcal{V},X,\phi^{*})$ is an
almost polynomial ring over ${\mathcal{O}}_{Y}(Y)$.

The following lemma allows to produce a nice class of suitable for $D$-lifting
schemes.

######  Lemma 4.0.8.

Let $({\mathfrak{X}},\phi,X)$ be a triple that satisfies the first two
conditions of Definition 4.0.1 and such that $\phi:T\times
Y{\,\longrightarrow\,}{\mathfrak{X}}\setminus\cup_{d}V(X_{d})$ is an open
embedding with big image. If moreover $Y$ is an irreducible
$\mathbb{C}$-scheme such that ${\mathcal{O}}_{Y}(Y)^{*}=\mathbb{C}^{*}$, and
there exist a point $y\in Y$ such that, for any $d\in D$
$\phi^{*}(X_{d})(e,y)=1$, then $({\mathfrak{X}},\phi,X)$ is suitable for
$D$-lifting.

###### Proof.

We just need to prove that, for any $d\in D$, $\phi^{*}(X_{d})=x_{d}\otimes 1$
and the collection of $x_{d}$ is a base of $X(T)$.

The argument at the beginning of the proof of Theorem 4.0.3 implies that
${\mathcal{O}}_{\mathfrak{X}}(T\times
Y)=\mathbb{C}[T]\otimes{\mathcal{O}}_{Y}(Y)$. Since ${\mathcal{O}}_{Y}(Y)$ is
a domain and ${\mathcal{O}}_{Y}(Y)^{*}=\mathbb{C}^{*}$, it follows that

${\mathcal{O}}_{\mathfrak{X}}(T\times Y)^{*}=\\{\lambda\otimes
c\,:\,\lambda\in X(T),c\in\mathbb{C}^{*}\\}.$

But $\phi^{*}(X_{d})\in{\mathcal{O}}_{\mathfrak{X}}(T\times Y)^{*}$ by
Hartogs’ lemma, moreover $\phi^{*}(X_{d})(e,y)=1$. Hence there exist $x_{d}\in
X(T)$ such that $\phi^{*}(X_{d})=x_{d}\otimes 1$. Let $\lambda\in X(T)^{D}$ be
a base of $X(T)$ and ${\operatorname{Div}}\in\mathbb{Z}^{D\times D}$ be
defined by

${\operatorname{Div}}_{d_{1},d_{2}}=\mathcal{V}_{d_{1}}(\lambda_{d_{2}}\otimes
1).$

Since $\lambda$ is a base of $X(T)$, there exists $A\in\mathbb{Z}^{D\times D}$
such that $\lambda^{A}=(x_{d})_{d}$. Hence ${\operatorname{Div}}\cdot
A={\operatorname{Id}}$. This implies that $(x_{d})_{d}$ is a base of $X(T).$

∎

From now on we suppose to be in the setting of Theorem 4.0.3. In particular,
${\mathfrak{X}}$, $t$ are fixed, $\nu$, $\upharpoonleft\kern-3.50006ptt^{D}$
denote respectively the minimal lifting matrix and the minimal monomial
lifting of $t$ associated to ${\mathfrak{X}}$.

The hypothesis of coprimality of Theorem 4.0.3 often holds. For example:

######  Lemma 4.0.9.

If $Y$ is affine and ${\mathcal{O}}_{Y}(Y)$ is factorial then, for any $i,j\in
I_{uf}$ with $i\neq j$, $x_{i}$ and $x_{j}$ are coprime on $Y$. Moreover, for
any $k\in I_{uf}$, $x_{k}$ and $x_{k}^{\prime}:=\mu_{k}(x_{k})$ are coprime on
$Y$.

###### Proof.

No mutable vertex of $t$ is completely disconnected because $t$ is of maximal
rank. Then, by Theorem 2.1.8, any non semi-frozen cluster variable $z$ is
irreducible in ${\mathcal{O}}_{Y}(Y)$, hence $(z)$ is prime. In particular,
two cluster variables $z$ and $z^{\prime}$ are coprime if and only if
$(z)\neq(z^{\prime})$. The lemma follows from the second part of Theorem
2.1.8. ∎

To the author’s best knowledge, no general coprimality criteria for cluster
variables in an upper cluster algebra is known. In [CKQ22], the authors
develop some usefull criteria to study factoriality in upper cluster algebras
of maximal rank.

The minimal monomial lifting is the only possible candidate to give a cluster
structure on ${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ compatible with
the one on ${\mathcal{O}}_{Y}(Y)$ in the following sense.

######  Theorem 4.0.10.

Let $\widetilde{t}$ be a seed of $\mathbb{C}({\mathfrak{X}})$,
$\mu\in\mathbb{Z}^{D\times I}$ and $\lambda\in\mathbb{Z}^{D\times I_{uf}}$
such that:

  1. 1.

$\widetilde{t}$ is a $D$-pointed field field extension of $t$.

  2. 2.

For any $i\in I$,
$\widetilde{x}_{i}=x_{i}\,{\widehat{x}}^{\mu_{\bullet,i}}_{D}.$

  3. 3.

