2.1 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 $D, D' \subset X$ , the Cartier divisor $D + D'$ is the closed subscheme with sheaf of ideals $I_D \otimes_{\mathcal{O}_X} I_{D'}$ whence the identity $\mathcal{O}_X(D + D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')$ . If the closed immersion $D \to X$ factors through D’, then $I_{D'} \subset I_{D}$ and the sheaf of ideals $I_{D'} \otimes_{\mathcal{O}_X} \mathcal{O}_X(D)$ defines the Cartier divisor $D ' - D$ . 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 $X' \to X$ is called a U-admissible blow-up if there exists a finitely presented closed subscheme $Z \subset X \smallsetminus U$ such that X’ is isomorphic to the blow-up of X along Z.

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 $\pi \colon X' \to X$ such that the strict transform of D in X’ 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 Theorem 1.7.

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 $f \times \textrm{id}_K$ .

One recovers Theorem 1.7 in the introduction by taking $Y = \operatorname{Spec} R$ . The hypothesis of Y being finitely presented over R is easily met. Indeed, given a finitely generated K-algebra $A_K$ and a surjection $\varphi \colon K [ t_1, \ldots, t_n ] \to A_K$ of K-algebras, the ideal $I := \operatorname{Ker} \varphi \cap R [ t_1, \dots, t_n ]$ is finitely generated by Lemma 2.8. Thus, the R-algebra $A := R [ t_1, \ldots, t_n ] / I$ is flat, finitely presented, and such that $A \otimes_R K$ is isomorphic to $A_K$ .

2.2 Proofs

The remainder of this section is devoted to the proof of Theorem 2.1. We begin with the following technical result.

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 $\varphi \otimes \textrm{id} \colon F(nZ) \to G(nZ)$ .

The key result at the core of the method is as follows.

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 $\pi \colon X' \to X$ be the blow-up of X along the scheme-theoretic intersection of $D_0$ and nZ, and $E \subset X'$ be the exceptional divisor.

Then, the strict transform of D in X’ is the effective Cartier divisor $\pi^{-1}(D_0) - E$ .

In particular, there is a U-admissible blow-up $\pi \colon X' \to X$ 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 $f_1, \ldots, f_r$ such that $f_0, \ldots, f_r$ generate the ideal of D. With this notation, note that the hypothesis implies that the ideals $f_0 A_g$ and $f_0 A_g + \cdots + f_r A_g$ coincide. Moreover, the blow-up X’ is covered by the following affine open subsets:

\begin{align*}X'_0 &= \operatorname{Spec} A[ f_0 / g^n], \quad\ A[ f_0 / g^n] = (A[t] / (f_0 - t g^n)) / (g\textrm{-torsion}),\\X'_1 &= \operatorname{Spec} A[ g^n / f_0], \quad\ A[ g^n / f_0] = (A[u] / (g^n - u f_0)) / (f_0\textrm{-torsion}).\end{align*}

Proof of the claim. The strict transform of D is the scheme-theoretic closure of $\pi^{-1}(D \smallsetminus Z)$ in X’. Now, the function $g^n$ is invertible on $X'_1 \smallsetminus \pi^{-1}(Z)$ . The equality $g^n = uf_0$ on $X'_1$ implies that $f_0$ is also invertible on $X'_1 \smallsetminus \pi^{-1}(Z)$ . Therefore,

\[ \pi^{-1}(D \smallsetminus Z) \cap X'_1 = \pi^{-1}(D) \cap (X'_1 \smallsetminus \pi^{-1}(Z)) = \emptyset. \]

It follows that D’ does not meet $X'_1$ .

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’ is the scheme-theoretic closure of $\pi^{-1}(D \smallsetminus Z)$ in $X'_0$ . Therefore, the ideal $I_{D'} \subset A[f_0/g^n]$ defining D’ is

\[ \operatorname{Ker} (A[f_0/g^n] \to A[f_0/g^n]_{g} / (f_0)).\]

Proving the claim amounts to showing that $I_{D'}$ is generated by t. Of course, t belongs to $I_{D'}$ 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 $a_0, \ldots, a_d \in A$ . 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 ( $i = 1, \ldots, r$ ). By the choice of n, there are $b_1, \ldots, b_r \in A$ such that $f_i = b_i f_0 / g^n$ . Written otherwise, for $i = 1, \ldots, r$ , 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 $f_0 \mathcal{O}_{X'}$ . The exceptional divisor E is given by the ideal $g^n A[f_0/g^n]$ on the affine chart $X'_0$ , whereas it is given by the ideal $f_0 A[g^n/f_0]$ on the affine chart $X'_1$ . 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 $X'_0$ and by the ideal generated by $f_0 / f_0 = 1$ on $X'_1$ . One concludes the proof by comparing this with the description of the strict transform D’ given by the above claims.

