# Affine vs. Stein in rigid geometry

Marco Maculan<EMAIL_ADDRESS>Institut de Mathématiques de Jussieu,
Sorbonne Université, 4 place Jussieu, F-75252 Paris  and  Jérôme Poineau
<EMAIL_ADDRESS>Laboratoire de mathématiques Nicolas Oresme,
Université de Caen Normandie, BP 5186, F-14032 Caen Cedex

###### Abstract.

We investigate the relationship between affine and Stein varieties in the
context of rigid geometry. We show that the two concepts are much more closely
related than in complex geometry, _e.g._ they are equivalent for surfaces.
This rests on the density of algebraic functions in analytic functions. One
key ingredient to prove such a density statement is an extension result for
Cartier divisors.

##  0\. Introduction

###  0.1. Motivation

Let $K$ be a complete non-trivially valued non-Archimedean field. A variety
$X$ over $K$, that is, a separated $K$-scheme of finite type, is Stein if the
$K$-analytic space $X^{\textup{an}}$ attached to $X$ admits a closed
$K$-analytic embedding in $\mathbb{A}^{n,\textup{an}}_{K}$ for some
$n\in\mathbb{N}$. Needless to say, affine varieties are Stein. The aim of the
present note is to investigate to which extent the converse statement holds
true. The complex analytic version of the question is a classical problem,
whose history (recalled below) consists mainly in negative results. For
instance, given an integer $d\geqslant 2$, there is a $d$-dimensional smooth
complex variety that is not affine but whose associated complex manifold is
Stein. In stark contrast, and breaking the persisting analogy with complex
geometry, we derive several general consequences of the Stein property for
non-Archimedean varieties summarized as follows (see theorems 8.1, LABEL:,
8.2, LABEL: and 8.3):

######  Theorem A.

A Stein variety $X$ is quasi-affine and, for any closed subscheme $Y$ of
dimension $\leqslant 1$, the restriction map
$\Gamma(X,\mathcal{O}_{X})\to\Gamma(Y,\mathcal{O}_{Y})$ is surjective.
Furthermore, if the ring $\Gamma(X,\mathcal{O}_{X})$ is Noetherian, then $X$
is affine.

Recall that a variety $X$ is quasi-affine if it admits an open embedding into
an affine variety. The $K$-algebra of regular functions on an algebraic group
is finitely generated [DG70, III.3.8]. Therefore as a consequence we obtain a
new proof of the following result previously proved in [Mac22]:

######  Corollary.

An algebraic group is Stein if and only if it is affine.

Also, a result of Goodman-Landman (see [GL73, proposition 3.13], or
proposition 7.12) permits to derive the equivalence in the case of surfaces
(see corollary 8.4):

######  Corollary.

An algebraic surface is Stein if and only if it is affine.

###  0.2. Complex background

As we are about to remind, the complex analogue of the previous statements is
false—not to throw too fine a pun in it, the behaviour in non-Archimedean
geometry is far more rigid. Let $M_{\textup{dR}}(E)$ be the moduli space of
line bundles endowed with a connection on a complex elliptic curve $E$. The
line bundles in question are necessarily of degree $0$, and forgetting the
connection gives rise to a surjective map of complex varieties $\pi\colon
M_{\textup{dR}}(E)\to\operatorname{Pic}^{0}(E)=E$. Tensor product of line
bundles makes $M_{\textup{dR}}(E)$ an algebraic group and $\pi$ a morphism of
algebraic groups. Its kernel is identified with the space of differential
forms on $E$:

$0\longrightarrow\mathbb{V}(\Gamma(E,\Omega^{1}_{E}))\longrightarrow
M_{\textup{dR}}(E)\longrightarrow E\longrightarrow 0.$

The algebraic group $M_{\textup{dR}}(E)$ is _not_ affine—the quotient of an
affine group by a subgroup is affine, whereas the elliptic curve $E$ is most
certainly not. Actually, more is true: every algebraic function on
$M_{\textup{dR}}(E)$ is constant (see [Bri09, proposition 2.3]). From the
transcendental perspective, Riemann-Hilbert’s correspondence sets up a
biholomorphism of $M_{\textup{dR}}(E)^{\textup{an}}$ with the group of
characters of the topological fundamental group of the elliptic curve

$\operatorname{Hom}_{\mathbb{Z}}(\pi_{1}(E(\mathbb{C}),0),\mathbb{C}^{\times})\cong\mathbb{C}^{\times}\times\mathbb{C}^{\times}.$

In particular, $M_{\textup{dR}}(E)^{\textup{an}}$ admits a closed holomorphic
embedding in $\mathbb{C}^{3}$. Summing up, the complex algebraic group
$M_{\textup{dR}}(E)$ is not affine, but the complex Lie group
$M_{\textup{dR}}(E)^{\textup{an}}$ is Stein. Since it is of dimension $2$,
this shows that the analogue of both corollaries to A does not hold over the
complex numbers.111Hartshorne attributes this example to Serre (see [Har70,
Chapter VI, Example 3.2]), but it seems that such a phenomenon was known
earlier by Severi and Conforto.

As remarked by Neeman [Nee88], a complex quasi-affine variety $X$ is affine if
and only if $X^{\textup{an}}$ is Stein and the $\mathbb{C}$-algebra
$\Gamma(X,\mathcal{O}_{X})$ is finitely generated. The equivalence has been
strengthened by Brenner [Bre02] (see corollary 7.21), replacing the finite
generation of $\Gamma(X,\mathcal{O}_{X})$ by being Noetherian. Keeping the
notation of the previous paragraph, let $H$ be an ample line bundle on $E$.
Consider the $\mathbb{G}_{m}$-principal bundle $X$ on $M:=M_{\textup{dR}}(E)$
associated with the line bundle $\pi^{\ast}H^{\vee}$ dual to the pull-back of
$H$ along $\pi\colon M\to E$. Ampleness of $H$ implies that $X$ is quasi-
affine. Since $M^{\textup{an}}$ is Stein, the $\mathbb{C}^{\times}$-bundle
$X^{\textup{an}}$ is also Stein [MM60]. However, the complex variety $X$
cannot be affine, as it would imply that $M$ is. It follows that the ring
$\Gamma(X,\mathcal{O}_{X})$ is not Noetherian.222This can be proved by purely
algebraic means ([Bri09, theorem 3.9]).

###  0.3. Main results

Over the non-Archimedean field $K$ the technical crux leading to A is the
following (see theorem 6.1):

######  Theorem B.

Let $F$ be a semi-reflexive coherent sheaf on a variety $X$. Then
$\Gamma(X,F)$ is dense in $\Gamma(X^{\textup{an}},F^{\textup{an}})$.

A series of comments:

  1. (1)

Within the framework of Berkovich spaces, the topological space underlying the
$K$-analytic space $X^{\textup{an}}$ is locally compact: when $X$ is reduced,
the topology on $\Gamma(X^{\textup{an}},\mathcal{O}_{X}^{\textup{an}})$ is
that of uniform convergence on compact subsets; when $X$ is not reduced, as in
complex analysis, the topology is finer.

  2. (2)

An $\mathcal{O}_{X}$-module $F$ is _semi-reflexive_ if the natural
homomorphism $F\to F^{\vee\vee}$ is injective. Therefore the previous
statement applies in particular to vector bundles and coherent sheaves of
ideals. However, we ignore whether it stays true for any coherent
$\mathcal{O}_{X}$-module.

  3. (3)

The techniques involved in the proof are fairly general and permit to carry on
the proof when $K$ is replaced by an affinoid $K$-algebra $A$ (in the sense of
Berkovich) and $X$ by a separated $A$-scheme of finite type.

  4. (4)

Let $X$ be a variety and $F$ a coherent $\mathcal{O}_{X}$-module. If $X$ is
affine, then the density of $\Gamma(X,F)$ in
$\Gamma(X^{\textup{an}},F^{\textup{an}})$ is rather easy to prove. When $X$ is
proper, the non-Archimedean GAGA theorem implies that the natural map
$\Gamma(X,F)\to\Gamma(X^{\textup{an}},F^{\textup{an}})$ is an isomorphism.
Therefore, B is most interesting when $X$ belongs to neither of these two
classes, and especially when $X$ does not admit a proper morphism onto an
affine variety.

  5. (5)

B has little to do with algebraic varieties: the only property we use is that
they can be “compactified”. The general statement concerns the complement of
an effective Cartier divisor in a proper $K$-analytic space (_infra_ theorem
5.9) or, even more generally, holomorphically convex (_infra_ theorem 5.2).

  6. (6)

The argument for deducing A from B is not exclusive to non-Archimedean
analysis and can be adapted over the complex numbers to yield the following
result: if $X$ is a complex variety such that $\Gamma(X,\mathcal{O}_{X})$ is
dense in $\Gamma(X^{\textup{an}},\mathcal{O}_{X}^{\textup{an}})$ and
$X^{\textup{an}}$ is Stein, then it satisfies the conclusions of A.

  7. (7)

Consider the moduli space $M_{\textup{dR}}(E)$ of line bundles endowed with a
connection on a complex elliptic curve $E$. Then, all of the algebraic
functions on $M_{\textup{dR}}(E)$ are constant, while $M_{\textup{dR}}(E)$ is
biholomorphic to $\mathbb{C}^{\times}\times\mathbb{C}^{\times}$. Therefore,
the complex analogue of B is false.

B marks quite a difference between complex and non-Archimedean analysis:
roughly speaking, there are fewer rigid analytic functions with essential
singularities than holomorphic ones.333Resembling phenomena were observed by
Cherry throughout his work in non-Archimedean Nevanlinna theory ([Che93],
[Che94], [CY97]). Mirroring this discrepancy, its proof has to rely on some
technique unavailable in the complex framework: with no originality, it is the
involvement of models over the ring of integers $R$ of $K$. To state the
result let $\varpi\in R\smallsetminus\\{0\\}$ be a topological nilpotent
element and for an $R$-scheme $S$ let $S_{n}$ be the closed subscheme
$S\times_{R}(R/\varpi^{n}R)$ of $S$. The density result lying at the core of
the method is the following (see theorem 3.3):

######  Theorem C.

Let $D$ be an effective divisor in a proper and flat $R$-scheme $X$. If $D$ is
flat over $R$, then for any semi-reflexive coherent $\mathcal{O}_{X}$-module
$F$ the natural map

$\Gamma(X\smallsetminus
D,F)\longrightarrow\projlim_{n\in\mathbb{N}}\Gamma(X_{n}\smallsetminus
D_{n},F),$

is injective, and has a dense image.

Let us relate this to B. Let $\hat{U}$ be the formal $R$-scheme obtained as
formal completion of $U=X\smallsetminus D$ along its special fiber and
$i\colon\hat{U}\hookrightarrow U$ the natural morphism of locally ringed
spaces. Then,

$\Gamma(\hat{U},i^{\ast}F)=\projlim_{n\in\mathbb{N}}\Gamma(U_{n},F).$

Let $U_{K}$ be the generic fiber of $U$ and $F_{K}$ the restriction of $F$ to
$U_{K}$. With this notation, the $K$-vector space
$\Gamma(\hat{U},i^{\ast}F)\otimes_{R}K$ is the space of global sections
$\Gamma(\hat{U}_{\eta},F_{K}^{\textup{an}})$ of the coherent sheaf
$F_{K}^{\textup{an}}$ on a certain compact subset $\hat{U}_{\eta}$ of
$U_{K}^{\textup{an}}$, called the _Raynaud generic fiber_ of the admissible
formal $R$-scheme $\hat{U}$. C states that any $K$-analytic section
$f\in\Gamma(\hat{U}_{\eta},F_{K}^{\textup{an}})$ can be approximated globally
on the compact subset $\hat{U}_{\eta}$ by algebraic sections
$g\in\Gamma(U_{K},F_{K})$.

######  Example.

In general, C fails to be true when the flatness of $D$ is discarded. Suppose
for simplicity $R$ discretely valued and let $k$ be the residue field of $R$.
Consider the blow-up $\pi\colon X\to\mathbb{P}^{1}_{R}$ in a $k$-rational
point of the special fiber. Let $D$ be the exceptional divisor. The completion
of the open subset $U:=X\smallsetminus D$ along its special fiber is
isomorphic to the completion of the affine line $\mathbb{A}^{1}_{R}$ along its
special fiber. Thus

$\projlim_{n\in\mathbb{N}}\Gamma(U_{n},\mathcal{O}_{X})\cong
R\\{z\\}:=\projlim_{n\in\mathbb{N}}R/\varpi^{n}R[z],$

where $z$ is the coordinate function on $\mathbb{A}^{1}_{R}$. However the
generic fiber of $U$ is $\mathbb{P}^{1}_{K}$ so by flat base change
$\Gamma(U,\mathcal{O}_{X})\hookrightarrow\Gamma(\mathbb{P}^{1}_{K},\mathcal{O}_{\mathbb{P}^{1}_{K}})=K$.
In particular, the image of

$R=\Gamma(U,\mathcal{O}_{X})\longrightarrow\projlim_{n\in\mathbb{N}}\Gamma(U_{n},\mathcal{O}_{X})\cong
R\\{z\\},$

is not dense.

Most of the comments for B have their counterpart for C:

  1. (1)

The topology on $\projlim_{n\in\mathbb{N}}\Gamma(U_{n},F)$ is the prodiscrete
one.

  2. (2)

Technically speaking, it is the flatness of $F$ and of $F_{\rvert D}$ that is
used during the proof rather than the semi-reflexivity of $F$ and the flatness
of $D$. However, in the deduction of B, we fail to see how to extend an
arbitrary coherent sheaf on the generic fiber to one on the model whose
restriction to $D$ is flat over $R$, while we manage to do this under the
semi-reflexivity assumption.

  3. (3)

The argument to prove C goes through when $R$ is replaced by a flat
$R$-algebra of finite type.

  4. (4)

When $U$ is affine, C follows at once from the surjectivity of the restriction
map $\Gamma(U,F)\to\Gamma(U_{n},F)$, in turn implied by the vanishing of
higher cohomology. A slight generalization of the previous argument involving
the formal GAGA theorem yields C when $U$ admits a proper morphism onto an
affine $R$-scheme of finite type. Therefore, C is more interesting when $U$
does not fall in the latter category.

  5. (5)

The natural setting for C is that of a proper formal $R$-scheme $X$ together
with an effective Cartier divisor $D\subset X$. The result then states the
density of sections over $X\smallsetminus D$ having polar singularities along
$D$ among all the sections over $X\smallsetminus D$ (_infra_ theorem 3.3).

For C to be of any use, given a variety $U_{K}$ over $K$, one has to be able
to produce a model $U$ of $U_{K}$ which is the complement of an effective
Cartier divisor $D$ flat over $R$ in a proper flat $R$-scheme $X$ (see theorem
1.2):

######  Theorem D.

Let $D_{K}$ be an effective Cartier divisor in a be a proper variety $X_{K}$.
Then, there are an effective Cartier divisor $D$ in a proper flat $R$-scheme
$X$ and an isomorphism of $K$-schemes $X_{K}\cong X\times_{R}K$ inducing an
isomorphism $D_{K}\cong D\times_{R}K$.

In order to construct such an $X$ and $D$, the first reflex is to take any
proper flat model $X$ of $X_{K}$ and the Zariski closure $D$ of $D_{K}$ in
$X$. The closed subscheme $D$ is without doubt flat over $R$ but may fail to
be a Cartier divisor. To repair such an issue, one considers the blow-up
$X^{\prime}$ of $X$ in $D$. The inverse image $D^{\prime}$ of $D$ in
$X^{\prime}$ is thus Cartier, but will not be flat over $R$ anymore in
general. Looking at the Zariski closure in $X^{\prime}$ of the generic fiber
of $D^{\prime}$ brings us back to square one. Not to be caught in a vicious
circle, one has to choose carefully the blow-up of $X$ to perform. Namely, if
$I$ is the sheaf of ideals of $\mathcal{O}_{X}$ defining the closed subscheme
$D$, then we consider the blow-up $\pi\colon X^{\prime}\to X$ of $X$ in the
ideal $I+\varpi^{n}\mathcal{O}_{X}$, for some $n\in\mathbb{N}$ big enough. The
strict transform $D^{\prime}$ of $D$ in $X^{\prime}$ is shown to be the
effective Cartier divisor $\pi^{\ast}D-E$, where $E$ is the exceptional
divisor and the minus sign stands for the difference of effective Cartier
divisors (_cf._ lemma 1.4). Gabber communicated us an alternative proof of D.
With D under the belt, the proof of B is achieved by arbitrarily enlarging the
compact subset obtained as Raynaud’s generic fiber (of the formal completion)
of the model. This is a routine operation (_cf._ lemma 5.11), except for the
extra care one has to employ not to lose the flatness over $R$ of the Cartier
divisor in question (_cf._ proposition 2.12).

### Acknowledgements

We thank O. Gabber for communicating an alternative proof of D and J.-B. Bost
and F. Charles for sharing an earlier version of their manuscript [BC21]. M.M.
was supported by ANR-18-CE40-0017. J.P. acknowledges support by the ERC
project TOSSIBERG, grant agreement 637027, and the Deutsche
Forschungsgemeinschaft, through TRR 326 _Geometry and Arithmetic of
Uniformized Structures_ , project number 444845124.

##  1\. Extending Cartier divisors: the algebraic case

###  1.1. Statements

We start by recalling some basic notions concerning Cartier divisors in a
scheme $X$. For a closed subscheme $Z\subset X$ let $I_{Z}$ denote the kernel
of the restriction map $\mathcal{O}_{X}\to\mathcal{O}_{Z}$. An _effective
Cartier divisor_ $D\subset X$ is a closed subscheme whose defining sheaf of
ideals $I_{D}$ is invertible. Let $\mathcal{O}_{X}(D)$ denote the dual of
$I_{D}$. The homomorphism $s_{D}\colon\mathcal{O}_{X}\to\mathcal{O}_{X}(D)$
dual to the inclusion $I_{D}\to\mathcal{O}_{X}$ is called the _canonical
section_ associated with $D$. For effective Cartier divisors
$D,D^{\prime}\subset X$, the Cartier divisor $D+D^{\prime}$ is the closed
subscheme with sheaf of ideals $I_{D}\otimes_{\mathcal{O}_{X}}I_{D^{\prime}}$
whence the identity
$\mathcal{O}_{X}(D+D^{\prime})=\mathcal{O}_{X}(D)\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}(D^{\prime})$.
If the closed immersion $D\to X$ factors through $D^{\prime}$, then
$I_{D^{\prime}}\subset I_{D}$ and the sheaf of ideals
$I_{D^{\prime}}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}(D)$ defines the
Cartier divisor $D^{\prime}-D$. A closed subscheme $Z$ of a scheme $X$ is
_finitely presented_ if the closed immersion $i\colon Z\to X$ is of finite
presentation. Let $U\subset X$ be an open subscheme. A morphism $X^{\prime}\to
X$ is called a _$U$ -admissible blow-up_ if there exists a finitely presented
closed subscheme $Z\subset X\smallsetminus U$ such that $X^{\prime}$ is
isomorphic to the blow-up of $X$ in $Z$.

######  Theorem 1.1.

Let $D$ be a closed subscheme of a Noetherian scheme $X$. Let $U\subset X$ be
an open subset such that

  1. (1)

$D\cap U$ is scheme-theoretically dense in $D$;

  2. (2)

$D\cap U$ is an effective Cartier divisor in $U$.

Then there is a $U$-admissible blow-up $\pi\colon X^{\prime}\to X$ such that
the strict transform of $D$ in $X^{\prime}$ is an effective Cartier divisor.

Over the ring of integers $R$ of a field $K$ with a non-trivial non-
Archimedean absolute value we can get round of the Noetherian hypothesis and
obtain the following generalization of D:

######  Theorem 1.2.

Let $Y$ be a finitely presented affine $R$-scheme, $f_{K}\colon X_{K}\to
Y\times_{R}K$ a proper morphism of $K$-schemes and $D_{K}\subset X_{K}$ an
effective Cartier divisor.

Then, there are a flat $R$-scheme $X$, a proper morphism of $R$-schemes
$f\colon X\to Y$, an effective Cartier divisor $D\subset X$ flat over $R$, and
an isomorphism $X_{K}\cong X\times_{R}K$ of $K$-schemes through which $D_{K}$
is identified with $D\times_{R}K$ and $f_{K}$ with
$f\times\operatorname{id}_{K}$.