For any $k\in I_{uf}$,
$\mu_{k}(\widetilde{x}_{k})=\mu_{k}(x_{k})\,{\widehat{x}}^{\lambda_{\bullet,k}}_{D}$

If
$\overline{\mathcal{A}}(\widetilde{t})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$,
then $\mu=\nu$ and $\widetilde{t}=\upharpoonleft\kern-3.50006ptt^{D}.$

###### Proof.

For any $d\in D$, $X_{d}\not\in{\mathcal{O}}_{X}(X)^{*}$. It follows that
$D\subseteq\widetilde{I}_{hf}$. We claim that, for any $d\in D$ and $i\in I$,
$\mathcal{V}_{d}(\widetilde{x}_{i})=0.$ This implies that $\mu=\nu$ because

$\mathcal{V}_{d}(\widetilde{x}_{i})=\mathcal{V}_{d}(x_{i}\otimes
1)+\mu_{d,i}=-\nu_{d,i}+\mu_{d,i}.$

If $i\in I_{sf}$, since $I_{sf}\subseteq\widetilde{I}_{sf}$, then
$\widetilde{x}_{i}\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})^{*}$. Then
the claim follows from the Hartogs’ lemma. Similarly, if $i\not\in I_{sf}$,
hence $i\not\in\widetilde{I}_{sf}$, then $\widetilde{x}_{i}$ is irreducible in
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$. Then the claim follows from
Hartogs’ lemma and the second hypothesis in Definition 4.0.1. For the final
statement, it’s sufficient to apply Proposition 3.0.9 to $\widetilde{t}_{D}$.
∎

######  Corollary 4.0.11.

Let $k\in I_{uf},$ $t^{\prime}=\mu_{k}(t)$ and $\nu^{\prime}=\mu_{k}(\nu)$. If
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$,
then $\nu^{\prime}$ and
$\mu_{k}((\upharpoonleft\kern-3.50006ptt^{\prime})^{D})$ are respectively the
minimal lifting matrix and the minimal monomial lifting of $t^{\prime}$
associated to ${\mathfrak{X}}$.

###### Proof.

Because of statement 5 of Lemma 2.4.2 and Lemma 3.0.8, we can apply the
previous theorem with
$\widetilde{t}=\mu_{k}(\upharpoonleft\kern-3.50006ptt^{D})$ (considered as an
extension of $t^{\prime}$), $\mu=\nu^{\prime}$ and
$\lambda_{\bullet,i}=\mu_{i}(\nu^{\prime})_{\bullet,i}$ ∎

We expect the answer to the following question to be negative, in general,
without the assumption that
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$.

######  Question 4.0.12.

If $k\in I_{uf}$, is $\mu_{k}(\nu)$ the minimal lifting matrix of $\mu_{k}(t)$
associated to ${\mathfrak{X}}$?

It’s interesting to notice that, reinterpreting the results of [Fei21] in this
light, as it is done in Section 8.2, one can construct an example where
$\mathcal{A}(t)=\overline{\mathcal{A}}(t)$,
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$
but
$\mathcal{A}(\upharpoonleft\kern-3.50006ptt^{D})\neq\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D}).$
See [Fei21][Example 8.3].

###  4.1 Equality conditions

In this section we suppose to be in the setting of Theorem 4.0.3. In
particular, ${\mathfrak{X}}$, $t$ are fixed, $\nu$ and
$\upharpoonleft\kern-3.50006ptt^{D}$ denote the minimal lifting matrix and the
minimal monomial lifting of $t$, associated to ${\mathfrak{X}}$. It’s easy to
see that we don’t always have equality between
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ and
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$. Because of Theorem 4.0.3, the
lack of equality should be caused by some bad behaviour along the divisors
$V(X_{d}).$

######  Example 4.1.1.

We make an affine example in the spirit of Example 4.0.2. Consider
$R=\mathbb{C}[y,z,X,\frac{y+z}{X}]$ where $y,z$ and $X$ are abstract
independent variables. Notice that $R$ is a polynomial ring in the variables
$y,X,\frac{y+z}{X}$ and we have an isomorphism
$R_{X}\simeq\mathbb{C}[y,z]\otimes\mathbb{C}[X^{\pm 1}]$. The ring
$\mathbb{C}[y,z]$ is the upper cluster algebra of the seed $t$ which has only
highly-frozen vertices and has $y,z$ as cluster variables. Here $D$ consists
of one element associated to the prime ideal generated by $X$. Applying the
minimal monomial lifting we get
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})=\mathbb{C}[y,z,X]$
which is not equal to $R$. In this very simple example we clearly see the
problem: the cluster valuation of $y+z$ along the new frozen vertex is zero.
If we had equality between
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})$ and $R$, it
should be one.

######  Lemma 4.1.2.

For any $f\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ and $d\in D$, we
have $\mathcal{V}_{d}(f)\geq\mathcal{C}\mathcal{V}_{d}(f)$. Here
$\mathcal{C}\mathcal{V}_{d}$ is the cluster valuation defined in 2.3.1.