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

Proof of Theorem 2.1. Endow the closed subset $Z := X \smallsetminus U$ with its reduced structure. The blow-up $\pi \colon X' \to X$ of X along 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 [Sta19, Lemma 080L].) Suppose Z is a Cartier divisor. The blow-up $\pi \colon X' \to X$ of X along 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 $D'_0 := \pi^{-1}(D)$ is an effective Cartier divisor, as well as the pre-image $Z' = \pi^{-1}(Z)$ of Z. Let D’ be the scheme-theoretic closure of $D'_0 \smallsetminus Z'$ in X’. We may now apply Lemma 2.4 to X’, D’, $D_0'$ and Z’. Indeed, let $U ' = X ' \smallsetminus Z'$ . The closed immersion $D' \cap U' \to D_0' \cap U'$ is an isomorphism. On the level of sheaves of ideals, this means that the natural inclusion $I_{D'_0 \rvert U} \to I_{D' \rvert U}$ is an isomorphism. According to Lemma 2.3, there is a non-negative integer N with the following property: given an integer $n \geqslant N$ , the injective homomorphism $I_{D_0'}(n Z') \to I_{D'}(n Z')$ is an isomorphism. In particular, Lemma 2.4 can be applied with every such integer n.

Let R the ring of integers of a field K endowed with a non-trivial non-Archimedean absolute value. Recall that an R-module M is flat if and only if it is torsion-free [Sta19, Lemma 0539].

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 of Theorem 2.2. By Nagata’s compactification theorem (see [Reference LütkebohmertLüt93, Reference ConradCon07, Reference ConradCon09] or [Reference TemkinTem11]), the morphism of K-analytic schemes $\sigma_K \colon X_K \to Y \times_R K$ 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 2.1 with Z being the special fibre of X. Note that the so-obtained blow-up $\pi \colon X' \to X$ is such that X’ 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’. In the general case, some argument is needed in order to ensure the hypothesis of finite presentation of the strict transform considered in the proof of Theorem 2.1.

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 fibre $X_K$ of X. Let $\pi \colon X' \to X$ the blow-up of X along the first Fitting ideal of $I_D$ . Since D is flat and finitely presented over R by Lemma 2.8, the ideal $I_D$ is finitely presented by Lemma 2.8, hence $\operatorname{Fit}_1 I_D$ is finitely presented by [Sta19, Lemma 07ZA (4)] and Lemma 2.8. Lemma 2.9 states that $\pi^{-1}(D)$ is an effective Cartier divisor. 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 facts, one deduces that $\pi \colon X' \to X$ is a U-admissible blow-up. Let $D_0'$ be the inverse image of D in X’: it is an effective Cartier divisor. Let D’ be the Zariski of closure of $\pi^{-1}(D_K)$ in X’. Again, thanks to Lemma 2.8, the closed subscheme D’ is finitely presented and one can apply Lemma 2.4 to X’, $D_0'$ , D’ and the closed subscheme Z’ of X’ defined by the vanishing of $\varpi$ . As in the discretely valued case, the blow-up $\pi' \colon X''\to X'$ given by Lemma 2.4 is such that X” is flat over R as well as the strict transform of D’ in X”.

In this proof we used the following fact.