One recovers D in the introduction by taking $Y=\operatorname{Spec}R$. Notice
that the hypothesis of $Y$ being finitely presented over $R$ is easily met.
Indeed, given a finitely generated $K$-algebra $A_{K}$ and a surjection
$\varphi\colon K[t_{1},\dots,t_{n}]\to A_{K}$ of $K$-algebras, the ideal the
ideal $I:=\operatorname{Ker}\varphi\cap R[t_{1},\dots,t_{n}]$ is finitely
generated by lemma 1.8. Thus the $R$-algebra $A:=R[t_{1},\dots,t_{n}]/I$ is
flat, finitely presented, and such that $A\otimes_{R}K$ is isomorphic to
$A_{K}$.

###  1.2. Proofs

The remainder of this section is devoted to the proof of theorems 1.1, LABEL:
and 1.2. We begin with the following technical result:

######  Lemma 1.3.

Let $X$ be a Noetherian scheme and $\varphi\colon F\to G$ a homomorphism of
coherent $\mathcal{O}_{X}$-modules such that $\varphi_{\rvert U}\colon
F_{\rvert U}\to G_{\rvert U}$ is surjective on the complement $U$ of an
effective Cartier divisor $Z\subset X$.

Then there is an integer $N\geqslant 1$ such that, for all $n\geqslant N$, the
image of $G\to G(nZ)$ is contained in the image of
$\varphi\otimes\operatorname{id}\colon F(nZ)\to G(nZ)$.

###### Proof.

By quasi-compactness of $X$, one may suppose that $X=\operatorname{Spec}A$ is
affine, and $Z$ is a principal Cartier divisor with equation $a=0$ for a
nonzerodivisor $a\in A$. The $A$-modules $\Gamma(X,F)$, $\Gamma(X,G)$ are
finitely generated and $U$ is the principal open subset
$D(a)=\operatorname{Spec}A_{a}$. Let $g_{1},\dots,g_{r}$ be generators of
$\Gamma(X,G)$. The hypothesis of $\varphi_{\rvert U}\colon F_{\rvert U}\to
G_{\rvert U}$ being surjective implies that, for $i=1,\dots,r$ there are non-
negative integers $n_{i},m_{i}\in\mathbb{N}$ and $f_{i}\in\Gamma(X,F)$ such
that $a^{m_{i}}(a^{n_{i}}g-\varphi(f_{i}))=0$. The integer
$N=\max\\{n_{1}+m_{1},\dots,n_{r}+m_{r}\\}$ does the job. ∎

The key result at the core of the method is the following:

######  Lemma 1.4.

Let $D$ be a finitely presented closed subscheme of a scheme $X$, $Z\subset X$
an effective Cartier divisor and $U=X\smallsetminus Z$ such that

  1. (1)

$D\cap U$ is scheme-theoretically dense in $D$;

  2. (2)

$D$ is contained in an effective Cartier divisor $D_{0}$ of $X$.

Assume there is an integer $n\geqslant 1$ such that the image of $I_{D}\to
I_{D}(nZ)$ is contained in the image of $I_{D_{0}}(nZ)\to I_{D}(nZ)$. Let
$\pi\colon X^{\prime}\to X$ be the blow-up of $X$ in the scheme-theoretic
intersection of $D_{0}$ and $nZ$, and $E\subset X^{\prime}$ be the exceptional
divisor.

Then, the strict transform of $D$ in $X^{\prime}$ is the effective Cartier
divisor $\pi^{-1}(D_{0})-E$.

In particular, there is a $U$-admissible blow-up $\pi\colon X^{\prime}\to X$
such that the strict transform of $D$ is an effective Cartier divisor.

###### Proof.

The question is local on $X$, thus one may assume that
$X=\operatorname{Spec}A$ is affine and that $D_{0}$, $Z$ are principal Cartier
divisors. Let $f_{0}\in A$ be a generator of the ideal of $D_{0}$, and let
$g\in A$ be a generator of the ideal of $Z$. Since $D$ is finitely presented,
there are $f_{1},\dots,f_{r}$ such that $f_{0},\dots,f_{r}$ generate the ideal
of $D$. The blow-up $X^{\prime}$ is covered by the following affine open
subsets:

$\displaystyle X^{\prime}_{0}$
$\displaystyle=\operatorname{Spec}A[f_{0}/g^{n}],$ $\displaystyle
A[f_{0}/g^{n}]$ $\displaystyle=(A[t]/(f_{0}-tg^{n}))/(g\textup{-torsion}),$
$\displaystyle X^{\prime}_{1}$
$\displaystyle=\operatorname{Spec}A[g^{n}/f_{0}],$ $\displaystyle
A[g^{n}/f_{0}]$
$\displaystyle=(A[u]/(g^{n}-uf_{0}))/(f_{0}\textup{-torsion}).$

######  Claim 1.5.

The strict transform $D^{\prime}$ of $D$ in $X^{\prime}$ does not meet
$X^{\prime}_{1}$.

###### Proof of the claim.

The strict transform of $D$ is the scheme-theoretic closure in $X^{\prime}$ of
$\pi^{-1}(D\smallsetminus Z)$. Now, the function $g^{n}$ is invertible on
$X^{\prime}_{1}\smallsetminus\pi^{-1}(Z)$. The equality $g^{n}=uf_{0}$ on
$X^{\prime}_{1}$ implies that $f_{0}$ is also invertible on
$X^{\prime}_{1}\smallsetminus\pi^{-1}(Z)$. Therefore,

$\pi^{-1}(D\smallsetminus Z)\cap
X^{\prime}_{1}=\pi^{-1}(D)\cap(X^{\prime}_{1}\smallsetminus\pi^{-1}(Z))=\emptyset.$

It follows that $D^{\prime}$ does not meet $X^{\prime}_{1}$. ∎

######  Claim 1.6.

The strict transform $D^{\prime}$, seen as closed subscheme of
$X^{\prime}_{0}$, is the Cartier divisor with equation $t=0$.

###### Proof of the claim.

Let $A[f_{0}/g^{n}]_{g}$ denote the localization of $A[f_{0}/g^{n}]$ with
respect to $g$: via the ring homomorphism $A\to A[f_{0}/g^{n}]$, it is
isomorphic to the localization of $A$ with respect to $g$. The strict
transform $D^{\prime}$ is the scheme-theoretic closure of
$\pi^{-1}(D\smallsetminus Z)$ in $X^{\prime}_{0}$. Therefore, the ideal
$I_{D^{\prime}}\subset A[f_{0}/g^{n}]$ defining $D^{\prime}$ is

$\operatorname{Ker}(A[f_{0}/g^{n}]\to A[f_{0}/g^{n}]_{g}/(f_{0})).$

Proving the claim amounts to showing that $I_{D^{\prime}}$ is generated by
$t$. Of course, $t$ belongs to $I_{D^{\prime}}$ as $t=f_{0}/g^{n}$ in
$A[f_{0}/g^{n}]_{g}$. On the other hand, any element of $A[f_{0}/g^{n}]$ can
be written as $\sum_{i=0}^{d}a_{i}t^{i}$ for some non-negative integer $d$ and
$a_{0},\dots,a_{d}\in A$. Such an element belongs to the ideal generated by
$t$ if and only if $a_{0}$ belongs to the ideal general by $t$. We are
therefore led back to prove the following: each $a\in A$ whose image in
$A[f_{0}/g^{n}]_{g}/(f_{0})$ is $0$ belongs to the ideal generated by $t$.
Since the homomorphism $A_{g}\to A[f_{0}/g^{n}]_{g}$ is an isomorphism, $a$ is
already $0$ in $A_{g}/(f_{0})$. The hypothesis of $D\cap U$ being scheme-
theoretically dense in $D$ is rephrased as the equality

$\operatorname{Ker}(A\to A_{g}/(f_{0}))=f_{0}A+\cdots+f_{r}A.$

(We used here that the ideals $f_{0}A_{g}$ and $f_{0}A_{g}+\cdots+f_{r}A_{g}$
coincide.) It follows that $a$ belongs to the ideal $f_{0}A+\cdots+f_{r}A$.
Thus, in order to conclude the proof, it suffices to express $f_{i}$ in terms
of $t$ ($i=1,\dots,r$). By the choice of $n$, there are $b_{1},\dots,b_{r}\in
A$ such that $f_{i}=b_{i}f_{0}/g^{n}$. Written otherwise, for $i=1,\dots,r$,
the equality $f_{i}=b_{i}t$ holds in $A[f_{0}/g^{n}]$, which concludes the
proof. ∎

The Cartier divisor $\pi^{-1}(D_{0})$ is given by the ideal
$f_{0}\mathcal{O}_{X^{\prime}}$. The exceptional divisor $E$ is given by the
ideal $g^{n}A[f_{0}/g^{n}]$ on the affine chart $X^{\prime}_{0}$, while it is
given by the ideal $f_{0}A[g^{n}/f_{0}]$ on the affine chart $X^{\prime}_{1}$.
It follows that the effective Cartier divisor $\pi^{-1}(D_{0})-E$ is given by
the ideal generated by $f_{0}/g^{n}=t$ on $X^{\prime}_{0}$ and by the ideal
generated by $f_{0}/f_{0}=1$ on $X^{\prime}_{1}$. One concludes the proof by
comparing this with the description of the strict transform $D^{\prime}$ given
by the above claims. ∎

######  Proposition 1.7.

Let $D,Z$ be effective Cartier divisors in a scheme $X$ whose intersection
$D\cap Z$ is scheme-theoretically dense in $D$. Let $\pi\colon X^{\prime}\to
X$ be the blow-up of $X$ in $D\cap Z$ and $E\subset X^{\prime}$ the
exceptional divisor.

Then, the strict transform of $D$ in $X^{\prime}$ is the effective Cartier
divisor $\pi^{-1}(D)-E$.

###### Proof.

Apply lemma 1.4 with $D=D_{0}$ and $n=1$. ∎

###### Proof of theorem 1.1.

Endowed the closed subset $Z:=X\smallsetminus U$ with its reduced structure.
The blow-up $\pi\colon X^{\prime}\to X$ of $X$ in $Z$ is $U$-admissible
because $Z$ is finitely presented. Up to replacing $Z$ by the Cartier divisor
$\pi^{-1}(Z)$, and $D$ by $\pi^{-1}(D)$, one may assume that $Z$ is a Cartier
divisor. (Note that the composition of $U$-admissible blow-ups is a
$U$-admissible blow-up [Stacks, Lemma 080L].) Suppose $Z$ is a Cartier
divisor. The blow-up $\pi\colon X^{\prime}\to X$ of $X$ in $D$ is
$U$-admissible because $D$ is finitely presented and, since $D\cap U$ is
supposed to be a Cartier divisor in $U$, the induced map
$\pi\colon\pi^{-1}(U)\to U$ is an isomorphism. The exceptional divisor
$D^{\prime}_{0}:=\pi^{-1}(D)$ is an effective Cartier divisor, as well as the
pre-image $Z^{\prime}=\pi^{-1}(Z)$ of $Z$. Let $D^{\prime}$ be the scheme-
theoretic closure of $D^{\prime}_{0}\smallsetminus Z$ in $X^{\prime}$. We may
now apply lemma 1.4 to $X^{\prime}$, $D^{\prime}$, $D_{0}^{\prime}$ and
$Z^{\prime}$. Indeed, let $U^{\prime}=X^{\prime}\smallsetminus Z^{\prime}$.
The closed immersion $D^{\prime}\cap U^{\prime}\to D_{0}^{\prime}\cap
U^{\prime}$ is an isomorphism. On the level of sheaves of ideals, this means
that the natural inclusion $I_{D^{\prime}_{0}\rvert U}\to I_{D^{\prime}\rvert
U}$ is an isomorphism. According to lemma 1.3, there is a non-negative integer
$N$ with the following property: given an integer $n\geqslant N$, the
injective homomorphism $I_{D_{0}^{\prime}}(nZ^{\prime})\to
I_{D^{\prime}}(nZ^{\prime})$ is an isomorphism. In particular, lemma 1.4 can
be applied with every such an integer $n$. ∎

######  Lemma 1.8.

Let $R$ the ring of integers of a field $K$ endowed with a non-trivial non-
Archimedean absolute value. Let $A$ be a flat $R$-algebra of finite type and
$M$ a finitely generated $A$-module flat over $R$.

  1. (1)

The $A$-module $M$ is of finite presentation.

  2. (2)

Let $N\subset M$ be a submodule such that $M/N$ is flat over $R$. Then, $N$ is
of finite presentation.

###### Proof.

(1) This is a particular case of [RG71, theorem 3.4.6].

(2) The $A$-module $M/N$ is finitely generated and flat over $R$ thus finitely
presented by (1). Since $M$ is finitely generated and $M/N$ finitely
presented, $N$ is finitely generated ([Stacks, Lemma 0519 (5)]). The
$A$-module $N$ is finitely generated and flat over $R$, thus, again thanks to
(1), it is finitely presented. ∎

###### Proof of theorem 1.2.

By Nagata’s Compactification theorem (see [Lüt93], [Con07] and [Con09], or
[Tem11]), the morphism of $K$-analytic spaces $\sigma_{K}\colon X_{K}\to
Y\times_{R}K$ can be extended to a proper morphism of $R$-schemes
$\sigma\colon X\to Y$. Furthermore, up to replacing $X$ by the scheme-
theoretic closure of $X_{K}$ in $X$, one may suppose that $X$ is flat over
$R$. Let $D$ be the scheme-theoretic closure of $D_{K}$. Note that $X$ and $D$
are flat over $R$. If $K$ is discretely valued, _i.e._ $R$ is Noetherian, then
one can blithely apply theorem 1.1 with $Z$ being the special fibre of $X$.
Note that the so-obtained blow-up $\pi\colon X^{\prime}\to X$ is such that
$X^{\prime}$ is flat over $R$. Moreover, the strict transform of $D$ is flat
over $R$ too, being the scheme-theoretic closure of $\pi^{-1}(D_{K})$ in
$X^{\prime}$. In the general case, some argument is needed in order to assure
the hypothesis of finite presentation of the strict transform considered in
the proof of theorem 1.1.

Let $\varpi\in R\smallsetminus\\{0\\}$ be topologically nilpotent and let $Z$
be the closed subscheme of $X$ given by the vanishing of $\varpi$. The open
subset $U=X\smallsetminus Z$ is the generic fiber $X_{K}$ of $X$. Let
$\pi\colon X^{\prime}\to X$ the blow-up of $X$ in $D$. According to lemma 1.8,
the closed subscheme $D$ is finitely presented. Moreover, since $D_{K}=D\cap
U$ is an effective Cartier divisor, the induced morphism of $R$-schemes
$\pi\colon\pi^{-1}(U)\to U$ is an isomorphism. By combining these two facts,
one deduces that $\pi\colon X^{\prime}\to X$ is a $U$-admissible blow-up. Let
$D_{0}^{\prime}$ be the inverse image of $D$ in $X^{\prime}$: it is an
effective Cartier divisor. Let $D^{\prime}$ be the Zariski of closure of
$\pi^{-1}(D_{K})$ in $X^{\prime}$. Again, thanks to lemma 1.8, the closed
subscheme $D^{\prime}$ is finitely presented and one can apply lemma 1.4 to
$X^{\prime}$, $D_{0}^{\prime}$, $D^{\prime}$ and the closed subscheme
$Z^{\prime}$ of $X^{\prime}$ defined by the vanishing of $\varpi$. As in the
discretely valued case, the blow-up $\pi^{\prime}\colon X^{\prime\prime}\to
X^{\prime}$ given by lemma 1.4 is such that $X^{\prime\prime}$ is flat over
$R$ as well as the strict transform of $D^{\prime}$ in $X^{\prime\prime}$. ∎

##  2\. Extending Cartier divisors: the formal case

Let $K$ be a complete non-trivially valued non-Archimedean field and $R\subset
K$ its ring of integers.

###  2.1. Preliminaries on formal schemes

Recall that a formal scheme over $R$ is _admissible_ if it is locally the
formal spectrum of a flat $R$-algebra which is topologically of finite
presentation. To such a formal scheme $X$ is attached its _(Raynaud) generic
fiber_ $X_{\eta}$ seen as a $K$-analytic space in the sense of Berkovich. The
generic fiber of an affine formal scheme $X=\operatorname{Spf}A$ where $A$ is
flat topologically of finite type is the Banach spectrum
$X_{\eta}=\mathcal{M}(A\otimes_{R}K)$ of the affinoid $K$-algebra
$A\otimes_{R}K$. Here we recall some facts that are repeatedly used
afterwards:

######  Lemma 2.1.

Let $X$ be a quasi-compact formal $R$-scheme topologically of finite type.
Then the topological space underlying $X$ is Noetherian.

###### Proof.

Let $k$ be the residue field of $R$. The topological space $|X|$ underlying
$X$ coincides with the topological space underlying the finite type $k$-scheme
$X_{\operatorname{red}}$: in particular $|X|$ is a Noetherian topological
space. ∎

######  Lemma 2.2.

Let $A$ be a flat $R$-algebra topologically of finite type and $M$ a finitely
generated $A$-module flat over $R$.

  1. (1)

The $A$-module $M$ is of finite presentation.

  2. (2)

If $N\subset M$ is a submodule such that $M/N$ is flat over $R$, then $N$ is
of finite presentation.

  3. (3)

The ring $A$ is coherent.

###### Proof.

(1) and (2) are respectively theorem 7.3.4 and lemma 7.3.6 in [Bos14], and (3)
follows from (2). ∎

######  Lemma 2.3.

Let $X$ be a quasi-compact formal $R$-scheme topologically of finite type and
$F$ a sheaf of $R$-modules on $X$. The presheaf
$U\mapsto\Gamma(U,F)\otimes_{R}K$ on $X$ is a sheaf.

###### Proof.

Let $U$ be an open subset of $X$. Since the topological space underlying $X$
is Noetherian (lemma 2.1), the open subset $U$ is quasi-compact. Let
$U=\bigcup_{i=1}^{n}U_{i}$ a finite open cover of $U$. The exact sequence

$0\longrightarrow
F(U)\stackrel{{\scriptstyle\delta_{0}}}{{\longrightarrow}}\prod_{i=1}^{n}F(U_{i})\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}\prod_{i,j=1}^{n}F(U_{i}\cap
U_{j}),$

where $\delta_{0}$ and $\delta_{1}$ are the usual coboundary operator of the
Čech cochain complex, stays exact after taking its tensor product with $K$
(because $K$ is a flat $R$-algebra):

$0\longrightarrow
F(U)\otimes_{R}K\stackrel{{\scriptstyle\delta_{0}}}{{\longrightarrow}}\left(\prod_{i=1}^{n}F(U_{i})\right)\otimes_{R}K\stackrel{{\scriptstyle\delta_{1}}}{{\longrightarrow}}\left(\prod_{i,j=1}^{n}F(U_{i}\cap
U_{j})\right)\otimes_{R}K.$

One concludes by noticing that tensor product commutes with finite products. ∎

######  Definition 2.4.

Let $X$ be a quasi-compact formal $R$-scheme topologically of finite type and
$F$ an $\mathcal{O}_{X}$-modules. We denote the sheaf
$U\mapsto\Gamma(U,F)\otimes_{R}K$ by $F_{K}$.

###  2.2. Admissible blow-ups

The central tool in this section is blowing-up in the context of admissible
formal schemes. Let us recall a few definitions. Let $X$ be an admissible
formal $R$-scheme, $\varpi\in R\smallsetminus\\{0\\}$ topologically nilpotent,
$I\subset\mathcal{O}_{X}$ a coherent sheaf of ideals such that, locally on
$X$, it contains a sheaf of ideals of the form $\mu\mathcal{O}_{X}$ for some
$\mu\in R\smallsetminus\\{0\\}$. The formal $R$-scheme

$X^{\prime}=\injlim_{n\in\mathbb{N}}\operatorname{Proj}\left(\bigoplus_{d\in\mathbb{N}}I^{d}\otimes_{\mathcal{O}_{X}}(\mathcal{O}_{X}/\varpi^{n}\mathcal{O}_{X})\right),$

with structural morphism $\pi\colon X^{\prime}\to X$, is called the _blow-up
of $X$ in $I$_. The _exceptional divisor_ of $\pi$ is the closed formal
subscheme of $X^{\prime}$ defined by the coherent sheaf of ideals
$I\mathcal{O}_{X^{\prime}}$.

######  Lemma 2.5.

With the notation above, let $Z\subset X$ be a closed admissible formal
subscheme defined by a coherent sheaf of ideals $J\subset\mathcal{O}_{X}$.
Then,

$J^{\prime}:=\operatorname{Ker}(\mathcal{O}_{X^{\prime}}\to\mathcal{O}_{X^{\prime},K}/J\mathcal{O}_{X^{\prime},K})$

is a coherent sheaf of ideals of $\mathcal{O}_{X^{\prime}}$.