###### Proof.

It’s clear from (24) that, for any
$i\in\,\upharpoonleft\kern-3.50006pt\kern-1.00006ptI$ and $d\in D$,
$\mathcal{V}_{d}(\upharpoonleft\kern-3.50006ptx_{i})=\mathcal{C}\mathcal{V}_{d}(\upharpoonleft\kern-3.50006ptx_{i}).$
Since $f\in\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)$ by Theorem
4.0.3, then we can write $f=\frac{P}{M}$ where $P$ is a polynomial in the
cluster variables of $\upharpoonleft\kern-3.50006ptx$, which is not divisible
by any unfrozen or semi-frozen variable of $\upharpoonleft\kern-3.50006ptt$,
and

$M=\displaystyle\prod_{i\in I_{uf}\sqcup I_{sf}\sqcup
D}\upharpoonleft\kern-3.50006ptx_{i}^{m_{i}}$

is a Laurent monomial. Note that, for any $d\in D$, we have
$\mathcal{C}\mathcal{V}_{d}(P)=0$. Moreover, since
$P\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ because of Theorem 4.0.3
and the Laurent phenomenon, then $\mathcal{V}_{d}(P)\geq 0.$ Then

$\mathcal{C}\mathcal{V}_{d}(f)=\mathcal{C}\mathcal{V}_{d}(P)-\mathcal{C}\mathcal{V}_{d}(M)=-m_{d}=-\mathcal{V}_{d}(M)\leq\mathcal{V}_{d}(P)-\mathcal{V}_{d}(M)=\mathcal{V}_{d}(f).$

∎

######  Proposition 4.1.3.

Suppose that $t^{*}\in\Delta(\upharpoonleft\kern-3.50006ptt)$. The following
are equivalent

  1. 1.

$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}}).$

  2. 2.

For any $d\in D$,
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})\subseteq{\mathcal{L}}((t^{*})^{\\{d\\}}\bigr{)}.$

  3. 3.

For any $d\in D$,
$\mathcal{C}\mathcal{V}_{d}({\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}}))\subseteq\mathbb{N}\cup\\{\infty\\}$.

  4. 4.

For any $d\in D$, $\mathcal{C}\mathcal{V}_{d}\geq\mathcal{V}_{d}$ over
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}}).$

  5. 5.

For any $d\in D$, $\mathcal{C}\mathcal{V}_{d}=\mathcal{V}_{d}$ over
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}}).$

###### Proof.

Condition 1 implies 2 and 2 implies 3 are obvious. For 3 implies 4, suppose
there exists $f\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ and $d\in D$
such that $n_{d}=\mathcal{V}_{d}(f)>\mathcal{C}\mathcal{V}_{d}(f)$. Then
$\frac{f}{X_{d}^{n_{d}}}\in{\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ and
$\mathcal{C}\mathcal{V}_{d}(f)<0$, which is a contradiction. Then 4 implies 5
because of Lemma 4.1.2. Finally, 5 implies that
$\mathcal{C}\mathcal{V}_{d}=\mathcal{V}_{d}$ holds over the fraction field of
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$, that we denote by
$\mathbb{F}$. By Theorem 4.0.3, $\mathbb{F}$ is also the fraction field of
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)$. Then Hartogs’ lemma
and Theorem 4.0.3 imply that
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})=\bigcap_{d\in
D}\mathcal{V}_{d}{{}^{-1}}(\mathbb{Z}_{\geq
0}\cup\\{\infty\\})\bigcap\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt)$.
Hence, using part 8 of Lemma 2.4.2, we deduce that 5 implies 1. ∎

######  Corollary 4.1.4 (of the proof).

Conditions 2 to 5 of Proposition 4.1.3 are equivalent at $d$ fixed, for any
$d\in D$.

######  Proposition 4.1.5.

Let $D^{\prime}\subseteq D$, then
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})\subseteq\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D^{\prime}})$
if and only if
$D^{\prime}\subseteq\\{d\,:\,\mathcal{V}_{d}=\mathcal{C}\mathcal{V}_{d}\\}.$

###### Proof.

Note that
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})\subseteq\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D^{\prime}})$
if and only if for any $t^{*}\in\Delta(\upharpoonleft\kern-3.50006ptt)$ and
$d\in D^{\prime}$,
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})\subseteq{\mathcal{L}}((t^{*})^{\\{d\\}})$.
Then the statement follows from Corollary 4.1.4. ∎

Observe that ${\mathcal{O}}_{Y}(Y)$ has two pole filtrations: one coming from
${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ and the other one coming from
the standard almost polynomial structure of
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D}).$ The one coming
from ${\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$ may not admit a lifting
graduation.

######  Corollary 4.1.6.

If
$\overline{\mathcal{A}}(\upharpoonleft\kern-3.50006ptt^{D})={\mathcal{O}}_{\mathfrak{X}}({\mathfrak{X}})$,
then the two pole filtrations on ${\mathcal{O}}_{Y}(Y)$ coincide. In