Proof. Set $I:= I_D$ . By [Sta19, Lemma 07ZA (3)] the kth Fitting ideal of $\pi^\ast I$ for $k \geqslant 0$ is the pull-back of that of I:

${Fit}_k(\pi^\ast I) = \operatorname{Fit}_k(I)\mathcal{O}_{X'}.$

Since $X \smallsetminus D$ is scheme-theoretically dense in X, the ideal $\operatorname{Fit}_0(I)$ vanishes altogether, thus so does $\operatorname{Fit}_0(\pi^\ast I)$ . On the other hand, by the defining property of the blow-up, the ideal sheaf $\operatorname{Fit}_1(I) \mathcal{O}_{X'}$ is invertible. It follows that the ideal $J:= \operatorname{Fit}_1(\pi^\ast I)$ is invertible. Set $M:= \pi^\ast I$ and let $M' \subset M$ be the submodule annihilated by J. We claim that the natural morphism $\varphi \colon \pi^\ast I \to \mathcal{O}_{X'}$ induces an isomorphism

$M/M' \stackrel{\sim}{\longrightarrow} I \mathcal{O}_{X'}.$

This will conclude the proof, as the $\mathcal{O}_{X'}$ -module $M / M'$ is invertible [Sta19, Lemma 0F7M]. To show the claim, first note that $I \mathcal{O}_{X'}$ is the image of $\varphi$ , so it suffices to prove $M' = \operatorname{Ker} \varphi$ . The inclusion $M' \subset \operatorname{Ker} \varphi$ is obvious because $\mathcal{O}_{X'}$ has no torsion with respect to J, the latter ideal being invertible. For the converse inclusion, $\pi$ induces an isomorphism outside the exceptional divisor $E \subset X'$ hence $\varphi$ is injective outside E. The ideal defining E is J, hence any local section of $\operatorname{Ker} \varphi$ is annihilated by some power of J. It follows from [Sta19, Lemma 0F7M (2)] that $\operatorname{Ker} \varphi$ is actually annihilated by J, which concludes the proof.

3.1 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 fibre $X_\eta$ seen as a K-analytic space in the sense of Berkovich. The generic fibre 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.

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. Let U be an open subset of X. Since the topological space underlying X is Noetherian (Lemma 3.1), the open subset U is quasi-compact. Let $U = \bigcup_{i = 1}^n U_i$ a finite open cover of U. The exact sequence

\[ 0 \longrightarrow F(U) \stackrel{\delta_0}{\longrightarrow} \prod_{i = 1}^n F(U_i) \stackrel{\delta_1}{\longrightarrow} \prod_{i, j = 1}^n F(U_i \cap U_j),\]

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):

\[ 0 \longrightarrow F(U) \otimes_R K\stackrel{\delta_0}{\longrightarrow} \bigg( \prod_{i = 1}^n F(U_i)\bigg) \otimes_R K \stackrel{\delta_1}{\longrightarrow} \bigg( \prod_{i, j = 1}^n F(U_i \cap U_j) \bigg)\otimes_R K.\]

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

3.2 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 $\mu \in R \smallsetminus \{ 0\}$ . Following [Reference BoschBos14, 8.2, Definition 3], the formal R-scheme

\[ X' = \mathop{\operatorname{colim}}\limits_{n \in \mathbb{N}} \textrm{Proj} \bigg(\bigoplus_{d \in \mathbb{N}} I^d \otimes_{\mathcal{O}_X}(\mathcal{O}_X / \varpi^n \mathcal{O}_X)\bigg),\]

with structural morphism $\pi \colon X' \to X$ , is called the blow-up of X along I. The exceptional divisor of $\pi$ is the closed formal subscheme of X’ defined by the coherent sheaf of ideals $I\mathcal{O}_{X'}$ . The formal scheme X’ is admissible by [Reference BoschBos14, 8.2, Corollary 8].

Lemma 3.5. With the previous notation, let $Z \subset X$ be a closed admissible formal subscheme defined by a coherent sheaf of ideals $J \subset \mathcal{O}_X$ . Then,

\[ J' := \operatorname{Ker}( \mathcal{O}_{X'} \to \mathcal{O}_{X', K} / J\mathcal{O}_{X', K} ),\]

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

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 (respectively, J) with the finitely presented ideal $\Gamma(X, I)$ (respectively, $\Gamma(X, J)$ ). Let $f_0, \ldots, f_r$ be generators of I. The blow-up X’ is covered by affine open subsets $X'_i$ , $i = 0, \ldots, r$ ,

\begin{align*}X'_i = \operatorname{Spf} A \{ f_j / f_i\}_{j \neq i},\quad\! 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\textrm{-torsion}).\end{align*}

By definition,

\[ \Gamma(X'_i, J') = \operatorname{Ker}( A \{ f_j / f_i\}_{j \neq i} \to [A \{ f_j / f_i\}_{j \neq i} / J A \{ f_j / f_i\}_{j \neq i}] \otimes_R K).\]

According to Lemma 3.2, $\Gamma(X'_i, J')$ is a finitely presented ideal of $A \{ f_j / f_i\}_{j \neq i}$ , which concludes the proof.