###### Proof.

The statement is local on $X$, therefore one may assume that
$X=\operatorname{Spf}A$ is affine, where $A$ is a topologically finitely
generated flat $R$-algebra. In order to simplify notation, identify $I$ (resp.
$J$) with the finitely presented ideal $\Gamma(X,I)$ (resp. $\Gamma(X,J)$).
Let $f_{0},\dots,f_{r}$ be generators of $I$. The blow-up $X^{\prime}$ is
covered by affine open subsets $X^{\prime}_{i}$, $i=0,\dots,r$,

$\displaystyle X^{\prime}_{i}=\operatorname{Spf}A\\{f_{j}/f_{i}\\}_{j\neq i},$
$\displaystyle A\\{f_{j}/f_{i}\\}_{j\neq i}=[A\\{t_{ij}\\}_{j\neq
i}/(f_{j}-t_{ij}f_{i})_{j\neq i}]/(f_{i}\textup{-torsion}).$

By definition,

$\Gamma(X^{\prime}_{i},J^{\prime})=\operatorname{Ker}(A\\{f_{j}/f_{i}\\}_{j\neq
i}\to[A\\{f_{j}/f_{i}\\}_{j\neq i}/JA\\{f_{j}/f_{i}\\}_{j\neq
i}]\otimes_{R}K).$

According to lemma 2.2, $\Gamma(X^{\prime}_{i},J^{\prime})$ is a finitely
presented ideal of $A\\{f_{j}/f_{i}\\}_{j\neq i}$, which concludes the proof.
∎

With the notation of above lemma the formal closed subscheme
$Z^{\prime}\subset X^{\prime}$ associated with $J^{\prime}$ is called the
_strict transform_ of $Z$.

###  2.3. Statements

We are now in position to state the main results of this section. To do this,
notice that the concepts in section 1.1 are translated _verbatim_ when $X$ is
a formal $R$-scheme or a $K$-analytic space. An effective Cartier divisor $D$
on a quasi-compact admissible formal $R$-scheme $X$ is _admissible_ if it is
admissible as a formal $R$-scheme.

######  Theorem 2.6.

Let $X$ be a quasi-compact admissible formal $R$-scheme and $D\subset X$ an
admissible formal closed subscheme such that $I_{D,K}$ is an invertible
$\mathcal{O}_{X,K}$-module.

Then, there exists an admissible blow-up $\pi\colon X^{\prime}\to X$ such that
the strict transform of $D$ in $X^{\prime}$ is an admissible effective Cartier
divisor.

######  Theorem 2.7.

Let $Y$ be an admissible affine formal $R$-scheme, $X_{K}$ a strict compact
quasi-separated $K$-analytic space, $f_{K}\colon X_{K}\to Y_{\eta}$ a proper
morphism of $K$-analytic spaces and $D_{K}\subset X_{K}$ an effective Cartier
divisor.

Then, there are a quasi-compact quasi-separated admissible formal $R$-scheme
$X$, a proper morphism $f\colon X\to Y$, an admissible effective Cartier
divisor $D$ in $X$, and an isomorphism $X_{K}\cong X_{\eta}$ through which
$D_{K}$ is identified with $D_{\eta}$ and $f_{K}$ with $f_{\eta}$.

Notice that the hypothesis of $Y$ being an admissible formal $R$-scheme is
easily achieved. Indeed let $A_{K}$ be a strictly affinoid $K$-algebra and
$\varphi\colon K\\{t_{1},\dots,t_{n}\\}\to A_{K}$ a surjective homomorphism of
Banach $K$-algebras. According to lemma 2.2, the ideal
$I:=\operatorname{Ker}\varphi\cap R\\{t_{1},\dots,t_{n}\\}$ is finitely
generated. Thus $A:=R\\{t_{1},\dots,t_{n}\\}/I$ is a flat, topologically
finitely presented $R$-algebra and $A\otimes_{R}K\cong A_{K}$.

###  2.4. Fitting ideals in formal geometry

In the proof of theorem 1.2 we started by blowing up $D$ so that we could
suppose that it was a Cartier divisor. Here it is not the right operation to
do, somewhat because a formal scheme does not have a true generic fiber. As we
will see the correct transformation to perform is to blow-up $X$ in the first
Fitting ideal of the coherent sheaf of ideals defining $D$. For lack of
references we recall here the definition of Fitting ideals. Let $X$ be an
admissible formal $R$-scheme and $F$ a coherent $\mathcal{O}_{X}$-module.

######  Lemma 2.8.

For an integer $r\geqslant 0$ let $\operatorname{Fit}_{r}F$ be the
sheafification of the presheaf associating to an affine open subset $U$ the
$r$-th Fitting ideal of $\Gamma(U,F)$. Then $\operatorname{Fit}_{r}F$ is a
coherent sheaf of ideals and, for an affine open subset $U$,

$\Gamma(U,\operatorname{Fit}_{r}F)=\operatorname{Fit}_{r}\Gamma(U,F).$

###### Proof.

Let $U=\operatorname{Spf}(A)$ be an affine open subset, where $A$ is a
finitely presented flat $R$-algebra, and $M:=\Gamma(U,F)$. We claim that
$\operatorname{Fit}_{r}(F)$ coincides with the $\mathcal{O}_{X}$-module
associated with the finitely generated ideal $\operatorname{Fit}_{r}M$.
Indeed, for $f\in A$, the following formula holds ([Stacks, lemma 07ZA (3)]):

$\operatorname{Fit}_{r}(M\otimes_{A}A_{\\{f\\}})=(\operatorname{Fit}_{r}M)A_{\\{f\\}}.$

On the other hand $A_{\\{f\\}}$ is flat444The ring $A_{\\{f\\}}$ is the
completion of $A_{f}$, therefore it is flat over $A_{f}$ ([Bos14, 8.2, lemma
2]). The ring $A_{f}$ is flat over $A$, hence $A_{\\{f\\}}$ is flat over $A$.
over $A$, thus the right-hand of the previous equality coincides with
$(\operatorname{Fit}_{r}M)\otimes_{A}A_{\\{f\\}}$. This concludes the proof. ∎

The sheaf $\operatorname{Fit}_{r}F$ is called the _$r$ -th Fitting ideal of
$F$_.

######  Lemma 2.9.

Let $D\subset X$ be an admissible closed formal subscheme such that $I_{D,K}$
is an invertible $\mathcal{O}_{X,K}$-module.

  1. (1)

The ideal $\operatorname{Fit}_{1}I_{D}$ locally contains a sheaf of the form
$\varpi\mathcal{O}_{X}$ with $\varpi\in R\smallsetminus\\{0\\}$.

  2. (2)

If $\pi\colon X^{\prime}\to X$ is the blow-up of $X$ in
$\operatorname{Fit}_{1}I_{D}$, then $I_{D}\mathcal{O}_{X^{\prime}}$ is an
invertible $\mathcal{O}_{X^{\prime}}$-module and $\pi^{-1}(D)$ is an effective
Cartier divisor.

  3. (3)

If $D^{\prime}$ be the strict transform of $D$ in $X^{\prime}$, then
$I_{D}\mathcal{O}_{X^{\prime},K}=I_{D^{\prime},K}$.

###### Proof.

All statements are local on $X$, therefore one may assume that
$X=\operatorname{Spf}A$ is affine, where $A$ is a finitely presented flat
$R$-algebra, and $I_{K}:=\Gamma(X,I)\otimes_{R}K$ is a principal ideal of
$A_{K}:=A\otimes_{R}K$.

(1) We have
$\operatorname{Fit}_{1}I_{K}=(\operatorname{Fit}_{1}I)A_{K}=(\operatorname{Fit}_{1}I)\otimes_{R}K$
by [Stacks, lemma 07ZA (3)] and the flatness of the $A$-algebra $A_{K}$. The
ideal $I_{K}$ is principal, thus $\operatorname{Fit}_{1}I_{K}=A_{K}$ by
[Stacks, lemma 07ZD]. This implies that $\operatorname{Fit}_{1}I$ contains
some $\varpi\in R\smallsetminus\\{0\\}$.

(2) The sheaf of ideals
$(\operatorname{Fit}_{1}I_{D})\mathcal{O}_{X^{\prime}}$ is an invertible
$\mathcal{O}_{X^{\prime}}$-module ([Bos14, 8.2, proposition 9]). On the other
hand, $(\operatorname{Fit}_{1}I_{D})\mathcal{O}_{X^{\prime}}$ is the Fitting
ideal of the $\mathcal{O}_{X^{\prime}}$-module $\pi^{\ast}I_{D}$ by [Stacks,
lemma 07ZA (3)]. It follows from [Stacks, lemma 080Z] that
$I_{D}\mathcal{O}_{X^{\prime}}$ is an invertible
$\mathcal{O}_{X^{\prime}}$-module.

(3) By definition
$I_{D^{\prime}}=\operatorname{Ker}(\mathcal{O}_{X^{\prime}}\to\mathcal{O}_{X^{\prime},K}/I_{D}\mathcal{O}_{X^{\prime},K})$
whence the statement. ∎

###  2.5. Proofs

After these preliminaries we can finally prove theorems 2.6, LABEL: and 2.7.
The arguments in the proof for theorem 2.6 are a routine adaptation of those
in the proofs of theorems 1.1, LABEL: and 1.2. For an admissible formal
$R$-scheme $X$, a coherent $\mathcal{O}_{X}$-module $F$ flat over $R$ and
$\varpi\in R\smallsetminus\\{0\\}$, let $\frac{1}{\varpi}F$ denote the image
of $F$ in $F_{K}$ through the map $f\mapsto f/\varpi$. The proofs of the
following three statements are obtained _mutatis mutandis_ respectively from
those of lemmas 1.3, LABEL:, 1.4, LABEL: and 1.7:

######  Lemma 2.10.

Let $\varphi\colon F\to G$ a homomorphism between coherent sheaves on a quasi-
compact admissible formal $R$-scheme $X$. If $\varphi_{K}\colon F_{K}\to
G_{K}$ is surjective, then there is $\varpi\in R\smallsetminus\\{0\\}$ such
that $\operatorname{Im}(G\to G_{K})$ is contained in
$\operatorname{Im}(\frac{1}{\varpi}F\to G_{K})$.

######  Lemma 2.11.

Let $D_{0}$ be an effective Cartier divisor in an admissible formal $R$-scheme
and $D\subset D_{0}$ an admissible formal closed subscheme. Assume that the
image of $I_{D}$ in $I_{D,K}$ is contained in $\frac{1}{\varpi}I_{D_{0},K}$
for some $\varpi\in R\smallsetminus\\{0\\}$. Let $\pi\colon X^{\prime}\to X$
be the blow-up of the ideal $\varpi\mathcal{O}_{X}+I_{D_{0}}$ and $E\subset
X^{\prime}$ the exceptional divisor.

Then, the strict transform of $D$ in $X^{\prime}$ is the effective Cartier
divisor $\pi^{-1}(D_{0})-E$.

######  Proposition 2.12.

Let $D$ be an effective Cartier divisor in a quasi-compact admissible formal
$R$-scheme $X$, $\varpi\in R\smallsetminus\\{0\\}$, $\pi\colon X^{\prime}\to
X$ the blow-up of $X$ in $I_{D}+\varpi\mathcal{O}_{X}$ and $E\subset
X^{\prime}$ the exceptional divisor.

Then, the strict transform of $D$ in $X^{\prime}$ is the effective Cartier
divisor $\pi^{-1}(D)-E$.

###### Proof of theorem 2.6.

Let $\pi\colon X^{\prime}\to X$ the blow-up of the first Fitting ideal
$\operatorname{Fit}_{1}(I_{D})$ of the coherent $\mathcal{O}_{X}$-module
$I_{D}$. According to lemma 2.9, the blow-up $\pi\colon X^{\prime}\to X$ is
admissible and $D_{0}=\pi^{-1}(D)$ is an effective Cartier divisor in
$X^{\prime}$. Moreover, if $D^{\prime}$ denotes the strict transform of $D$ in
$X^{\prime}$, then $I_{D^{\prime},K^{\prime}}=I_{D_{0},K^{\prime}}$. Thanks to
lemma 2.10 one can apply lemma 2.11 to $X^{\prime}$, $D^{\prime}$ and
$D_{0}^{\prime}$. ∎

######  Lemma 2.13.

Let $X$ be a quasi-compact admissible formal $R$-scheme and $D_{K}\subset
X_{\eta}$ an effective Cartier divisor. Then, there are an admissible blow-up
$\pi\colon X^{\prime}\to X$ and an admissible closed formal subscheme
$D^{\prime}\subset X^{\prime}$ such that $I_{D^{\prime},K}$ is an invertible
$\mathcal{O}_{X^{\prime},K}$-module and
$\pi_{\eta}^{-1}(D_{K})=D^{\prime}_{\eta}$.

###### Proof.

Let $X_{K}=U_{1,K}\cup\cdots\cup U_{n,K}$ be a finite $G$-covering of $X_{K}$
by affinoid domains such that, for all $i=1,\dots,n$, the effective Cartier
divisor $U_{i,K}\cap D_{K}$ in $U_{i,K}$ is principal. By [Bos14, 8.4, Lemma
5], there is an admissible blow-up $\pi\colon X^{\prime}\to X$ and an open
covering $X^{\prime}=U^{\prime}_{1}\cup\cdots\cup U^{\prime}_{n}$ such that
$U^{\prime}_{i,\eta}=\pi_{\eta}^{-1}(U_{i,K})$ for all $i=1,\dots,n$. Let
$I\subset\mathcal{O}_{X^{\prime}}$ be the sheaf of ideals whose sections on an
open subset $V\subset X$ are

$\operatorname{Ker}(\mathcal{O}_{X^{\prime}}(V)\to\Gamma(\pi_{\eta}(V_{\eta}),\mathcal{O}_{X_{\eta}}/I_{D_{K}})).$

(Note that $\pi_{\eta}\colon X^{\prime}_{\eta}\to X_{\eta}$ is an isomorphism,
thus an open map.)

######  Claim 2.14.

The sheaf of ideals $I$ is coherent.

###### Proof of the claim.

In order to simplify the notation, for an open subset $V\subset X^{\prime}$
write $I_{D_{K}}(V_{\eta})$ for $\Gamma(\pi_{\eta}(V_{\eta}),I_{D_{K}})$, and
$I(V)$ for $\Gamma(V,I)$. Let $V=\operatorname{Spf}(A)$ be an affine open
subset of $X^{\prime}$, where $A$ is a flat topologically finitely presented
$R$-algebra. By lemma 2.2 (2) the ideal $I(V)=\operatorname{Ker}(A\to
A_{K}/I_{D_{K}}(V_{\eta}))$ is finitely presented. In order to prove that $I$
is coherent it is enough to show that $I(D(f))=I(V)\otimes_{A}A_{\\{f\\}}$ for
each $f\in A$.Now the sheaf of ideals $I_{D_{K}}$ is coherent thus

$I_{D_{K}}(D(f)_{\eta})=I_{D_{K}}(V_{\eta})\otimes_{A_{K}}A_{K}\\{f^{-1}\\}=I_{D_{K}}(V_{\eta})A_{K}\\{f^{-1}\\},$

where $A_{K}\\{f^{-1}\\}:=A_{K}\\{t\\}/(1-tf)$ and the second equality holds
by flatness of the $A_{K}$-algebra $A_{K}\\{f^{-1}\\}$. Using the equality
$A_{K}\\{f^{-1}\\}=A_{\\{f\\}}\otimes_{R}K$,

$A_{\\{f\\}}\otimes_{A}(A_{K}/I_{D_{K}}(V_{\eta}))=A_{K}\\{f^{-1}\\}/I_{D_{K}}(D(f)_{\eta}).$

Finally, the flatness of the $A$-algebra $A_{\\{f\\}}$ yields

$\operatorname{Ker}(A\to
A_{K}/I_{D_{K}}(V_{\eta}))\otimes_{A}A_{\\{f\\}}=\operatorname{Ker}(A_{\\{f\\}}\to
A_{K}\\{f^{-1}\\}/I_{D_{K}}(D(f)_{\eta})),$

which concludes the proof of the claim. ∎

######  Claim 2.15.

The sheaf of $\mathcal{O}_{X^{\prime},K}$-modules $I_{K}$ is invertible.

###### Proof.

Let $i\in\\{1,\dots,n\\}$. The sheaf of ideals $I_{D_{K}}$ is principal on
$U^{\prime}_{i,\eta}$, that is, there is
$f_{i}\in\Gamma(U^{\prime}_{i},\mathcal{O}_{X^{\prime}})$ such that
$I_{D_{K}\rvert U^{\prime}_{i,\eta}}=f_{i}\mathcal{O}_{U^{\prime}_{i,\eta}}$.
Let $V=\operatorname{Spf}(A)$ be an affine open subset of $U^{\prime}_{i}$,
where $A$ is a flat topologically finitely presented $R$-algebra. Then,

$\Gamma(V,I_{K}):=\Gamma(V,I)\otimes_{R}K=\Gamma(V_{\eta},I_{D_{K}})=f_{i}(A\otimes_{R}K)=f_{i}\Gamma(V,\mathcal{O}_{X^{\prime},K}),$

where the second equality holds by construction of $I$. This shows that
$I_{K\rvert U^{\prime}_{i}}$ is a sheaf of principal ideals of
$\mathcal{O}_{X^{\prime},K}$. ∎

One concludes the proof by taking $D^{\prime}$ to be the closed admissible
formal subscheme of $X^{\prime}$ defined by the sheaf of ideals $I$. ∎

###### Proof of theorem 2.7.

Raynaud’s point of view on rigid analytic spaces relies upon the insight that
every (strict, paracompact, and quasi-separated) $K$-analytic space, and
morphisms between, can be obtained as the generic fibre of an admissible
formal $R$-scheme ([Bos14, 8.4, lemma 4]). Let $X$ be an admissible formal
$R$-scheme with generic fibre $X_{K}$. Up to taking a suitable formal
admissible blow-up of $X$ the morphism $f_{K}$ extends to a morphism of formal
$R$-schemes $f\colon X\to Y$. The morphism $f$ is proper by [Lüt90, theorem
3.1] in the discretely valued case and [Tem00, Corollary 4.4] in the general
case. According to lemma 2.13, there is an admissible blow-up $\pi\colon
X^{\prime}\to X$ and an admissible closed formal subscheme $D^{\prime}$ in
$X^{\prime}$ with generic fibre $\pi_{\eta}^{-1}(D_{K})$ and such that
$I_{D^{\prime},K}$ is an invertible $\mathcal{O}_{X^{\prime},K}$-module. We
conclude by applying theorem 2.6 to $X^{\prime}$ and $D^{\prime}$. ∎

##  3\. Approximating essential singularities by poles: the formal case

Let $R$ be the ring of integers of a field $K$ complete with respect to a on-
trivial non-Archimedean absolute value.

###  3.1. Functions with polar singularities

The main result of this section concerns the approximation of formal functions
having essential singularities at infinity by those having polar
singularities. More precisely, let $D$ be an effective divisor on a scheme
$X$. Given an $\mathcal{O}_{X}$-module $F$, for $d\in\mathbb{N}$ set
$F(dD):=F\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}(dD)$ and let

$F(\ast D):=\injlim_{d\in\mathbb{N}}F(dD)$

be the sheaf of sections of $F$ with polar singularities along $D$, where the
transition maps are induced by the canonical section
$s_{D}\colon\mathcal{O}_{X}\to\mathcal{O}_{X}(D)$. The
$\mathcal{O}_{X}$-module $F$ is _$D$ -torsion free_ if
$\operatorname{id}_{F}\otimes s_{D}\colon F\to F(D)$ is injective. Let
$U=X\smallsetminus D$ and let $j\colon U\to X$ the open immersion. The
canonical section induces an isomorphism
$\mathcal{O}_{U}\cong\mathcal{O}_{X}(D)_{\rvert U}$ hence an isomorphism
$j^{\ast}F\cong j^{\ast}F(dD)$ for each $d\in\mathbb{N}$. By taking the
injective limit of the homomorphisms $F(dD)\to j_{\ast}j^{\ast}F(dD)\to
j_{\ast}j^{\ast}F$, the latter map being the inverse of previous isomorphism,
one obtains a homomorphism $F(\ast D)\to j_{\ast}j^{\ast}F$ that will be
referred to as the _restriction map_.

######  Lemma 3.1.

Let $D$ be an effective Cartier divisor in a quasi-compact scheme $X$ and $F$
a quasi-coherent $\mathcal{O}_{X}$-module. Then the restriction map induces an
isomorphism

$\injlim_{d\in\mathbb{N}}\Gamma(X,F(dD))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Gamma(X\smallsetminus
D,F).$

Moreover if $X$ is quasi-separated, then the map $F(\ast D)\to
j_{\ast}j^{\ast}F$ given by restriction is an isomorphism where $j\colon
X\smallsetminus D\to X$ is the open immersion.

###### Proof.

For an $\mathcal{O}_{X}$-module $E$ let
$\Gamma(E,D):=\bigoplus_{d\in\mathbb{N}}\Gamma(X,E(dD))$. With this notation
$\injlim_{d\in\mathbb{N}}\Gamma(X,F(dD))$ is the homogeneous localization of
the $\Gamma(\mathcal{O}_{X},D)$-graded module $\Gamma(F,D)$ with respect to
the canonical section $s_{D}$, which is a homogeneous element of degree $1$.
The statement then follows from [EGA III1, proposition 1.4.5]. For the latter
statement notice that on a quasi-compact scheme, every open cover may be
refined by a finite open affine cover. Moreover, if the scheme is quasi-
separated, then the intersection of affine open subsets is a finite union of
open affine subsets, hence quasi-compact. Thus the natural map
$\injlim_{d\in\mathbb{N}}\Gamma(V,F(dD))\to\Gamma(V,F(\ast D))$ is an
isomorphism for any quasi-compact open subset $V\subset X$ by [Stacks, lemma
009F (4)]. Applying statement (1) to $V$, the composite map

$\injlim_{d\in\mathbb{N}}\Gamma(V,F(dD))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Gamma(V,F(\ast
D))\longrightarrow\Gamma(U\cap V,F)=\Gamma(V,j_{\ast}j^{\ast}F)$

is an isomorphism. The result follows because quasi-compact open subsets form
a basis for the topology of $X$. ∎

######  Lemma 3.2.

Let $D$ be an effective Cartier divisor in a separated quasi-compact formal
$R$-scheme $X$ topologically of finite type. Then, for any sheaf of
$\mathcal{O}_{X}$-modules $F$, then the restriction induces an isomorphism

$\injlim_{d\in\mathbb{N}}\Gamma(X,F(dD))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Gamma(X,F(\ast
D)).$

###### Proof.

Since the topological space underlying $X$ is Noetherian (lemma 2.1) the
statement follows from [Stacks, lemma 009F (4)]. ∎

###  3.2. Statements

In order to state the main result of this section, say that an admissible
formal $R$-scheme $X$ is _proper over affine_ if there are an affine
admissible formal $R$-scheme $Y$ and a proper morphism of formal $R$-schemes
$X\to Y$.

######  Theorem 3.3.

Let $D$ be an effective Cartier divisor in an admissible formal $R$-scheme $X$
which is proper over affine. Let $F$ be a coherent sheaf on $X$, which is
$D$-torsion free, and such that $F$ and $F_{\rvert D}$ are flat over $R$.
Then, the natural map

$\Gamma(X,F(\ast D))\longrightarrow\Gamma(X\smallsetminus D,F),$

is injective, and has a dense image.

Recall that the topology on the global section of a coherent sheaf $F$ on a
quasi-compact admissible formal $R$-scheme $X$ is defined as follows. For any
topologically nilpotent $\varpi\in R\smallsetminus\\{0\\}$ we have

$\Gamma(X,F)=\projlim_{n\in\mathbb{N}}\Gamma(X,F/\varpi^{n}F)$

and the prodisrete topology on the left-hand side is given by the discrete
topology on each $\Gamma(X,F/\varpi^{n}F)$. See also [Stacks, Section 0AHY].
The remainder of this section is devoted to the proof of this result.

###  3.3. Norms and modules

We begin by recalling some facts about the topology on an $R$-module $M$. Its
_completion_ is the $R$-module
$\hat{M}:=\projlim_{n\in\mathbb{N}}M/\varpi^{n}M$ where $\varpi\in
R\smallsetminus\\{0\\}$ is topologically nilpotent. Notice that $\hat{M}$ does
not depend on the chosen $\varpi$. The $R$-module $M$ is said to be
_separated_ (resp. _complete_) if the natural map $\varepsilon\colon
M\to\hat{M}$ is injective (resp. _bijective_). Suppose $M$ torsion-free and
identify $M$ with $\operatorname{Im}(M\hookrightarrow M\otimes_{R}K)$. For
$m\in M\otimes_{R}K$, let

$\|m\|_{M}:=\inf\\{|\varpi|:\varpi\in K^{\times},m/\varpi\in M\\}.$

This defines a semi-norm $\|\cdot\|_{M}\colon M\otimes_{R}K\to\mathbb{R}_{+}$.
The $R$-module $\hat{M}$ is torsion-free and
$\|\varepsilon(m)\|_{\hat{M}}=\|m\|_{M}$ for each $m\in M$. Moreover the
homomorphism $\varepsilon$ extends to an isomorphism between the completion of
$M$ with respect to the semi-norm $\|\cdot\|_{M}$ and $\hat{M}$. The
construction of the semi-norm $\|\cdot\|_{M}$ is functorial. Let $f\colon M\to
M^{\prime}$ be a homomorphism between torsion-free $R$-modules. Then
$\|f(m)\|_{M^{\prime}}\leqslant\|m\|_{M}$ for each $m\in M$. If $f$ injective,
then $\operatorname{Coker}f$ is torsion-free if and only if
$\|f(m)\|_{M^{\prime}}=\|m\|_{M}$ for each $m\in M$.

######  Lemma 3.4.

Let $X$ be an admissible formal $R$-scheme and $F$ a coherent
$\mathcal{O}_{X}$-module flat over $R$. Then $\Gamma(X,F)$ is complete.

###### Proof.

Let $\varpi\in R\smallsetminus\\{0\\}$ be topologically nilpotent and forq
$n\in\mathbb{N}$ let $X_{n}$ be the closed subscheme of $X$ defined by the
condition $\varpi^{n}=0$. By definition,

$\Gamma(X,F)=\projlim_{n\in\mathbb{N}}\Gamma(X_{n},F).$

Since $F$ is flat over $R$, the natural map
$\Gamma(X,F)/\varpi^{n}\Gamma(X,F)\to\Gamma(X_{n},F)$ is injective. Since the
projective limit is left-exact, the natural map

$\projlim_{n\in\mathbb{N}}\Gamma(X,F)/\varpi^{n}\Gamma(X,F)\longrightarrow\projlim_{n\in\mathbb{N}}\Gamma(X_{n},F).$

is injective. Therefore the map
$\Gamma(X,F)\to\projlim_{n\in\mathbb{N}}\Gamma(X,F)/\varpi^{n}\Gamma(X,F)$ is
bijective, that is, $\Gamma(X,F)$ is complete. ∎

###  3.4. Proof

Let $S$ be an $R$-scheme, $G$ a quasi-coherent $\mathcal{O}_{X}$-module, and
$\varpi$ a nonzero topologically nilpotent element of $R$. Then, the sequence
of $R/\varpi R$-modules

$0\longrightarrow\Gamma(S,M)/\Gamma(S,\varpi
M)\longrightarrow\Gamma(S,M/\varpi M)\longrightarrow\Gamma(S,M/\varpi
M)/\Gamma(S,M)\longrightarrow 0$

is short exact. For a nonzero topologically nilpotent element
$\varpi^{\prime}$ such that $|\varpi^{\prime}|\leqslant|\varpi|$ and a
homomorphism of $\mathcal{O}_{S}$-modules $\varphi\colon M\to M^{\prime}$, the
above discussion, applied with $S=S_{\varpi^{\prime}}$, yields the following
commutative and exact diagram of $R$-modules:

${0}$${\displaystyle\frac{\Gamma(S_{\varpi^{\prime}},M)}{\Gamma(S_{\varpi^{\prime}},\varpi
M)}}$${\Gamma(S_{\varpi},M)}$${\displaystyle\frac{\Gamma(S_{\varpi},M)}{\Gamma(S_{\varpi^{\prime}},M)}}$${0}$${0}$${\displaystyle\frac{\Gamma(S_{\varpi^{\prime}},M^{\prime})}{\Gamma(S_{\varpi^{\prime}},\varpi
M^{\prime})}}$${\Gamma(S_{\varpi},M^{\prime})}$${\displaystyle\frac{\Gamma(S_{\varpi},M^{\prime})}{\Gamma(S_{\varpi^{\prime}},M^{\prime})}}$${0.}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

######  Lemma 3.5.

Let $D$ be an effective Cartier divisor in an admissible formal $R$-scheme $X$
with canonical section $s\colon\mathcal{O}_{X}\to\mathcal{O}_{X}(D)$. Let $F$
be a coherent $\mathcal{O}_{X}$-module which is $D$-torsion free and such that
$F$, $F_{\rvert D}$ are flat over $R$. Then, for any $i,d\in\mathbb{N}$ and
any $\varpi\in R\smallsetminus\\{0\\}$,

  1. (1)

the map $s^{i}\colon\Gamma(X_{\varpi},F(dD))\to\Gamma(X_{\varpi},F((d+i)D))$
is injective;

  2. (2)

given $\varpi^{\prime}\in R\smallsetminus\\{0\\}$ such that
$|\varpi^{\prime}|\leqslant|\varpi|$, the map $s^{i}$ induces an injection:

$s^{i}\colon\frac{\Gamma(X_{\varpi},F(dD))}{\Gamma(X_{\varpi^{\prime}},F(dD))}\hookrightarrow\frac{\Gamma(X_{\varpi},F((d+i)D))}{\Gamma(X_{\varpi^{\prime}},F((d+i)D))}.$

###### Proof.

Up to replacing $F$ by $F(dD)$ one may assume $d=0$. Moreover, by induction on
$i$ one reduces to the case $i=1$.

(1) Since $s\colon F\to F(D)$ is injective by assumption, the following
sequence of coherent $\mathcal{O}_{X}$-modules is exact:

(3.1) $0\longrightarrow F\stackrel{{\scriptstyle
s}}{{\longrightarrow}}F(D)\longrightarrow F(D)_{\rvert D}\longrightarrow 0.$

The coherent sheaf $F(D)_{\rvert D}$ is flat over $R$. Therefore, the previous
short exact sequence stays exact after taking its tensor product with any
$R$-module $M$. In particular, taking $M=R/\varpi R$ yields the following
short exact sequence of coherent $\mathcal{O}_{X_{\varpi}}$-modules

$0\longrightarrow F_{\rvert X_{\varpi}}\stackrel{{\scriptstyle
s}}{{\longrightarrow}}F(D)_{\rvert X_{\varpi}}\longrightarrow F(D)_{\rvert
D_{\varpi}}\longrightarrow 0.$

The statement follows by taking global sections.

(2) The short exact sequence of $\mathcal{O}_{X}$-modules (3.1) yields the
following commutative diagram:

${\displaystyle\frac{\Gamma(X_{\varpi^{\prime}},F)}{\Gamma(X_{\varpi^{\prime}},\varpi
F)}}$${\displaystyle\frac{\Gamma(X_{\varpi^{\prime}},F(D))}{\Gamma(X_{\varpi^{\prime}},\varpi
F(D))}}$${\displaystyle\frac{\Gamma(D_{\varpi^{\prime}},F(D))}{\Gamma(D_{\varpi^{\prime}},\varpi
F(D))}}$${\Gamma(X_{\varpi},F)}$${\Gamma(X_{\varpi},F(D))}$${\Gamma(D_{\varpi},F(D)),}$$\hookleftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{s}$$\hookleftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{s}$$\leftarrow$$\rightarrow$

where $s\colon\Gamma(X_{\varpi},F)\to\Gamma(X_{\varpi},F(D))$ is injective
because of (1), and the vertical arrows are injective because of the
discussion before the lemma. The statement then follows by the Snake lemma. ∎

######  Lemma 3.6.

Let $D$ be an effective Cartier divisor in a quasi-compact separated
admissible formal $R$-scheme $X$. Let $F$ be a $D$-torsion free coherent
$\mathcal{O}_{X}$-module such that $F$ and $F_{\rvert D}$ are flat over $R$.
Let $j\colon X\smallsetminus D\to X$ be the open immersion. Then,

  1. (1)

the natural homomorphisms $\alpha\colon F\to F(\ast D)$, $\beta\colon F\to
j_{\ast}j^{\ast}F$ are injective with cokernel flat over $R$;

  2. (2)

the restriction homomorphism $\rho\colon F(\ast D)\to j_{\ast}j^{\ast}F$ is
injective;

  3. (3)

the cokernel of $\rho$ is a sheaf of $K$-vector spaces;

  4. (4)

the restriction map $\rho\colon\Gamma(X,F)\to\Gamma(X\smallsetminus D,F)$ is
an isometric embedding and induces a homeomorphism of $\Gamma(X,F)$ with a
closed subset of $\Gamma(X\smallsetminus D,F)$.

###### Proof.

The statement is local on $X$, therefore we may suppose that
$X=\operatorname{Spf}(A)$ is affine, for a topologically finitely presented
flat $R$-algebra $A$, and $D$ is a principal Cartier divisor. Let $f\in A$ be
a generator of the ideal of $D$. Let $M:=\Gamma(X,F)$. With this notation,

(3.2) $\displaystyle\Gamma(X,F(\ast D))=M[t]/(1-tf),$
$\displaystyle\Gamma(U,F)=M\\{t\\}/(1-tf).$

(2) Let $m(t)\in M[t]$. Write $m(t)=\sum_{i=0}^{d}m_{i}t^{i}$ with $m_{i}\in
M$. Suppose its image in $\Gamma(U,F)$ vanishes. This means that there exists
$n(t)\in M\\{t\\}$ such that $m(t)=(1-ft)n(t)$. Write
$n(t)=\sum_{i=0}^{\infty}n_{i}t^{i}$ with $n_{i}\to 0$ as $i\to\infty$. By
comparing Taylor expansions, for $i,j\in\mathbb{N}$ with $i\geqslant d$,
$n_{i+j}=f^{j}n_{i}$. Let $\varpi\in R\smallsetminus\\{0\\}$ be topologically
nilpotent. Because of convergence, there is $I_{\varpi}\in\mathbb{N}$ such
that, for $i\geqslant I_{\varpi}$,

$n_{i}\equiv 0\pmod{\varpi M}.$

On the other hand, because of lemma 3.5 (1), multiplication by $f$ is
injective on $M/\varpi M$. One concludes $n_{i}=0$ for all $i\geqslant d$.
Therefore $n(t)\in M[t]$ and the image of $m(t)$ in $\Gamma(X,F(\ast D))$ is
$0$.

(3) If $T$ denotes the cokernel of $\rho$, then the following exact sequence
of $A$-modules is exact:

$0\longrightarrow\Gamma(X,F(\ast
D))\longrightarrow\Gamma(U,F)\longrightarrow\Gamma(X,T)\longrightarrow 0.$

Indeed it suffices to show that the homomorphism $\Gamma(U,F)\to\Gamma(X,T)$
is surjective. By quasi-compactness and (quasi-)separation of $X$,

$\textup{H}^{1}(X,F(\ast D))=\injlim_{d\in\mathbb{N}}\textup{H}^{1}(X,F(dD)).$

For $d\in\mathbb{N}$, since $X$ is affine and $F(dD)$ coherent, the cohomology
group $\textup{H}^{1}(X,F(dD))$ vanishes for each $d\in\mathbb{N}$. Equation
(3.2) can be rewritten as the following commutative and exact diagram of
$A$-modules:

${0}$${M[t]}$${M[t]}$${\Gamma(X,F(\ast
D))}$${0}$${0}$${M\\{t\\}}$${M\\{t\\}}$${\Gamma(U,F)}$${0}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{1-ft}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{1-ft}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

Since taking global sections is left exact, statement (2) implies that the
right-most vertical arrow is injective. According to the previous claim, the
Snake Lemma gives the following short exact sequence:

$0\longrightarrow M\\{t\\}/M[t]\stackrel{{\scriptstyle
1-ft}}{{\longrightarrow}}M\\{t\\}/M[t]\longrightarrow\Gamma(X,T)\longrightarrow
0.$

Let $\varpi\in R\smallsetminus\\{0\\}$ be topologically nilpotent. An
$R$-module is a $K$-vector space if and only if multiplication by $\varpi$ is
bijective. Because of the previous short exact sequence, it suffices to show
that $M\\{t\\}/M[t]$ is a $K$-vector space. In order to do so, for $m(t)\in
M\\{t\\}$, write $m(t)=\sum_{i=0}^{\infty}m_{i}t^{i}$ with $m_{i}\in M$ such
that $m_{i}\to 0$ as $i\to\infty$.

(Injective.) Suppose $\varpi m(t)\in M[t]$. Let $d\in\mathbb{N}$ be its
degree. For $i>d$, one has $\varpi m_{i}=0$, thus $m_{i}=0$ because $F$ is
flat over $R$. In other words, $m(t)\in M[t]$.

(Surjective.) Because of the convergence of $m(t)$, there is $d$ such that,
for $i\geqslant d$, $m_{i}$ is divisible by $\varpi$. For $i\geqslant d$ let
$\tilde{m}_{i}:=m_{i}/\varpi$ and
$\tilde{m}(t)=\sum_{i=d}^{\infty}\tilde{m}_{i}t^{i}$. Then,

$m(t)\equiv\varpi\tilde{m}(t)\pmod{M[t]},$

which concludes the proof.

(1) Because of statement (2), if $\alpha$ is injective, then $\beta$ is. In
order to prove that $\alpha$ in injective, let $m\in M$ and $n(t)\in M[t]$
such that $m=(1-ft)n(t)$. Write $n(t)=\sum_{i=0}^{d}n_{i}t^{i}$ with $n_{i}\in
M$. By comparing the Taylor expansions, one obtains $m=n_{0}$,
$n_{i}=fn_{i-1}$ for $i=1,\dots,d$ and $fn_{d}=0$. Since multiplication by $f$
on $M$ is injective, one obtains $n_{0}=0$, thus $m=0$. This proves that
$\alpha$ is injective.

The natural map $\operatorname{Coker}\alpha\to\operatorname{Coker}\beta$ is
injective by (2). Therefore is suffices to prove the statement for $\beta$. In
order to do so, suppose there are $m(t)\in M\\{t\\}$ and $\varpi\in
R\smallsetminus\\{0\\}$ such that $\varpi m(t)$ belongs to $(1-ft)M\\{t\\}+M$.
Let $n(t)\in M\\{t\\}$ and $p\in M$ be such that $\varpi m(t)=p+(1-ft)n(t)$.
Write $m(t)=\sum_{i=0}^{\infty}m_{i}t^{i}$ and
$n(t)=\sum_{i=0}^{\infty}n_{i}t^{i}$ with $m_{i},n_{i}\in M$. By comparing
Taylor expansions, $\varpi m_{0}=p+n_{0}$ and, for $i\geqslant 1$, $\varpi
m_{i}=n_{i}-fn_{i-1}$. In other words, modulo $\varpi M$, the following
congruences hold:

$\displaystyle n_{0}\equiv-p,$ $\displaystyle fn_{i-1}\equiv
n_{i},\quad(i\geqslant 1).$

Moreover, by convergence, there is $i_{0}\in\mathbb{N}$ such that $n_{i}$ is
divisible by $\varpi$. On the other hand, because of lemma 3.5 (1),
multiplication by $f$ is injective on $M/\varpi M$. It follows that $n_{i}$ is
divisible by $\varpi$ for all $i\in\mathbb{N}$, thus $p$ is also divisible by
$\varpi$. Set $\tilde{p}=p/\varpi$, $\tilde{n}_{i}=n_{i}/\varpi$ and
$\tilde{n}(t)=\sum_{i=0}^{\infty}\tilde{n}_{i}t^{i}$. (Note that this makes
sense because $M$ is torsion free). Since the $R$-module $M\\{t\\}$ is
torsion-free,

$m(t)=\tilde{p}+(1-ft)\tilde{n}(t),$

that is, the image of $m(t)$ in the cokernel of $\beta$ is $0$.