With the notation of Lemma 3.5, the formal closed subscheme $Z' \subset X'$ associated with J’ is called the strict transform of Z.

3.3 Statements

We are now in position to state the main results of this section. To do this, note that the concepts in § 2.1 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.

Then, there exists an admissible blow-up $\pi \colon X' \to X$ such that the strict transform of D in X’ is an admissible 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 of K-analytic space $X_K \cong X_\eta$ through which $D_K$ is identified with $D_\eta$ and $f_K$ with $f_\eta$ .

Note 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 $\varphi \colon K \{ t_1, \ldots, t_n \} \to A_K$ a surjective homomorphism of Banach K-algebras. According to Lemma 3.2, the ideal $I := \operatorname{Ker} \varphi \cap R \{ t_1, \ldots, t_n \}$ is finitely generated. Thus, $A := R \{ t_1, \ldots, t_n \} / I$ is a flat, topologically finitely presented R-algebra and $A \otimes_R K \cong A_K$ .

3.4 Fitting ideals in formal geometry

In the proof of Theorem 2.2 we started by blowing up D so that we could suppose that it was a Cartier divisor. Here it is not the correct operation, somewhat because a formal scheme does not have a true generic fibre. As we will see, the correct transformation to perform is to blow-up X along 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 3.8. For an integer $r \geqslant 0$ let $\operatorname{Fit}_rF$ be the sheafification of the presheaf associating to an affine open subset U the rth Fitting ideal of $\Gamma(U, F)$ . Then $\operatorname{Fit}_rF$ 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 [Sta19, Lemma 07ZA (3)]:

\[ \operatorname{Fit}_r (M \otimes_A A_{\{f\}}) = (\operatorname{Fit}_r M) A_{\{f\}}. \]

On the other hand, $A_{\{f\}}$ is flatFootnote ⁴ 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 rth Fitting ideal of F.

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 $\pi \colon X' \to X$ is the blow-up of X along $\operatorname{Fit}_1 I_D$ , then $I_D \mathcal{O}_{X'}$ is an invertible $\mathcal{O}_{X'}$ -module and $\pi^{-1}(D)$ is an effective Cartier divisor.

3. (3) If D’ be the strict transform of D in X’, then $I_{D} \mathcal{O}_{X', K} = I_{D', K}$ .

1. (1) We have $ \operatorname{Fit}_1 I_K = (\operatorname{Fit}_1 I) A_K = (\operatorname{Fit}_1 I) \otimes_R K$ by [Sta19, Lemma 07ZA (3)] and flatness of the A-algebra $A_K$ . The ideal $I_K$ is principal, thus $\operatorname{Fit}_1 I_K = A_K$ by [Sta19, Lemma 07ZD]. This implies that $\operatorname{Fit}_1 I$ contains some $\varpi \in R \smallsetminus \{ 0\}$ .

2. (2) The proof of Lemma 2.9 can be translated in the formal setting with no changes, and we do not repeat it here.

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

3.5 Proofs

After these preliminaries we can finally prove Theorems 3.6 and 3.7. The arguments in the proof for Theorem 3.6 are a routine adaptation of those in the proofs of Theorems 2.1 and 2.2. 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 from those of Lemmas 2.3, 2.4 and 2.7, respectively.

Then, the strict transform of D in X’ is the effective Cartier divisor $\pi^{-1}(D_0) - E$ .

Proof of Theorem 3.6. Let $\pi \colon X' \to X$ be the blow-up along the first Fitting ideal $\operatorname{Fit}_1(I_D)$ of the coherent $\mathcal{O}_X$ -module $I_D$ . According to Lemma 3.9, the blow-up $\pi \colon X' \to X$ is admissible and $D_0 = \pi^{-1}(D)$ is an effective Cartier divisor in X’. Moreover, if D’ denotes the strict transform of D in X’, then $I_{D', K'} = I_{D_0, K'}$ . Thanks to Lemma 3.10 one can apply Lemma 3.11 to X’, D’ and $D_0'$ .