(4) Lemma 3.6 (3) implies $\|\rho(f)\|_{\Gamma(U,F)}=\|f\|_{\Gamma(X,F)}$ for
each $f\in\Gamma(X,F)$. According to lemma 3.4, $\rho$ is an isometric
embedding between complete metric spaces. It follows that the image is closed.
∎

###### Proof of theorem 3.3.

Let $j\colon U\to X$ be the open immersion of $U$ in $X$. According to lemma
3.6, the natural map $F(\ast D)\to j_{\ast}j^{\ast}F$ is injective. By taking
global sections, one sees that the map $\rho\colon\Gamma(X,F(\ast
D))\to\Gamma(U,F)$ is injective. Let $\varpi\in R\smallsetminus\\{0\\}$ be
topologically nilpotent. For a $R$-scheme $S$, let $S_{n}$ denote its base-
change to $R/\varpi^{n}R$. Let $s\colon\mathcal{O}_{X}\to\mathcal{O}_{X}(D)$
be the canonical section. In order to show that the image of $\rho$ is dense,
one has to prove that given $f\in\Gamma(U,F)$ and an integer $n\in\mathbb{N}$,
there are a non-negative integer $d\in\mathbb{N}$ and $g\in\Gamma(X,F(dD))$
such that $f_{\rvert U_{n}}=g_{\rvert U_{n}}$. Let $f\in\Gamma(U,F)$ and
$n\in\mathbb{N}$. Because of lemma 3.5 (1), for each $d\in\mathbb{N}$, the map
$\alpha_{d,n}\colon\Gamma(X_{n},F(dD))\to\Gamma(U_{n},F)$ is injective. Let
$d_{n}$ be the smallest integer $d$ such that there is a section
$g\in\Gamma(X_{n},F(dD))$ mapping to $f_{\rvert U_{n}}$, and let
$f_{n}:=\alpha_{d_{n},n}^{-1}(f_{\rvert U_{n}})$. For non-negative integers
$m\geqslant n$, the commutativity of the diagram

${\Gamma(X_{m},F(d_{m}D))}$${\Gamma(U_{m},F)}$${\Gamma(X_{n},F(d_{m}D))}$${\Gamma(U_{n},F)}$$\leftarrow$$\rightarrow$$\scriptstyle{\alpha_{d,m}}$$\leftarrow$$\rightarrow$$\scriptstyle{\cdot_{\rvert
X_{n}}}$$\leftarrow$$\rightarrow$$\scriptstyle{\cdot_{\rvert
U_{n}}}$$\leftarrow$$\rightarrow$$\scriptstyle{\alpha_{d,n}}$

implies $d_{m}\geqslant d_{n}$. Since
$s^{d_{m}-d_{n}}\colon\Gamma(X_{n},F(d_{n}D))\to\Gamma(X_{n},F(d_{m}D))$ is
injective (lemma 3.5 (1)), one has $s^{d_{m}-d_{n}}f_{n}=f_{m\rvert X_{n}}$.

######  Claim 3.7.

For $f\in\Gamma(U,F)$ and $m,n\in\mathbb{N}$ with $m\geqslant n$, there is
$g_{m}\in\Gamma(X_{m},F(d_{n}D))$ such that $g_{m\rvert X_{n}}=f_{n}$.

###### Proof of the claim.

Let $i:=d_{m}-d_{n}$. Because of the equality $f_{m\rvert X_{n}}=s^{i}f_{n}$,
the image of the section $s^{i}f_{n}$ in
$\Gamma(X_{n},F(d_{m}D))/\Gamma(X_{m},F(d_{m}D))$ is zero. However, by lemma
3.5, the map

$s^{i}\colon\frac{\Gamma(X_{n},F(d_{n}D))}{\Gamma(X_{m},F(d_{n}D))}\longrightarrow\frac{\Gamma(X_{n},F(d_{m}D))}{\Gamma(X_{m},F(d_{m}D))},$

is injective. Therefore, the image of $f_{n}$ in
$\Gamma(X_{n},F(d_{n}D))/\Gamma(X_{m},F(d_{n}D))$ is also zero. In other
words, there is $g_{m}\in\Gamma(X_{m},F(d_{n}D))$ such that $g_{m\rvert
X_{n}}=f_{n}$. ∎

By hypothesis there are an affine admissible formal $R$-scheme $Y$ and proper
morphism of formal $R$-schemes $\sigma\colon X\to Y$. Since $\sigma$ is proper
and $Y$ is affine, the projective system
$(\Gamma(X_{m},F(d_{n}D)))_{m\geqslant n}$ satisfies the Mittlag-Leffler
condition. In the discretely valued case, this is one the key ingredients of
the proof of the formal GAGA theorem in [EGA III1, Corollaire 4.1.7] (see also
[FGI+05, 8.2.7, p. 191]); the general case can be found in [Abb10, Corollaire
2.11.7] and [FK18, proposition I.11.3.3 and Lemma 0.8.8.5 (2)]. For
$m\geqslant n$ let
$\rho_{m}\colon\Gamma(X_{m},F(d_{n}D))\to\Gamma(X_{n},F(d_{n}D))$ be the
restriction map. According to the previous claim, the set
$E_{m}=\rho_{m}^{-1}(f_{n})$ is non-empty, so that the projective limit of the
projective system $(E_{m})_{m\geqslant n}$ is non-empty ([Stacks, lemma
0597]). In other words, there is
$g_{\infty}\in\projlim_{m\in\mathbb{N}}\Gamma(X_{m},F(d_{n}D))$ such that
$g_{\infty\rvert X_{n}}=f_{n}$. By using again that $\sigma$ is proper and $Y$
is affine, the natural map

$\varphi\colon\Gamma(X,F(d_{n}D))\longrightarrow\projlim_{m\geqslant
n}\Gamma(X_{m},F(d_{n}D)),$

is an isomorphism. One concludes the proof by taking
$g:=\varphi^{-1}(g_{\infty})$. ∎

##  4\. A Remmert factorization theorem with boundary

Let $K$ be a complete non-trivially valued non-Archimedean field . The
analogue of theorem 3.3 in the context of analytic spaces is naturally stated
for holomorphically convex spaces. In absence of boundary these spaces have
been characterized in [MP21, theorem 6.1] as being proper over boundaryless
Stein spaces—this is what is called the Remmert factorization theorem in
complex analysis. However affinoid spaces do have boundary and in order to
include them we present here a version of Remmert factorization for spaces
with boundary.

###  4.1. Reminder on Stein spaces

We begin with a reminder on Stein spaces in the non-Archimedean framework, see
also [MP21]. Recall that a $K$-analytic space $X$ is said to _holomorphically
separable_ if, for $x,x^{\prime}\in X$ distinct, there is
$f\in\Gamma(X,\mathcal{O}_{X})$ such that $|f(x)|\neq|f(x^{\prime})|$. It is
said _holomorphically convex_ if, for each $C\subset X$ compact, the
holomorphically convex hull of $C$,

$\hat{C}_{X}:=\\{x\in
X:|f(x)|\leqslant\|f\|_{C},f\in\Gamma(X,\mathcal{O}_{X})\\}$

is compact, where $\|f\|_{C}=\sup_{x\in C}|f(x)|$. An _affinoid Stein
exhaustion_ of $X$ is a G-cover $\\{X_{i}\\}_{i\in\mathbb{N}}$ of $X$ made of
affinoid domains such that, for $i\in\mathbb{N}$, $X_{i}$ is a Weierstrass
domain in $X_{i+1}$. Here we say that $X$ is _Stein_ if it admits an affinoid
Stein exhaustion. This is what is called being being W-exhausted by affinoid
domains in [MP21]. A $K$-analytic space Stein $X$ is holomorphically separable
and holomorphically convex by [MP21, theorem 1.1]. Let $X$ be a $K$-analytic
space admitting an affinoid Stein exhaustion $\\{X_{i}\\}_{i\in\mathbb{N}}$.
Then, for each coherent $\mathcal{O}_{X}$-module $F$,

  1. (1)

for all $i\in\mathbb{N}$, the restriction map $\Gamma(X,F)\to\Gamma(X_{i},F)$
has dense image;

  2. (2)

for all $q\geqslant 1$, the cohomology group $\textup{H}^{q}(X,F)$ vanishes.

Given a $K$-affinoid space $S$, closed analytic subspaces of
$\mathbb{A}^{n,\textup{an}}_{K}\times_{K}S$ are Stein. Conversely, let $X$ be
a $K$-analytic space without boundary such that

$\sup_{x\in
X}\dim_{\mathcal{H}(x)}\Omega_{X,x}\otimes_{\mathcal{O}_{X,x}}\mathcal{H}(x)<+\infty.$

Then, there exist $n\in\mathbb{N}$ and a closed embedding
$X\hookrightarrow\mathbb{A}^{n,\textup{an}}_{K}$. In particular for a
$K$-scheme of finite type $X$ the $K$-analytic space $X^{\textup{an}}$ is
Stein if and only if there is a closed embedding closed embedding
$X^{\textup{an}}\hookrightarrow\mathbb{A}^{n,\textup{an}}_{K}$ for some
$n\in\mathbb{N}$.

###  4.2. Statement

Let $\pi\colon X\to Y$ be a morphism of $K$-analytic spaces. A _Remmert
factorization_ of $\pi$ is a triple $(S,\sigma,\tilde{\pi})$ made of a Stein
$K$-analytic space $S$ and morphisms $\sigma\colon X\to S$, $\tilde{\pi}\colon
S\to Y$ such that $\pi=\tilde{\pi}\circ\sigma$ with the following properties:

  * RF1.

the morphism $\sigma$ is proper, surjective, with geometrically connected
fibers, and the homomorphism
$\sigma^{\sharp}\colon\mathcal{O}_{S}\to\sigma_{\ast}\mathcal{O}_{X}$ induced
by $\sigma$ an isomorphism;

  * RF2.

given a $K$-analytic space $T$ and a morphism of $K$-analytic spaces
$\tau\colon X\to T$ set-theoretically constant on the fibers of $\sigma$,
there is a unique morphism of $K$-analytic spaces $\tilde{\tau}\colon S\to T$
such that $\tau=\tilde{\tau}\circ\sigma$.

######  Lemma 4.1.

Let $\pi\colon X\to Y$ be a morphism of $K$-analytic spaces. Assume there is a
Remmert factorization $(S,\sigma,\tilde{\pi})$ of $\pi$. Let $T$ be a
holomorphically separable $K$-analytic space, and $\tau\colon X\to T$ a
morphism. Then, there exists a unique morphism $\tilde{\tau}\colon S\to T$
such that $\tau=\tilde{\tau}\circ\sigma$.

###### Proof.

Let $x,x^{\prime}\in X$ be such that $\sigma(x)=\sigma(x^{\prime})$. In view
of property RF2, it suffices to prove that $\tau(x)$ and $\tau(x^{\prime})$
coincide. Suppose, by contradiction, that this is not the case. Since $T$ is
holomorphically separable, there is a $K$-analytic function $f$ on $T$ such
that $|f(\tau(x))|\neq|f(\tau(x^{\prime}))|$. Because of the isomorphism
$\mathcal{O}_{S}\to\sigma_{\ast}\mathcal{O}_{X}$ we have
$\tau^{\ast}f=\sigma^{\ast}g$ for some $K$-analytic function $g$ on $S$. Then
$|g(\sigma(x))|\neq|g(\sigma(x^{\prime}))|$, whereas
$\sigma(x)=\sigma(x^{\prime})$. Contradiction. ∎

As usual, if it exists, a Remmert factorization is unique up to a unique
isomorphism.

######  Theorem 4.2.

Let $\pi\colon X\to Y$ be a morphism of $K$-analytic spaces where $X$ is
separated, holomorphically convex and countable at infinity, $Y$ is Stein, and
$\pi$ is without boundary.

Then, the Remmert factorization $(S,\sigma,\tilde{\pi})$ of $\pi$ exists and
the morphism $\tilde{\pi}$ is without boundary.

As an immediate consequence of theorem 4.2 one obtains:

######  Corollary 4.3.

Let $\pi\colon X\to S$ a morphism without boundary of $K$-analytic spaces with
$X$ is separated and countable at infinity, and $S$ Stein. Then $X$ is Stein
if and only if it is holomorphically separable and holomorphically convex.

In the rest of this section is devoted to the proof of theorem 4.2.

###  4.3. Proofs

The proof of theorem 4.2 follows closely that of [MP21, theorem 6.1], in turn
inspired by the proof of the complex Remmert factorization theorem. In order
to control boundary, we will need the following:

######  Lemma 4.4.

Let $\pi\colon X\to S$ and $f\colon X\to Y$ be morphisms between $K$-analytic
spaces, and let $V\subset X$ and $W\subset Y\times_{K}S$ be open subsets.
Consider the morphism of $K$-analytic spaces $F=(\pi,f)\colon X\to
Y\times_{K}S$. If $X$ is separated and $\pi$ is without boundary, then the
induced morphism $F\colon V\cap F^{-1}(W)\to W$ is without boundary.

###### Proof.

This follows by using repeatedly [Ber90, proposition 3.1.3 (iii)]. The
morphism $\pi\times\operatorname{id}\colon X\times_{K}Y\to S\times_{K}Y$ is
without boundary. Since $X$ is separated the graph
$(\operatorname{id},f)\colon X\to X\times_{K}Y$ is a closed immersion, hence
without boundary. Thus the composite morphism
$F:=(\pi,f)=(\pi\times\operatorname{id})\circ(\operatorname{id},f)\colon X\to
S\times_{K}Y$ is without boundary hence so is $F\colon F^{-1}(W)\to W$. In
particular restriction of $\pi$ to $V\cap F^{-1}(W)$ is without boundary too
because an open immersion is without boundary. ∎

We first prove theorem 4.2 when the target is affinoid:

######  Lemma 4.5.

Let $\pi\colon X\to Y$ be a morphism of $K$-analytic spaces. Suppose that the
$K$-analytic space $X$ is separated, holomorphically convex and countable at
infinity, the $K$-analytic space $Y$ is affinoid, and the morphism $\pi$ is
without boundary.

Then, the Remmert factorization $(S,\sigma,\tilde{\pi})$ of $\pi$ exists and
the morphism $\tilde{\pi}$ is without boundary.

###### Proof.

Let $\\{C_{i}\\}_{i\geqslant 0}$ be an exhaustion of $X$ by compact subsets
such that, for each $i$, $C_{i}$ is the closure of its interior and $C_{i}$ is
contained in the interior of $C_{i+1}$. Set

$n_{-1}=0,\qquad X_{-1}=U_{-1}=V_{-1}=\emptyset.$

For $i\geqslant 0$, we construct inductively an integer $n_{i}\geqslant 0$, a
relatively compact open subset $U_{i}$ of $X$, a compact analytic domain
$X_{i}$ of $U_{i}$, an open subset $W_{i}$ of
$\mathbb{A}^{n_{i},\textup{an}}_{Y}$ and $K$-analytic functions
$f_{n_{i-1}+1},\dots,f_{n_{i}}$ on $X$ such that:

  1. (1)

the morphism $(f_{1},\dots,f_{n_{i}},\pi)\colon
X\to\mathbb{A}^{n_{i},\textup{an}}_{Y}$ induces a proper morphism $F_{i}\colon
U_{i}\to W_{i}$;

  2. (2)

$\mathbb{D}_{i}:=\mathbb{D}(\|f_{1}\|_{K_{i}\cup\overline{U}_{i-1}},\dots,\|f_{n_{i}}\|_{K_{i}\cup\overline{U}_{i-1}})\times_{K}Y$
is contained in $W_{i}$;

  3. (3)

$X_{i}:=F_{i}^{-1}(\mathbb{D}_{i})\cap U_{i}$ contains the holomorphically
convex hull of the compact subset $K_{i}\cup\overline{U}_{i-1}$.

Set $C^{\prime}_{i}:=C_{i}\cup\overline{U}_{i-1}$ and consider a relatively
compact open subset $V_{i}\subset X$ containing the holomorphically convex
hull of $C^{\prime}_{i}$ in $X$ (which is compact since $X$ is holomorphically
convex). Let $\partial V_{i}$ be the topological boundary of $V_{i}$ in $X$.
Since $\partial V_{i}$ is compact, there are $K$-analytic functions
$f_{n_{i-1}+1},\dots,f_{n_{i}}$ on $X$ such that, for all $x\in\partial
V_{i}$,

$\max_{j=1,\dots,n_{i}}\frac{|f_{j}(x)|}{\|f_{j}\|_{C^{\prime}_{i}}}>1.$

Set $W_{i}:=\mathbb{A}^{n_{i},\textup{an}}_{T}\smallsetminus F_{i}(\partial
V_{i})$ and $U_{i}=V_{i}\cap F_{i}^{-1}(W_{i})$. Property (1) holds because
the morphism $F_{i}\colon U_{i}\to W_{i}$ induced by
$(f_{1},\dots,f_{n_{i}},\pi)\colon X\to\mathbb{A}^{n_{i},\textup{an}}_{Y}$ is
without boundary (lemma 4.4) and $F_{i}$ is topologically proper by
construction, thus proper [Ber90, p. 50]. Properties (2) and (3) hold by
construction. Let $X_{i}$ denote $\mathbb{D}_{i}\times_{W_{i}}U_{i}$. The
morphism $X_{i}\to\mathbb{D}_{i}$ induced by $F_{i}$ is proper. The “Stein
factorization theorem” ([Ber90, proposition 3.3.7]) states that the Remmert
factorization $(S_{i},\sigma_{i},\tilde{F}_{i})$ of $F_{i}$ exists. Indeed,
the morphism $\tilde{F}_{i}\colon S_{i}\to\mathbb{D}_{i}$ is finite, so that
$S_{i}$ is affinoid (and, in particular, Stein). Also, in _loc. cit._ the
fibers are only stated to be connected. However, because the construction of
the Stein factorization is compatible with extension of scalars, fibers are
actually geometrically connected. It follows that, for each $i\in\mathbb{N}$,
the Banach $K$-algebra $\mathcal{O}(X_{i})$ is affinoid. Moreover, for
$i\in\mathbb{N}$, the restriction map
$\mathcal{O}(X_{i+1})\to\mathcal{O}(X_{i})$ is a bounded homomorphism of
$K$-affinoid algebras and it induces a morphism of $K$-affinoid spaces
$\varepsilon_{i}\colon S_{i}\to S_{i+1}$.

######  Claim 4.6.

The map $\varepsilon_{i}$ identifies $S_{i}$ with a Weierstrass domain of
$S_{i+1}$.

The proof of the claim is by any means analogous to the that of [MP21, claim
6.3] and we do not repeat it here. Consider the $K$-analytic space
$S=\bigcup_{i\in\mathbb{N}}S_{i}$ and the morphism $\sigma\colon X\to S$
obtained by glueing the morphisms $\sigma_{i}$. The $K$-analytic space $S$ is
Stein by definition. Properties (RF1) and (RF2) are deduced from the
properties of $\sigma_{i}$, for $i\in\mathbb{N}$. (Note that, for each
$i\in\mathbb{N}$, the preimage $\sigma^{-1}(S_{i})$ of $S_{i}$ is $X_{i}$.)
The morphism $\tilde{\pi}$ is seen to be without boundary because $\sigma$ is
surjective and the morphism $\pi$ is without boundary ([Ber90, proposition
3.1.3 (ii)]). ∎

Let $\operatorname{Hom}_{K}(Y,X)$ denote the set of morphisms betwee some
$K$-analytic spaces $X$ and $Y$. If $X$ and $Y$ are countable at infinity, let
$\operatorname{Hom}_{K\textup{-alg}}(\mathcal{O}(X),\mathcal{O}(Y))$ be the
set of $K$-algebra homomorphisms $\mathcal{O}(X)\to\mathcal{O}(Y)$ that are
continuous with respect the natural Fréchet topologies on $\mathcal{O}(X)$ and
$\mathcal{O}(Y)$.

######  Proposition 4.7.

Let $X$ and $Y$ be $K$-analytic spaces countable at infinity with $X$ Stein.
Then the following map is bijective:

$\operatorname{Hom}_{K}(Y,X)\longrightarrow\operatorname{Hom}_{K\textup{-alg}}(\mathcal{O}(X),\mathcal{O}(Y)),\quad
f\colon Y\to X\longmapsto f^{\sharp}\colon\mathcal{O}(X)\to\mathcal{O}(Y).$

###### Proof.

By the very definition of morphism between $K$-analytic spaces, the statements
holds when $X$ is a $K$-affinoid space.

(Injective.) Let $f_{1},f_{2}\colon Y\to X$ be morphisms of $K$-analytic
spaces. Let $\\{Y_{i}\\}_{i\in\mathbb{N}}$ be a G-cover of $Y$ by compact
analytic domains such $Y_{i}\subset Y_{i+1}$ for all $i\in\mathbb{N}$. Let
$\\{X_{i}\\}_{i\in\mathbb{N}}$ be an affinoid Stein exhaustion of $X$ such
that, for $\alpha=1,2$ and $i\in\mathbb{N}$, the compact subset
$f_{\alpha}(Y_{i})$ is contained in $X_{i}$. The morphism $f_{\alpha}$ then
induces a morphism $f_{\alpha,i}\colon Y_{i}\to X_{i}$. Suppose
$f_{1}^{\sharp}=f_{2}^{\sharp}$. For $i\in\mathbb{N}$, the $K$-algebra
$\mathcal{O}(X)$ is dense in the affinoid $K$-algebra $\mathcal{O}(X_{i})$.
Therefore, the continuous homomorphisms

$f_{1,i}^{\sharp},f_{2,i}^{\sharp}\colon\mathcal{O}(X_{i})\longrightarrow\mathcal{O}(Y_{i}),$

induced by $f_{1,i}$ and $f_{2,i}$ coincide. By the affinoid case, the
morphisms $f_{1,i}$ and $f_{2,i}$ are the same. Since $i\in\mathbb{N}$ is are
arbitrary, one deduces $f_{1}=f_{2}$.

(Surjective.) Let $\varphi\colon\mathcal{O}(X)\to\mathcal{O}(Y)$ be a
continuous $K$-algebra homomorphism. Let $\\{Y_{i}\\}_{i\in\mathbb{N}}$ be a
G-cover of $Y$ by compact analytic domains such $Y_{i}\subset Y_{i+1}$ for all
$i\in\mathbb{N}$. For $i\in\mathbb{N}$, let $\|\cdot\|_{Y,i}$ be a norm on the
Banach $K$-algebra $\mathcal{O}(Y_{i})$ defining its topology. By the
definition of the topology on $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ and the
continuity of $\varphi$, for each $i\in\mathbb{N}$, there is a real number
$R_{i}>0$, a compact analytic domain $X_{i}$ of $X$ and a norm
$\|\cdot\|_{X,i}$ defining the topology of $\mathcal{O}(X_{i})$ such that, for
all $f\in\mathcal{O}(X)$,

$\|\varphi(f)\|_{Y,i}\leqslant R_{i}\|f\|_{X,i}.$

Arguing by induction and by possibly enlarging $X_{i}$, we may assume that the
collection $\\{X_{i}\\}_{i\in\mathbb{N}}$ is an affinoid Stein exhaustion. By
density of $\mathcal{O}(X)$ in $\mathcal{O}(X_{i})$, the homomorphism
$\varphi$ induces a continuous homomorphism of Banach $K$-algebras
$\varphi_{i}\colon\mathcal{O}(X_{i})\to\mathcal{O}(Y_{i})$. By the affinoid
case, for each $i\in\mathbb{N}$, there is a unique morphism of $K$-analytic
spaces $f_{i}\colon Y_{i}\to X_{i}$ such that the induced homomorphism
$f_{i}^{\sharp}\colon\mathcal{O}(X_{i})\to\mathcal{O}(Y_{i})$ is $\varphi$.
Moreover, by density of $\mathcal{O}(X)$ in $\mathcal{O}(X_{i})$ and
$\mathcal{O}(X_{i+1})$, the following diagram is commutative:

${Y_{i}}$${X_{i}}$${Y_{i+1}}$${X_{i+1},}$$\leftarrow$$\rightarrow$$\scriptstyle{f_{i}}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{f_{i+1}}$

where the vertical arrows are the inclusions. Therefore the morphisms $f_{i}$
glue to a morphism $f\colon Y\to X$ such that $f^{\sharp}=\varphi$. ∎

######  Lemma 4.8.

Let $\pi\colon X\to Y$ be a morphism of $K$-analytic spaces with $X$ Stein and
$Y$ affinoid. Let $Y^{\prime}\subset Y$ be a Weierstrass domain and
$X^{\prime}:=\pi^{-1}(Y^{\prime})$. Then,

  1. (1)

the $K$-analytic space $X^{\prime}$ is Stein;

  2. (2)

for any affinoid Stein exhaustion $\\{X^{\prime}_{i}\\}_{i\in\mathbb{N}}$ of
$X^{\prime}$, there is an affinoid Stein exhaustion
$\\{X_{i}\\}_{i\in\mathbb{N}}$ of $X$ such that, for each $i\in\mathbb{N}$,
$X^{\prime}_{i}$ is a Weierstrass domain of $X_{i}$.

###### Proof.

By definition of a Weierstrass domain there are
$f_{1},\dots,f_{n}\in\Gamma(Y,\mathcal{O}_{Y})$ and real numbers
$r_{1},\dots,r_{n}>0$ such that $Y^{\prime}=\\{y\in Y:|f_{j}(y)|\leqslant
r_{j},j=1,\dots,n\\}$. Given an affinoid Stein exhaustion
$\\{W_{i}\\}_{i\in\mathbb{N}}$ of $X$ define, for $i\in\mathbb{N}$,

$W^{\prime}_{i}=\\{x\in W_{i}:|f_{j}(\pi(x))|\leqslant r_{j},j=1,\dots,n\\}.$

(1) The collection $\\{W^{\prime}_{i}\\}_{i\in\mathbb{N}}$ is an affinoid
Stein exhaustion of $X^{\prime}$.

(2) Note that, for each $i\in\mathbb{N}$, the $K$-affinoid space
$W^{\prime}_{i}$ is a Weierstrass domain of $W_{i}$. Let
$\\{X_{i}^{\prime}\\}_{i\in\mathbb{N}}$ be affinoid Stein exhaustion of
$X^{\prime}$. Then, there is an increasing function
$\tau\colon\mathbb{N}\to\mathbb{N}$ such that, for each $i\in\mathbb{N}$,
$X^{\prime}_{i}$ is a Weierstrass domain in $W^{\prime}_{\tau(i)}$. For the
latter note that it suffices that $X_{i}^{\prime}$ is contained in
$W^{\prime}_{\tau(i)}$ because the restriction map
$\mathcal{O}(X^{\prime})\to\mathcal{O}(X^{\prime}_{i})$ has a dense image,
thus _a fortiori_ $\mathcal{O}(W^{\prime}_{i})\to\mathcal{O}(X^{\prime}_{i})$.
In particular, the affinoid Stein exhaustion
$\\{W_{\tau(i)}\\}_{i\in\mathbb{N}}$ does the job. ∎

######  Lemma 4.9.

Under the hypotheses of lemma 4.5 let $(S,\sigma,\tilde{\pi})$ be the Remmert
factorization of $\pi$ and $Y^{\prime}\subset Y$ a Weierstrass domain. Then
$(S\times_{Y}Y^{\prime},\sigma_{\rvert
X\times_{Y}Y^{\prime}},\tilde{\pi}_{\rvert S\times_{Y}Y^{\prime}})$ is the
Remmert factorization of $\pi_{\rvert X^{\prime}}$.

###### Proof.

Set $X^{\prime}:=X\times_{Y}Y^{\prime}$. Since $X^{\prime}\subset X$ is closed
and $X$ is holomorphically convex, the $K$-analytic space $X^{\prime}$ is
holomorphically convex. The morphism $\pi^{\prime}\colon X^{\prime}\to
Y^{\prime}$ is without boundary and $Y^{\prime}$ is affinoid. Therefore, by
lemma 4.5, the Remmert factorization
$(S^{\prime},\sigma^{\prime},\tilde{\pi}^{\prime})$ of $\pi^{\prime}$ exists.
By lemma 4.1, there is a unique morphism $\varphi\colon S^{\prime}\to S$ such
that $\sigma_{\rvert X^{\prime}}=\varphi\circ\sigma^{\prime}$. By the Open
Mapping theorem of Fréchet spaces [Sch02, corollary 8.7] the continuous
isomorphisms $\sigma_{\rvert
X^{\prime}}^{\sharp}\colon\mathcal{O}(S\times_{Y}Y^{\prime})\to\mathcal{O}(X^{\prime})$
and
$\sigma^{\prime\sharp}\colon\mathcal{O}(S^{\prime})\to\mathcal{O}(X^{\prime})$
induced respectively by the proper morphisms $\sigma_{\rvert X^{\prime}}$ and
$\sigma^{\prime}$ are homeomomorphisms. It follows that the homomorphism
$\varphi^{\sharp}\colon\mathcal{O}(S\times_{Y}Y^{\prime})\to\mathcal{O}(S^{\prime})$
induced by $\varphi$ is an isomorphism of Fréchet algebras. Since
$S\times_{Y}Y^{\prime}$ and $S^{\prime}$ are Stein (lemma 4.8), proposition
4.7 implies that $\varphi$ is an isomorphism. ∎

###### Proof of theorem 4.2.

Let $\\{Y_{i}\\}_{i\in\mathbb{N}}$ be an affinoid Stein exhaustion of $Y$. For
each $i\in\mathbb{N}$, set $X_{i}:=\pi^{-1}(Y_{i})$. By lemma 4.5, the Remmert
factorization $(S_{i},\sigma_{i},\tilde{\pi}_{i})$ of $\pi\colon X_{i}\to
Y_{i}$ exists. Moreover, by lemma 4.9, the Remmert factorization
$(S_{i},\sigma_{i},\tilde{\pi}_{i})$ is canonically isomorphic to
$(S^{\prime}_{i},\sigma_{i+1\rvert S^{\prime}_{i}},\tilde{\pi}_{i+1\rvert
S^{\prime}_{i}})$, where $S^{\prime}_{i}:=\tilde{\pi}^{-1}(Y_{i+1})$. In what
follows, we identify $S_{i}$ with $S_{i}^{\prime}$ through the preceding
canonical isomorphism. Thanks to lemma 4.8, construct by induction for each
$i\in\mathbb{N}$, a W-exhaustion by Weierstrass domain
$\\{S_{ij}\\}_{j\in\mathbb{N}}$ of $S_{i}$ such that, for $i^{\prime}\leqslant
i$ and $j\in\mathbb{N}$, the affinoid space $S_{i^{\prime}j}$ is a Weierstrass
domain in $S_{ij}$. It follows that $\\{S_{ii}\\}_{i\in\mathbb{N}}$ is an
affinoid Stein exhaustion of the $K$-analytic space
$S=\bigcup_{i\in\mathbb{N}}S_{i}$. Properties (RF1) and (RF2) are deduced from
those of $\sigma_{i}$, for $i\in\mathbb{N}$. Analogously, the morphism $\pi$
is seen to be without boundary. ∎

##  5\. Approximating essential singularities by poles: the analytic case

Let $R$ be the ring of integers of a field $K$ complete with respect to a on-
trivial non-Archimedean absolute value.

###  5.1. Statements

The main result of this section is the analogue of theorem 3.3 in the context
of analytic spaces. In order to state them we need to construct the sheaf of
functions with polar singularities:

######  Lemma 5.1.

Let $D$ be an effective Cartier divisor in a $K$-analytic space $X$ and $F$ a
coherent sheaf of $\mathcal{O}_{X}$-modules. Let $j\colon X\smallsetminus D\to
X$ be the open immersion.

  1. (1)

The restriction map $F(\ast D)\to j_{\ast}j^{\ast}F$ is injective.

  2. (2)

If $X$ is compact and $F$ is $D$-torsion free, then the restriction induces an
isomorphism

$\injlim_{d\in\mathbb{N}}\Gamma(X,F(dD))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Gamma(X,F(\ast
D)).$

###### Proof.

(1) The statement is local on $X$, thus we may suppose $X$ affinoid and
$I_{D}$ is principal. Let $A=\Gamma(X,\mathcal{O}_{X})$, $f\in A$ a generator
of $\Gamma(X,I_{D})$, and $Y$ the support of a section $t\in\Gamma(X,F)$, that
is, the support of the coherent $\mathcal{O}_{X}$-submodules
$t\mathcal{O}_{X}$ of $F$. If $J=\operatorname{Ann}(t)$ is the annihilator
ideal of $t$, then $Y=\mathcal{M}(A/J)$. Suppose $s$ vanishes identically on
$U$. Then $Y$ is set-theoretically contained in $D$, hence $f$ vanishes
identically on it. According to [Ber90, proposition 2.1.4 (i)], there is a
positive integer $d\geqslant 1$ such that $f^{d}\in J$. In other words, the
image $f^{d}t$ of $t$ in $\Gamma(X,F(dD))$ vanishes.

(2) Since $F$ is supposed to be $D$-torsion free, for each $d\in\mathbb{N}$,
the multiplication by the canonical section $s_{D}\colon F(dD)\to F((d+1)D)$
is injective. The $K$-analytic space being compact (thus quasi-compact), the
statement follows from [Stacks, lemma 009F (3)]. ∎

We are now in position to state the main result. Recall that for a ringed
space $X$ recall that an $\mathcal{O}_{X}$-module is said to be _semi-
reflexive_ if the homomorphism $F\to F^{\vee\vee}$ is injective.

######  Theorem 5.2.

Let $D$ be an effective Cartier divisor in a separated holomorphically convex
$K$-analytic space $X$. Assume there is a morphism without boundary $X\to S$
of $K$-analytic spaces with $S$ Stein. Then for any semi-reflexive coherent
$\mathcal{O}_{X}$-module $F$ the natural map

$\Gamma(X,F(\ast D))\longrightarrow\Gamma(X\smallsetminus D,F)$

is injective, and has a dense image.

Recall that the topology on the global sections of a coherent sheaf $F$ on a
$K$-analytic space $X$ is defined as follows. If $X=\mathcal{M}(A)$ is the
Banach spectrum of a $K$-affinoid algebra $A$, then the $A$-module
$\Gamma(X,F)$ is finitely generated. The topology on $\Gamma(X,F)$ is induced
via some surjective homomorphism $A^{n}\to\Gamma(X,F)$ of $A$-modules by the
one on $A$. It does not depend on the choice of said surjection. When $X$ is
arbitrary, the topology on $\Gamma(X,F)$ is the coarsest one for which, given
an affinoid domain $V\subset X$, the restriction map
$\Gamma(X,F)\to\Gamma(V,F)$ is continuous. If $X$ is reduced and
$F=\mathcal{O}_{X}$, then the so-defined topology coincides with the topology
of uniform convergence over compact subsets of $X$.

It is sometimes useful to combine theorem 5.2 with the following non-density
result. To state it, say that an open subset $U$ in a $K$-analytic space $X$
is _scheme-theoretically dense_ if the homomorphism of
$\mathcal{O}_{X}$-modules $\mathcal{O}_{X}\to j_{\ast}\mathcal{O}_{U}$ is
injective, where $j\colon U\to X$ is the open immersion.

######  Proposition 5.3.

Let $Z$ be a closed analytic subspace of a quasi-separated $K$-analytic space
$X$ which is countable at infinity. If $X\smallsetminus Z$ is scheme-
theoretically dense, then for any semi-reflexive coherent
$\mathcal{O}_{X}$-module $F$ the restriction map

$\rho\colon\Gamma(X,F)\longrightarrow\Gamma(X\smallsetminus Z,F)$

is a homeomorphism onto its image and its image is a closed subset.

The rest of this section is devoted to the proof of these results.

###  5.2. Extension of semi-reflexive modules

When $X$ is proper theorem 5.2 will be deduced from theorem 3.3 by passing to
the generic fiber. However to do so we need to extend modules from the generic
fiber to the formal scheme preserving the hypotheses needed to apply theorem
3.3. To start with recall that coherent sheaves can be extended from the
generic fiber to the formal scheme. More precisely, let $X$ be a quasi-compact
admissible formal $R$-scheme and $F_{K}$ be a coherent
$\mathcal{O}_{X_{\eta}}$-module. Then there is a coherent
$\mathcal{O}_{X}$-module $F$ flat over $R$ and an isomorphism $F_{\eta}\cong
F_{K}$. See [Lüt90, lemma 2.2] in the discretely valued case and [BL93,
proposition 5.6], [Abb10, proposition 5.6] or [FK18, Corollary II.5.3.3] in
the general case. Note that, a coherent sheaf $F$ on a $X$, the natural
homomorphism $\Gamma(X,F)\otimes_{R}K\to\Gamma(X_{\eta},F_{\eta})$ is an
isomorphism. The task here is the preserve the semi-reflexivity and the
flatness on the Cartier divisor. We start with:

######  Lemma 5.4.

Let $F$ be a coherent sheaf flat over $R$ on an admissible formal $R$-scheme
$X$. If $F_{\eta}$ is semi-reflexive, then $F$ is semi-reflexive.

###### Proof.

The statement is local on $X$, thus one may suppose that
$X=\operatorname{Spf}(A)$ is affine, where $A$ is a topologically finitely
presented flat $R$-algebra. Let $M:=\Gamma(X,F)$. Then
$\Gamma(X_{\eta},F_{\eta})=M\otimes_{R}K$ and
$\Gamma(X_{\eta},F_{\eta}^{\vee\vee})=(M\otimes_{R}K)^{\vee\vee}$. Since $M$
is finitely generated and $K$ is flat over $R$, the natural homomorphism
$M^{\vee\vee}\otimes_{R}K\to(M\otimes_{R}K)^{\vee\vee}$ is an isomorphism.
Consider the following commutative diagram:

${M}$${M^{\vee\vee}}$${M\otimes_{R}K}$${M^{\vee\vee}\otimes_{R}K.}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\alpha}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\beta}$

The homomorphism $\alpha$ is injective because $M$ is flat over $R$, while
$\beta$ is injective because $F_{\eta}$ is semi-reflexive. It follows that
$M\to M^{\vee\vee}$ is injective. ∎

######  Lemma 5.5.

Let $X$ be a quasi-compact admissible formal $R$-scheme. Let $D$ be an
_admissible_ effective Cartier divisor in $X$. Let $F$ be a coherent
$\mathcal{O}_{X}$-module.

Then, $F^{\vee}$ is $D$-torsion free, and $F^{\vee}$ and $F^{\vee}_{\rvert D}$
are flat over $R$.

###### Proof.

The statement is local on $X$. Therefore one may assume that
$X=\operatorname{Spf}(A)$ is affine, for a topologically finitely presented
flat $R$-algebra $A$, and the divisor $D$ is given by the ideal $fA$ for some
nonzerodivisor $f\in A$. Let $M:=\Gamma(X,F)$. The $A$-module
$\operatorname{Hom}_{A}(M,A)$ is $D$-torsion free because $f$ is a
nonzerodivisor in $A$. In order to show that the $A$-modules
$\operatorname{Hom}_{A}(M,A)$ and $\operatorname{Hom}_{A}(M,A)\otimes_{A}A/fA$
are flat, it suffices to show that they have no $\varpi$-torsion, for any
$\varpi\in R\smallsetminus\\{0\\}$. Let $\varpi\in R\smallsetminus\\{0\\}$.
The $A$-module $\operatorname{Hom}_{A}(M,A)$ is $\varpi$-torsion-free because
$\varpi$ is a nonzerodivisor in $A$. Concerning
$\operatorname{Hom}_{A}(M,A)\otimes_{A}A/fA$, given a homomorphism of
$A$-modules $\varphi\colon M\to A$ such that $\varpi\varphi=f\psi$ for some
$\psi\in\operatorname{Hom}_{A}(M,A)$, one needs to show that there is
$\varphi^{\prime}\in\operatorname{Hom}_{A}(M,A)$ such that
$\varphi=f\varphi^{\prime}$. Let $\bar{\varphi}\colon M/fM\to A/fA$ be the
homomorphism induced by $\varphi$. If $\varpi\varphi=f\psi$ for some
$\psi\in\operatorname{Hom}_{A}(M,A)$, then $\varpi\bar{\varphi}$ is $0$.
However, multiplication by $\varpi$ on $A/fA$ is injective because the Cartier
divisor $D$ is admissible. Therefore, the homomorphism $\bar{\varphi}$ is $0$.
Since the multiplication by $f$ on $A$ is injective, this means that $\varphi$
can be written as $f\varphi^{\prime}$ for some
$\varphi^{\prime}\in\operatorname{Hom}_{A}(M,A)$. ∎

######  Lemma 5.6.

Let $D$ be an effective Cartier divisor in a quasi-compact admissible formal
$R$-scheme $X$. Let $F$ be a coherent $\mathcal{O}_{X}$-module which is
$D$-torsion free, flat over $R$, and such that $F_{\rvert D}$ is flat over
$R$. Let $E\subset F$ be a coherent $\mathcal{O}_{X}$-submodule. Then, the
$\mathcal{O}_{X}$-module

$E^{\textup{sat}}:=\operatorname{Ker}(F\to[F(\ast D)/E(\ast D)]_{K})$

is coherent, $D$-torsion free, flat over $R$, $E^{\textup{sat}}_{\rvert D}$ is
flat over $R$, $E_{\eta}(\ast D_{\eta})\cong E_{\eta}^{\textup{sat}}(\ast
D_{\eta})$ and $E_{\rvert(X\smallsetminus D)_{\eta}}\cong
E^{\textup{sat}}_{\rvert(X\smallsetminus D)_{\eta}}$.

###### Proof.

The statement is local on $X$. Therefore one may assume that
$X=\operatorname{Spf}(A)$ is affine, where $A$ is a topologically finitely
presented flat $R$-algebra, and $D$ is a principal Cartier divisor whose ideal
is $fA$ for a nonzerodivisor $f\in A$. Let $M:=\Gamma(X,E)$,
$M^{\prime}:=\Gamma(X,E^{\textup{sat}})$ and $N:=\Gamma(X,F)$. Let
$A_{K}:=A\otimes_{R}K$. With this notation,

$M^{\prime}=\operatorname{Ker}(N\to(N/M)\otimes_{A}A_{K}[1/f]).$

The $A$-module $N$ has no $R$-torsion, thus $M^{\prime}$ has no $R$-torsion
too. It follows that $M^{\prime}$ is flat over $R$. Analogously, the
multiplication by $f$ on $N$ is injective, thus it is injective also on
$M^{\prime}$. By definition, $N/M^{\prime}$ injects into
$(N/M)\otimes_{A}A_{K}[1/f]$, hence $N/M^{\prime}$ is flat over $R$. It
follows from lemma 2.2 (2) that $M^{\prime}$ is finitely generated. It remains
to prove that $M^{\prime}/fM^{\prime}$ is $R$-torsion free. Suppose there are
$m,m^{\prime}\in M^{\prime}$ and $\varpi\in R\smallsetminus\\{0\\}$ such that
$\varpi m=fm^{\prime}$. We want to show that there is $m^{\prime\prime}\in
M^{\prime}$ such that $m=fm^{\prime\prime}$. Now, multiplication by $\varpi$
on $N/fN$ is injective, therefore there is $n\in N$ such that $m=fn$. Since
$f$ is invertible in $A_{K}[1/f]$, $n=m/f$ belongs to
$M\otimes_{A}A_{K}[1/f]$. Therefore $n$ belongs to $M^{\prime}$, and
$m^{\prime\prime}=n$ does the job. The first isomorphism follows from the
equality $M\otimes_{A}A_{K}[1/f]=M^{\prime}\otimes_{A}A_{K}[1/f]$, which holds
by definition of $M^{\prime}$. To prove the second isomorphism, one has to
show that the homomorphism $i_{K}\colon M\otimes_{A}A_{K}\\{1/f\\}\to
M^{\prime}\otimes_{A}A_{K}\\{1/f\\}$ is an isomorphism. The homomorphism
$i_{K}$ is injective because the $A$-algebra $A_{K}\\{1/f\\}$ is flat. Let
$m^{\prime}\in M^{\prime}$. By definition of $M^{\prime}$, there are
$\varpi\in R\smallsetminus\\{0\\}$ and a non-negative integer $n\in\mathbb{N}$
such that $\varpi f^{n}m^{\prime}$ belongs to $M$. Since both $\varpi$ and $f$
are invertible in $A_{K}\\{1/f\\}$, one can consider the element $m:=\varpi
f^{n}m^{\prime}\otimes 1/(\varpi f^{n})$ of $M\otimes_{A}A_{K}\\{1/f\\}$.
Clearly, $m$ maps to $m^{\prime}$ via $i_{K}$. This shows that $i_{K}$ is
surjective. ∎

######  Lemma 5.7.

Let $D$ be an admissible effective Cartier divisor in a quasi-compact
admissible formal $R$-scheme $X$ and $F_{K}$ a semi-reflexive coherent sheaf
on $X_{\eta}$.

Then, there exist a coherent $\mathcal{O}_{X}$-module $F$ flat over $R$ and
$D$-torsion free such that $F_{\rvert D}$ is flat over $R$, and an injective
homomorphism of $\mathcal{O}_{X_{\eta}}$-modules $F_{K}\hookrightarrow
F_{\eta}$ inducing isomorphisms $F_{K}(\ast D_{\eta})\cong F_{\eta}(\ast
D_{\eta})$ and $F_{K\rvert U_{\eta}}\cong F_{\eta\rvert U_{\eta}}$.

###### Proof.

Let $E$ be a coherent $\mathcal{O}_{X}$-module flat over $R$ such that
$E_{\eta}\cong F_{K}$. By lemma 5.4 $E$ is semi-reflexive because $F_{K}$. By
lemma 5.5 the coherent $\mathcal{O}_{X}$-module $E^{\vee\vee}$ is $D$-torsion
free, flat over $R$, and $E^{\vee\vee}_{\rvert D}$ is flat over $R$. According
to lemma 5.6, the coherent $\mathcal{O}_{X}$-module
$F:=\operatorname{Ker}(E^{\vee\vee}\to[E^{\vee\vee}(\ast D)/E(\ast D)]_{K})$
does the job. ∎

###  5.3. Proof of proposition 5.3

We begin with the proof of proposition 5.3 which is somewhat easier and does
not make use of theorem 3.3. To reduce to strict spaces we will use the
following:

######  Lemma 5.8.

Let $F$ be a coherent sheaf on a $K$-analytic space $X$ and $K^{\prime}$ a
complete valued extension of $K$. Then, the natural homomorphism
$\Gamma(X,F)\to\Gamma(X^{\prime},F^{\prime})$, where $X^{\prime}$ and
$F^{\prime}$ are obtained from $X$ and $F$ by extending scalars, is a
homeomorphism onto its image and its image is closed.

###### Proof.

Because of the definition of the topology on the global sections of a coherent
sheaf, by taking an affinoid G-cover one reduces to the case where $X$ is
affinoid. Once $X$ is supposed affinoid, by embedding it as a closed subspace
of a closed disc, one reduces to the case $X=\mathbb{D}(r_{1},\dots,r_{n})$
for $r_{1},\dots,r_{n}\in\mathbb{R}_{>0}$. By taking a surjective homomorphism
of $\mathcal{O}_{X}$-modules $\mathcal{O}_{X}^{r}\to F$, one reduces to the
case $F=\mathcal{O}_{X}$. The statement for the map

$\Gamma(X,\mathcal{O}_{X})=K\\{t_{1}/r_{1},\dots,t_{n}/r_{n}\\}\longrightarrow\Gamma(X^{\prime},\mathcal{O}_{X}^{\prime})=K^{\prime}\\{t_{1}/r_{1},\dots,t_{n}/r_{n}\\},$

is clearly true, the topology on both sides being defined by Gauss norm with
respect to the radii $r_{1},\dots,r_{n}$. This concludes the proof. ∎

###### Proof of proposition 5.3.

Write $X_{K}$, $Z_{K}$ and $F_{K}$ instead of $X$, $Z$ and $F$. According to
lemma 5.8 one may suppose that the $K$-analytic space $X_{K}$ is strict. In
this case the proof consists of three steps of increasing generality.

_Step 1._ Suppose $X_{K}$ compact and $Z_{K}$ an effective Cartier divisor. It
suffices to prove the following statement. Given a compact analytic domain
$C\subset X_{K}\smallsetminus Z_{K}$, there is a compact analytic domain
$C^{\prime}\subset X_{K}\smallsetminus Z_{K}$ containing $C$ such that the
restriction map $\Gamma(X_{K},F_{K})\to\Gamma(C^{\prime},F_{K})$ is a
homeomorphism onto its image. According to theorem 2.7, there are a quasi-
compact quasi-separated admissible formal $R$-scheme $X$ and an admissible
effective Cartier divisor $D$ in $X$ whose generic fibers are respectively
$X_{K}$ and $Z_{K}$. Up to replacing $X$ by its blow-up in
$\varpi\mathcal{O}_{X}+I_{D}$ for some $\varpi\in R\smallsetminus\\{0\\}$, one
may assume $C=U_{\eta}$ (lemma 5.11 and proposition 2.12). According to lemma
5.7, there are a coherent $\mathcal{O}_{X}$-module $F$ and an injective
homomorphism of $\mathcal{O}_{X_{K}}$-modules $\iota\colon F_{K}\to F_{\eta}$
inducing an isomorphism of $\mathcal{O}_{U_{\eta}}$-modules $F_{K\rvert
U_{\eta}}\simeq F_{\eta\rvert U_{\eta}}$. Now
$\Gamma(X_{K},F_{\eta})=\Gamma(X,F)\otimes_{R}K$ and
$\Gamma(U_{\eta},F_{\eta})=\Gamma(U,F)\otimes_{R}K$ because $X$ and $U$ are
quasi-compact. Therefore by lemma 3.6 (4) the restriction homomorphism
$\Gamma(X_{K},F_{\eta})\to\Gamma(U_{\eta},F_{\eta})$ is an isometric
embedding. On the other hand, the exact sequence

$0\longrightarrow\Gamma(X_{K},F_{K})\longrightarrow\Gamma(X_{K},F_{\eta})\longrightarrow\Gamma(X_{K},\operatorname{Coker}\iota)$

show that the image of
$\iota\colon\Gamma(X_{K},F_{K})\to\Gamma(X_{K},F_{\eta})$ is closed. Banach’s
Open Mapping theorem implies $\iota$ maps $\Gamma(X_{K},F_{K})$ into a closed
subset of $\Gamma(X_{K},F_{\eta})$. This concludes the proof in this case.

_Step 2._ Suppose $X_{K}$ compact. Let $\pi\colon X^{\prime}_{K}\to X_{K}$ be
the blow-up of $X_{K}$ in $Z_{K}$. Let $Z^{\prime}_{K}=\pi^{-1}(Z_{K})$, and
$U^{\prime}_{K}=X^{\prime}_{K}\smallsetminus Z^{\prime}_{K}$, so that the
induced map $\pi\colon U^{\prime}_{K}\to U_{K}$ is an isomorphism. The
coherent $\mathcal{O}_{X^{\prime}_{K}}$-module $\pi^{\ast}F_{K}$ may fail to
be semi-reflexive. Nonetheless the image $F^{\prime}_{K}$ in
$(\pi^{\ast}F_{K})^{\vee\vee}$ is semi-reflexive, being a
sub-$\mathcal{O}_{X^{\prime}_{K}}$-module of a semi-reflexive one. Now
$Z^{\prime}_{K}$ is an effective Cartier divisor in $X^{\prime}_{K}$, and the
preceding step implies that the restriction homomorphism
$\rho^{\prime}\colon\Gamma(X^{\prime}_{K},F^{\prime}_{K})\longrightarrow\Gamma(X^{\prime}_{K}\smallsetminus
Z^{\prime}_{K},F^{\prime}_{K})$ is injective, has a closed image, and induces
a homeomorphism of $\Gamma(X^{\prime}_{K},F^{\prime}_{K})$ with its image. The
composite homomorphism of $\mathcal{O}_{X_{K}}$-modules $\varphi\colon
F_{K}\to\pi_{\ast}\pi^{\ast}F_{K}\to\pi_{\ast}F^{\prime}_{K}$ is injective.
Indeed, let $j\colon U_{K}\to X_{K}$, $j^{\prime}\colon U^{\prime}_{K}\to
X^{\prime}_{K}$ be the open immersions. Then the following diagram of
$\mathcal{O}_{X_{K}}$-modules is commutative

${F_{K}}$${\pi_{\ast}F^{\prime}_{K}}$${j_{\ast}j^{\ast}F_{K}}$${\pi_{\ast}j^{\prime}_{\ast}j^{\prime\ast}F^{\prime}_{K}}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{\sim}$

where the vertical arrows are injective because $U_{K}$ (resp.
$U^{\prime}_{K}$) is scheme-theoretically dense in $X_{K}$ (resp.
$X^{\prime}_{K}$) and $F_{K}$ (resp. $F^{\prime}_{K}$) is semi-reflexive, and
the lower horizontal arrow is an isomorphism because $\pi\colon
U^{\prime}_{K}\to U_{K}$ is. The $\mathcal{O}_{X_{K}}$-module
$\pi_{\ast}F^{\prime}_{K}$ is coherent because $\pi$ is proper. From the exact
sequence

$0\longrightarrow\Gamma(X_{K},F_{K})\longrightarrow\Gamma(X^{\prime}_{K},F^{\prime}_{K})\longrightarrow\Gamma(X_{K},\operatorname{Coker}\varphi)$

one deduces that $\Gamma(X_{K},F_{K})\to\Gamma(X^{\prime}_{K},F^{\prime}_{K})$
is injective and has closed image. By Banach’s Open Mapping theorem, it
induces a homeomorphism of $\Gamma(X_{K},F_{K})$ with its image.

_Step 3_. Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be a increasing G-cover of
$X_{K}$ made of compact analytic domains $X_{i}$ of $X_{K}$. For
$i\in\mathbb{N}$, write $Z_{i}=Z_{K}\times_{X_{K}}X_{i}$. Then
$\\{X_{i}\smallsetminus Z_{i}\\}_{i\in\mathbb{N}}$ is an increasing G-cover of
$X_{K}\smallsetminus Z_{K}$ by analytic domains. Then,

$\displaystyle\Gamma(X_{K},F_{K})$
$\displaystyle=\projlim_{i\in\mathbb{N}}\Gamma(X_{i},F_{K}),$
$\displaystyle\Gamma(X_{K}\smallsetminus Z_{K},F_{K})$
$\displaystyle=\projlim_{i\in\mathbb{N}}\Gamma(X_{i}\smallsetminus
Z_{i},F_{K}).$

The statement follows by applying the preceding case. ∎

###  5.4. Proof of theorem 5.2 in the compact case

We will first prove theorem 5.2 when the $K$-analytic space is compact. To do
this it is convenient to say that a $K$-analytic space $X$ is _proper over
affinoid_ if there is a proper morphism of $K$-analytic spaces $X\to Y$ with
$Y$ affinoid. Here we will prove the following:

######  Theorem 5.9.

Let $D$ be an effective Cartier divisor in a $K$-analytic space $X$ which is
proper over affinoid. Then, for any semi-reflexive coherent
$\mathcal{O}_{X}$-module $F$, the natural map

$\Gamma(X,F(\ast D))\longrightarrow\Gamma(X\smallsetminus D,F)$

is injective, and has a dense image.

We reduce first of all to case where the $K$-analytic space in question is
strict. For, consider $r_{1},\dots,r_{n}>0$ linearly independent in the
$\mathbb{Q}$-vector space $\mathbb{R}_{>0}/\sqrt{|K^{\times}|}$. Then the
completion $K_{r}$ of the $K$-algebra $K[t_{1}^{\pm},\dots,t_{n}^{\pm}]$ with
respect to the norm

$\left\|\sum_{a\in\mathbb{Z}^{n}}c_{a}t^{a}\right\|_{r}:=\max_{a\in\mathbb{Z}^{n}}|c_{a}|r^{a}$

is a field by [Ber90, p. 22] where $t^{a}=t_{1}^{a_{1}}\cdots t_{n}^{a_{n}}$
and $r^{a}=r_{1}^{a_{1}}\cdots r_{n}^{a_{n}}$. For a $K$-analytic space $X$
let $X_{r}$ be the $K_{r}$-analytic space obtained from $X$ by extending
scalars to $K_{r}$. Given a coherent sheaf $F$ on $X$ let $F_{r}$ the coherent
sheaf on $X_{r}$ deduced from $F$. Suppose $X$ compact and let $\|\cdot\|_{X}$
be a norm defining the topology of $\Gamma(X,F)$. Then and
$\Gamma(X_{r},F_{r})=\Gamma(X,F)\operatorname*{\hat{\otimes}}_{K}K_{r}$ and a
norm $\|\cdot\|_{X_{r}}$ defining the topology of $\Gamma(X_{r},F_{r})$ is
given by

$\left\|\sum_{a\in\mathbb{Z}^{n}}f_{a}t^{a}\right\|_{X_{r}}=\max_{a\in\mathbb{Z}^{n}}\|f_{a}\|_{X}r^{a}.$

To reduce to strict spaces we need to be able to descend density in global
sections:

######  Lemma 5.10.

Let $D$ be an effective Cartier divisor in a compact $K$-analytic space $X$
and $F$ a coherent $\mathcal{O}_{X}$-module. If the restriction
$\Gamma(X_{r},F_{r}(\ast D_{r}))\to\Gamma(X_{r}\smallsetminus D_{r},F_{r})$ is
injective and has a dense image, then so does $\Gamma(X,F(\ast
D))\to\Gamma(X\smallsetminus D,F)$.

###### Proof.

Let $U=X\smallsetminus D$, $f\in\Gamma(U,F)$ and $C\subset U$ a compact
analytic domain. For $d\in\mathbb{N}$ let $\|\cdot\|_{X,d}$ and
$\|\cdot\|_{C}$) be norms defining the topology respectively on
$\Gamma(X,F(dD))$ and on $\Gamma(C,F)$. Via the isomorphism
$\Gamma(C,F)\cong\Gamma(C,F(dD))$ one deduces a norm on $\Gamma(C,F(dD))$
defining its topology: with an abuse of notation, denote it again
$\|\cdot\|_{C}$. Let $\varepsilon>0$. By hypotheses, there exists
$g\in\Gamma(X_{r},F_{r}(\ast D_{r}))$ such that $\|f-g\|_{C_{r}}<\varepsilon$.
Now $g\in\Gamma(X_{r},F_{r}(dD_{r}))$ for some $d\in\mathbb{N}$ because

$\Gamma(X_{r},F_{r}(\ast
D_{r}))=\injlim_{d\in\mathbb{N}}\Gamma(X_{r},F(dD_{r}))$

By compactness of $X_{r}$ and [Stacks, lemma 009F (4)]. Write
$g=\sum_{a\in\mathbb{Z}^{n}}g_{a}t^{a}$ with sections
$g_{a}\in\Gamma(X,F(dD))$ such that $\|g_{a}\|_{X,d}r^{a}\to 0$ as
$|a|\to\infty$. Since the restriction map $\Gamma(X,F(dD))\to\Gamma(C,F(dD))$
is continuous, there is a real number $R>0$, such that $\|g_{a}\|_{C}\leqslant
R\|g_{a}\|_{X,d}$ for all $a\in\mathbb{Z}^{n}$. Thus the series
$\sum_{a\in\mathbb{Z}^{n}}g_{a}t^{a}$ converges on $C_{r}$ and

$\|f-g\|_{C_{r}}=\max_{a\in\mathbb{Z}^{n}\smallsetminus\\{0\\}}\left\\{\|f-g_{0}\|_{C},\|g_{a}\|_{C}r^{a}\right\\}.$

In particular $\|f-g\|_{C_{r}}<\varepsilon$ implies
$\|f-g_{0}\|_{C}<\varepsilon$. This shows that $\Gamma(X,F(\ast D))$ is dense
in $\Gamma(U,F)$. ∎

The following lemma permits to enlarge your domain by suitably blowing up the
formal scheme.

######  Lemma 5.11.

Let $X$ be a quasi-compact admissible formal $R$-scheme. Let $Z$ be a closed
admissible formal subscheme of $X$. Let $C$ be a compact subset of
$X_{\eta}\smallsetminus Z_{\eta}$.

Then, there is $\varepsilon>0$ such that every $\varpi\in
R\smallsetminus\\{0\\}$ with $|\varpi|<\varepsilon$ has the following
property: if $\pi\colon X^{\prime}\to X$ denotes the admissible blow-up of $X$
in $\varpi\mathcal{O}_{X}+I_{Z}$ and $Z^{\prime}$ the strict transform of $Z$
in $X^{\prime}$, then $\pi_{\eta}^{-1}(C)\subset(X^{\prime}\smallsetminus
Z^{\prime})_{\eta}$.

###### Proof.

By taking a finite open affine covering of $X$, one reduces to the case where
$X=\operatorname{Spf}(A)$ is the formal spectrum of a flat topologically
finitely presented $R$-algebra $A$. Let $f_{1},\dots,f_{r}$ be generators of
$I:=\Gamma(X,I_{Z})$. Given $x\in X_{\eta}\smallsetminus Z_{\eta}$ there is at
least one among $f_{1},\dots,f_{r}$ which does not vanishes at $x$, thus by
compactness of $C$ we have $\varepsilon:=\min_{x\in
C}\max\\{|f_{1}(x)|,\dots,|f_{r}(x)|\\}>0$. Let $\varpi\in
R\smallsetminus\\{0\\}$ be such that $|\varpi|<\varepsilon$. Let $\pi\colon
X^{\prime}\to X$ be the admissible blow-up of $X$ in
$\varpi\mathcal{O}_{X}+I_{Z}$. By writing $f_{0}=\varpi$ the blow-up
$X^{\prime}$ is covered by the affine open subsets
$X^{\prime}_{i}=\operatorname{Spf}A_{i}$ with

$A_{i}:=A\\{f_{j}/f_{i}\\}_{j\neq i}=[A\\{t_{ij}\\}_{j\neq
i}/(f_{j}-t_{ij}f_{i})_{j\neq i}]/(f_{i}\textup{-torsion}).$

Let $Z^{\prime}$ be the strict transform of $Z$ in $X^{\prime}$.

######  Claim 5.12.

The strict transform $Z^{\prime}$ is contained in $X_{0}^{\prime}$.

###### Proof of the claim.

The intersection of $Z^{\prime}$ with $X_{i}^{\prime}$ is the closed subscheme
corresponding to the ideal
$I_{i}=\operatorname{Ker}(A_{i}\to[A_{i}/IA_{i}]\otimes_{R}K)$. If $i\geqslant
1$, then $\varpi$ belongs to $I_{i}$ because it can be written as
$\varpi=t_{i0}f_{i}\in I_{i}$. On the other hand, $\varpi$ is invertible in
$A_{i}\otimes_{R}K$, thus $1$ belongs to $I_{i}$. This means that $Z^{\prime}$
does not meet $X_{i}^{\prime}$ for all $i\geqslant 1$ and, consequently,
$Z^{\prime}$ is contained in $X^{\prime}_{0}$. ∎

######  Claim 5.13.

The compact subset $\pi_{\eta}^{-1}(C)$ does not meet
$(X^{\prime}_{0})_{\eta}$.

###### Proof of the claim.

For $x\in(X^{\prime}_{0})_{\eta}$, one has
$|f_{i}(x)|\leqslant|\varpi|<\varepsilon$ for all $i\geqslant 1$. Instead, by
the choice of $\varepsilon$, given a point $x\in\pi_{\eta}^{-1}(C)$ there
exists $i\in\\{1,\dots,n\\}$ such that $|f_{i}(x)|\geqslant\varepsilon$. ∎

From the previous claims one draws the following chain of inclusions:

$\pi_{\eta}^{-1}(C)\subset(X^{\prime}_{1}\cup\cdots\cup
X^{\prime}_{r})_{\eta}\subset(X^{\prime}\smallsetminus Z^{\prime})_{\eta},$

which proves the statement. ∎

###### Proof of theorem 5.9.

Write $X_{K}$ instead of $X$ and similarly for $D$ and $F$. Thanks to lemma
5.10, up to extending scalars, one may suppose that $X_{K}$ admits a proper
morphism with strictly affinoid target. Let $C\subset X_{K}\smallsetminus
D_{K}$ be a compact analytic domain. We want to show that
$\Gamma(X_{K},F_{K}(\ast D_{K}))$ is dense in $\Gamma(X_{K}\smallsetminus
D_{K},F_{K})$ with respect to a norm $\|\cdot\|_{C}$ defining the topology of
$\Gamma(C,F_{K})$. According to theorem 2.7 there are a proper over affine
admissible formal $R$-scheme, an admissible effective Cartier divisor $D$ in
$X$ and an isomorphism of $K$-analytic spaces $\varphi\colon X_{\eta}\to
X_{K}$ inducing an isomorphism $D_{\eta}\cong D_{K}$. Let $U=X\smallsetminus
D$. Thanks to lemma 5.7, there are a coherent $\mathcal{O}_{X}$-module $F$,
flat over $R$, $D$-torsion free and such that $F_{\rvert D}$ is flat over $R$,
and an injective homomorphism of $\mathcal{O}_{X_{\eta}}$-modules
$F_{K}\hookrightarrow F_{\eta}$ inducing isomorphisms $F_{K}(\ast
D_{\eta})\cong F_{\eta}(\ast D_{\eta})$ and $F_{K\rvert U_{\eta}}\cong
F_{\eta\rvert U_{\eta}}$ is an isomorphism. For $d\in\mathbb{N}$, the natural
homomorphisms

$\Gamma(X,F(dD))\otimes_{R}K\longrightarrow\Gamma(X_{K},F_{K}(dD_{K})),\quad\Gamma(U,F)\otimes_{R}K\longrightarrow\Gamma(U_{\eta},F_{K}),$

are isomorphisms. In what follows, these identifications are understood.

######  Claim 5.14.

Up to replacing $X$ by its blow-up in $\varpi\mathcal{O}_{X}+I_{D}$ for some
$\varpi\in R\smallsetminus\\{0\\}$, one may assume $C=U_{\eta}$ and
$\|\cdot\|_{C}$ to be the norm induced by the $R$-module $\Gamma(U,F)$.

###### Proof of the claim.

According to lemma 5.11, there is $\varpi\in R\smallsetminus\\{0\\}$ with the
following property. Let $\pi\colon X^{\prime}\to X$ be the admissible blow-up
of $X$ in $\varpi\mathcal{O}_{X}+I_{D}$. Let $D^{\prime}\subset X^{\prime}$ be
the strict transform of $D$ and $U^{\prime}=X^{\prime}\smallsetminus
D^{\prime}$. Then $\pi_{\eta}^{-1}(C)$ is contained in $U^{\prime}_{\eta}$.
Since the restriction map
$\Gamma(U^{\prime}_{\eta},\pi^{\ast}F_{K})\to\Gamma(\pi_{\eta}^{-1}(C),\pi^{\ast}F_{K})$
is continuous, one may suppose $\pi_{\eta}^{-1}(C)=U^{\prime}_{\eta}$.
Moreover, by proposition 2.12, the strict transform $D^{\prime}$ of $D$ in
$X^{\prime}$ in an admissible effective Cartier divisor. Concerning the choice
of the norm, density in $\Gamma(U^{\prime}_{\eta},\pi^{\ast}F_{K})$ only
depends on its topology. Therefore, it suffices to test density for a norm
defining its topology: the norm induced by the $R$-module
$\Gamma(U^{\prime},\pi^{\ast}F)$ definitely falls into this category. ∎

Henceforth suppose $C=U_{\eta}$. The natural map $\Gamma(X,F(\ast
D))\to\Gamma(U,F)$ is injective and has dense image (theorem 3.3). By
extending scalars to $K$, one deduces that the map $\Gamma(X_{K},F_{K}(\ast
D_{K}))\to\Gamma(U_{K},F_{K})$ is injective with dense image. This concludes
the proof. ∎

###  5.5. Proof of theorem 5.2 in the general case

We can now complete the proof of theorem 5.2.

######  Lemma 5.15.

Let $D$ be an effective Cartier divisor in a $K$-analytic space $X$. Then the
restriction homomorphism $\rho\colon\Gamma(X,F(\ast
D))\to\Gamma(X\smallsetminus D,F)$ is injective for any semi-reflexive
$\mathcal{O}_{X}$-module $F$.

###### Proof.

By taking an affinoid G-cover it suffices to prove the statement when $X$ is
compact. In this case $\Gamma(X,F(\ast
D))=\injlim_{d\in\mathbb{N}}\Gamma(X,F(dD))$ by compactness of $X$ and
[Stacks, lemma 009F]. For $d\in\mathbb{N}$ the $\mathcal{O}_{X}$-module
$F(dD)=F\otimes\mathcal{O}_{X}(dD)$ is semi-reflexive because $F$ is. The
complement $U:=X\smallsetminus D$ is scheme-theoretically dense in $X$ thus
the restriction homomorphism $\rho_{d}\colon\Gamma(X,F(dD))\to\Gamma(U,F(dD))$
is injective by proposition 5.3. For an integer $d^{\prime}\geqslant d$, one
has the following commutative diagram

${\Gamma(X,F(dD))}$${\Gamma(X,F(d^{\prime}D))}$${\Gamma(U,F(dD))}$${\Gamma(U,F(d^{\prime}D)),}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\rho_{d}}$$\hookleftarrow$$\rightarrow$$\scriptstyle{\rho_{d^{\prime}}}$$\leftarrow$$\rightarrow$$\scriptstyle{\sim}$

the lower horizontal arrow being an isomorphism. By taking the injective limit
one sees that $\rho$ is injective. This concludes the proof. ∎

###### Proof of theorem 5.2.

The restriction homomorphism $\Gamma(X,F(\ast D))\to\Gamma(X\smallsetminus
D,F)$ is injective by lemma 5.15. Let $U:=X\smallsetminus D$,
$s\in\Gamma(U,F)$, $C\subset U$ a compact analytic domain, $\|\cdot\|$ a norm
on $\Gamma(C,F)$ defining its topology. In order to prove the statement we
have to show that for each $\varepsilon>0$ there are $d\in\mathbb{N}$ and
$t\in\Gamma(X,F(dD))$ such that $\|s-t\|<\varepsilon$. According to theorem
4.2, there are a Stein $K$-analytic space $S$ and a proper morphism
$\sigma\colon X\to S$. Let $\\{S_{i}\\}_{i\in\mathbb{N}}$ be an affinoid Stein
exhaustion of $S$. For $i\in\mathbb{N}$ set $X_{i}:=X\times_{S}S_{i}$. For
$i\in\mathbb{N}$ big enough the analytic domain $X_{i}$ contains $C$.
Therefore, we may suppose $C=X_{i}$. The induced morphism $\sigma\colon
X_{i}\to S_{i}$ is proper. The closed analytic subspace
$D_{i}:=D\times_{X}X_{i}$ of $X_{i}$ is an effective Cartier divisor. This
allows to apply theorem 5.9 to $X_{i}$. It follows that there are
$d\in\mathbb{N}$ and $t\in\Gamma(X_{i},F(dD))$ such that
$\|t-s\|<\varepsilon$. In order to conclude, it suffices to remark that given
a coherent $\mathcal{O}_{X}$-module $M$ the restriction map
$\Gamma(X,M)\to\Gamma(X_{i},M)$ has a dense image. ∎

##  6\. Approximating analytic functions by rational functions

Let $K$ be a complete non-trivially valued non-Archimedean field.

###  6.1. Statements

In this section we deduce density results of algebraic functions by using the
results of the previous sections. Let $X$ be a separated finite type scheme
over an affinoid $K$-algebra $A$ and $F$ a semi-reflexive coherent
$\mathcal{O}_{X}$-module. The main result is the following:

######  Theorem 6.1.

The subset $\Gamma(X,F)\subset\Gamma(X^{\textup{an}},F^{\textup{an}})$ is
dense.

######  Corollary 6.2.

For a a scheme-theoretically dense open subset $U\subset X$ the following are
equivalent:

  1. (1)

$\Gamma(X,F)\subset\Gamma(U^{\textup{an}},F^{\textup{an}})$ is dense;

  2. (2)

$\Gamma(X^{\textup{an}},F^{\textup{an}})\subset\Gamma(U^{\textup{an}},F^{\textup{an}})$
is dense;

  3. (3)

$\Gamma(X^{\textup{an}},F^{\textup{an}})=\Gamma(U^{\textup{an}},F^{\textup{an}})$;

  4. (4)

$\Gamma(X,F)=\Gamma(U,F)$.

###  6.2. Scheme-theoretic closure and analytification

We need some results on the analytification of quasi-coherent sheaves. In this
section let $f\colon X\to Y$ be a separated morphism between $A$-schemes of
finite type. For an $\mathcal{O}_{X}$-module $E$ let

$\theta_{E,f}\colon(f_{\ast}E)^{\textup{an}}\longrightarrow
f^{\textup{an}}_{\ast}E^{\textup{an}}$

be the canonical homomorphism of $\mathcal{O}_{Y^{\textup{an}}}$-module.

######  Lemma 6.3.

Let $\\{E_{i}\\}_{i\in\mathbb{N}}$ be a directed system of
$\mathcal{O}_{X}$-modules and $E=\injlim E_{i}$. If $\theta_{E_{i},f}$ is
injective for all $i\in\mathbb{N}$, then $\theta_{E,f}$ is injective.

###### Proof.

The homomorphism of $\mathcal{O}_{Y^{\textup{an}}}$-modules

$\tilde{\theta}:=\injlim_{i\in\mathbb{N}}\theta_{E_{i},f}\colon\injlim_{i\in\mathbb{N}}(f_{\ast}E_{i})^{\textup{an}}\longrightarrow\injlim_{i\in\mathbb{N}}f^{\textup{an}}_{\ast}E^{\textup{an}}_{i}$

is injective. Since $f$ is quasi-compact and quasi-separated, pushing-forward
along $f$ commutes with injective limits [Stacks, lemma 009E] thus

$\injlim_{i\in\mathbb{N}}(f_{\ast}E_{i})^{\textup{an}}=(\injlim_{i\in\mathbb{N}}f_{\ast}E_{i})^{\textup{an}}=(f_{\ast}E)^{\textup{an}}.$

On the other hand $\tilde{\theta}=\psi\circ\theta_{E,f}$ where $\psi\colon
f^{\textup{an}}_{\ast}E^{\textup{an}}\to\injlim_{i\in\mathbb{N}}f^{\textup{an}}_{\ast}E^{\textup{an}}_{i}$
is the canonical morphism. Since $\tilde{\theta}$ is injective it follows that
$\theta_{E,f}$ is injective. ∎

######  Lemma 6.4.

If $E$ is quasi-coherent, then $\theta_{E,f}$ is injective.

###### Proof.

The module $E$ is the injective limit of its coherent
$\mathcal{O}_{X}$-submodules by [Stacks, lemma 01PG]. By lemma 6.3 one may
suppose $E$ coherent henceforth. When $f$ is proper $\theta_{E,f}$ is an
isomorphism by the non-Archimedean GAGA theorem ([Köp74], [Con06, Example
3.2.6], [Poi10, Théorème A.1]). In the general case, by Nagata’s
compactification theorem (see [Lüt93], [Con07] and [Con09], or [Tem11]) there
are an $A$-scheme $X$ of finite type together with a proper morphism
$\pi\colon X^{\prime}\to Y$ and open immersion $i\colon X\to X^{\prime}$ such
that $f=\pi\circ i$. Up to blowing up, one may assume that
$X^{\prime}\smallsetminus i(X)$ supports an effective Cartier divisor $D$. Let
$E^{\prime}$ be a coherent sheaf extending $E$ ([Stacks, lemma 01PI]). Then
$i_{\ast}E=\injlim_{d\in\mathbb{N}}E^{\prime}(dD)$. Therefore
$(i_{\ast}E)^{\textup{an}}=\injlim_{d\in\mathbb{N}}E^{\prime}(dD)^{\textup{an}}$
is the subsheaf of sections of $E^{\textup{an}}$ with polar singularities
along $D$. In particular, the natural map

$\theta_{E,i}\colon(i_{\ast}E)^{\textup{an}}=\injlim_{d\in\mathbb{N}}E^{\prime}(dD)^{\textup{an}}\longrightarrow
i^{\textup{an}}_{\ast}i^{\textup{an}\ast}E^{\prime\textup{an}}=i^{\textup{an}}_{\ast}E^{\textup{an}},$

is injective by lemma 5.1. According to the proper case, the natural map

$\theta_{i_{\ast}E,\pi}\colon(f_{\ast}E)^{\textup{an}}=(\pi_{\ast}i_{\ast}E)^{\textup{an}}\longrightarrow\pi^{\textup{an}}_{\ast}(i_{\ast}E)^{\textup{an}}$

is injective. Since pushing forward is a left exact functor
$\pi^{\textup{an}}_{\ast}\theta_{E,i}$ is injective, thus so is
$\theta_{E,f}=\pi^{\textup{an}}_{\ast}\theta_{E,i}\circ\theta_{i_{\ast}E,\pi}$.
∎

######  Lemma 6.5.

For an $\mathcal{O}_{Y}$-module $F$ we have a natural isomorphism:

$\operatorname{Ker}(F\to
f_{\ast}f^{\ast}F)^{\textup{an}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ker}(F^{\textup{an}}\to
f^{\textup{an}}_{\ast}f^{\textup{an}\ast}F^{\textup{an}}).$

###### Proof.

Let $K:=\operatorname{Ker}(F\to f_{\ast}f^{\ast}F)$. The functor
$M\rightsquigarrow M^{\textup{an}}$ on $\mathcal{O}_{Y}$-modules is exact
because the morphism of locally ringed spaces $Y^{\textup{an}}\to Y$ is
surjective and flat by [Ber93, 2.6.2] thus
$K^{\textup{an}}=\operatorname{Ker}(F^{\textup{an}}\to(f_{\ast}f^{\ast}F)^{\textup{an}})$.
This yields the following commutative diagram of
$\mathcal{O}_{Y^{\textup{an}}}$-modules:

${0}$${K^{\textup{an}}}$${F^{\textup{an}}}$${(f_{\ast}f^{\ast}F)^{\textup{an}}}$${0}$${K^{\prime}}$${F^{\textup{an}}}$${f^{\textup{an}}_{\ast}f^{\textup{an}\ast}F^{\textup{an}}}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\Leftarrow$$\Leftarrow$$\leftarrow$$\rightarrow$$\scriptstyle{\theta_{f^{\ast}F,f}}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

where $K^{\prime}:=\operatorname{Ker}(F^{\textup{an}}\to
f^{\textup{an}}_{\ast}f^{\textup{an}\ast}F^{\textup{an}})$. Since
$\theta_{f^{\ast}F,f}$ is injective by lemma 6.4, one concludes by the Snake
lemma. ∎

######  Corollary 6.6.

Let $U$ be an open subset of an $A$-scheme of finite type $X$. Let $F$ be a
quasi-coherent $\mathcal{O}_{X}$-module such that $F\to j_{\ast}j^{\ast}F$ is
injective where $j\colon U\to X$ is the open immersion. If
$f\in\Gamma(X^{\textup{an}},F^{\textup{an}})$ is such that $f_{\rvert
U^{\textup{an}}}\in\Gamma(U,F)$, then $f\in\Gamma(X,F)$.

###### Proof.

Let $\pi\colon X^{\textup{an}}\to X$ be the canonical morphism of locally
ringed spaces. For a sheaf of abelian groups $E$ on $X$ let $\pi^{-1}E$ be the
pull-back of $E$ as sheaf of abelian groups. For each $x\in X^{\textup{an}}$,
$\mathcal{O}_{X^{\textup{an}},x}$ is a faithfully flat
$\mathcal{O}_{X,\pi(x)}$-algebra ([Ber93, 2.6.2]). Given an
$\mathcal{O}_{X}$-module $E$, the natural homomorphism
$\varphi_{E}\colon\pi^{-1}E\to E^{\textup{an}}$ is injective. Let $C$ be the
cokernel of the homomorphism $F\to j_{\ast}j^{\ast}F$. Then the following
diagram of $\pi^{-1}\mathcal{O}_{X}$-modules is commutative and exact:

${0}$${\pi^{-1}F}$${\pi^{-1}j_{\ast}j^{\ast}F}$${\pi^{-1}C}$${0}$${0}$${F^{\textup{an}}}$${(j_{\ast}j^{\ast}F)^{\textup{an}}}$${C^{\textup{an}}}$${0}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\varphi_{F}}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\varphi_{j_{\ast}j^{\ast}F}}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\varphi_{C}}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

The Snake lemma implies that the homomorphism
$\operatorname{Coker}\varphi_{F}\to\operatorname{Coker}\varphi_{j_{\ast}j^{\ast}F}$
is injective. lemma 6.4 applied to the quasi-coherent $\mathcal{O}_{U}$-module
$j^{\ast}F$ and to the morphism $j$ states the homomorphism
$(j_{\ast}j^{\ast}F)^{\textup{an}}\to
j^{\textup{an}}_{\ast}j^{\textup{an}\ast}F^{\textup{an}}$ is injective. In
particular, the natural maps $F^{\textup{an}}\to
j^{\textup{an}}_{\ast}j^{\textup{an}\ast}F^{\textup{an}}$,

$\operatorname{Coker}\varphi_{F}\longrightarrow\operatorname{Coker}(\pi^{-1}j_{\ast}j^{\ast}F\to
j^{\textup{an}}_{\ast}j^{\textup{an}\ast}F^{\textup{an}})$

are injective. This concludes the proof. ∎

###  6.3. Proofs

We begin with a result of extension of semi-reflexive sheaves:

######  Lemma 6.7.

Let $D$ be a Cartier divisor in a Noetherian scheme $X$ and $U=X\smallsetminus
D$. Let $F$ be a coherent $\mathcal{O}_{X}$-module such that $F_{\rvert U}$ is
semi-reflexive. Then the $\mathcal{O}_{X}$-module

$F^{\textup{sat}}=\operatorname{Ker}(F\to(F^{\vee\vee}/F)_{\rvert U})$

is $D$-torsion free semi-reflexive and such that $F_{\rvert U}\cong
F^{\textup{sat}}_{\rvert U}$.

###### Proof.

The statement is local on $X$, therefore one may assume that
$X=\operatorname{Spec}A$ is affine and the Cartier divisor $D$ is given by the