Let X be a quasi-compact admissible formal R-scheme and $Z_K \subset X_\eta$ a closed analytic subspace. Then, there is a unique closed formal subscheme Z flat over R such that $Z_\eta = Z_K$ . We call Z the Zariski closure of $Z_K$ in analogy with the situation for schemes. When $X = \operatorname{Spf}(A)$ is affine, Z is defined by the ideal $I \cap A$ where I is the kernel of the evaluation morphism $A \otimes_R K = \Gamma(X_\eta, \mathcal{O}_{X_\eta}) \to \Gamma(Z_K, \mathcal{O}_{X_\eta})$ . Note that I is finitely presented by Lemma 3.2.

Proof of Theorem 3.7. 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 [Reference BoschBos14, 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 [Reference LütkebohmertLüt90, Theorem 3.1] in the discretely valued case and [Reference TemkinTem00, Corollary 4.4] in the general case. According to Lemma 3.9, there is an admissible blow-up $\pi \colon X' \to X$ and an admissible closed formal subscheme D’ in X’ with generic fibre $\pi_\eta^{-1}(D_K)$ and such that $I_{D', K}$ is an invertible $\mathcal{O}_{X', K}$ -module. We conclude by applying Theorem 3.1 to X’ and D’.

4.1 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 Cartier 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

$F(\ast D) := \mathop{\operatorname{colim}}\limits_{d \in \mathbb{N}} F(dD),$

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 $\textrm{id}_F \otimes s_D \colon F \to F(D)$ is injective. Let $U = X \smallsetminus D$ and let $j \colon U \to X$ be 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 4.1. 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

\[ \mathop{\operatorname{colim}}\limits_{d \in \mathbb{N}} \Gamma(X, F(dD)) \stackrel{\sim}{\longrightarrow} \Gamma(X \smallsetminus D, F).\]

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 $\operatorname{colim}_{d \in \mathbb{N}} \Gamma(X, F(dD))$ 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 [Reference GrothendieckEGA III, Proposition 1.4.5]. For the latter statement note 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 $ \operatorname{colim}_{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 [Sta19, Lemma 009F (4)]. Applying statement (1) to V, the composite map

\[ \mathop{\operatorname{colim}}\limits_{d \in \mathbb{N}} \Gamma(V, F(dD)) \stackrel{\sim}{\longrightarrow} \Gamma(V, F(\ast D)) \longrightarrow \Gamma( V \cap (X \smallsetminus D), F) = \Gamma( V, j_\ast j^\ast F) \]

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

Lemma 4.2. 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, the maps $F(dD) \to F(\ast D)$ induce an isomorphism

$\mathop{\textrm{colim}}\limits_{d \in \mathbb{N}} \Gamma(X, F(dD)) \stackrel{\sim}{\longrightarrow} \Gamma(X, F(\ast D)).$

4.2 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 4.3. 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

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

and the prodiscrete topology on the left-hand side is given by the discrete topology on each $\Gamma(X, F/\varpi^n F)$ . See also [Sta19, Section 0AHY]. When X is affine and D is principal, Theorem 4.3 is rather elementary and boils down to approximating power series by polynomials. The strength of Theorem 4.3 comes from the non-affine case, where the proof becomes more involved. We do not know whether Theorem 4.3 holds without the assumption proper over affine. However, note that the conclusion of Theorem 4.3 holds more generally when X can be compactified into a proper admissible formal scheme over R, by applying Theorem 4.3 to the compactification. The remainder of this section is devoted to the proof of Theorem 4.3.

4.3 Norms and modules

We begin by recalling some facts about the topology on an R-module M. Its completion is the R-module $\hat{M} := \lim_{n \in \mathbb{N}} M / \varpi^n M$ where $\varpi \in R \smallsetminus \{ 0 \}$ is topologically nilpotent. Note that $\hat{M}$ does not depend on the chosen $\varpi$ . The R-module M is said to be separated (respectively, complete) if the natural map $\varepsilon \colon M \to \hat{M}$ is injective (respectively, bijective). Suppose M torsion-free and identify M with $\textrm{Im}(M \hookrightarrow M \otimes_R K)$ . 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 $\| \cdot \|_M \colon M \otimes_R K \to\mathbb{R}_+$ ; it is a norm if and only if M is separated. The R-module $\hat{M}$ is torsion-free and $\| \varepsilon(m)\|_{\hat{M}} = \| m\|_M$ for each $m \in M$ . Moreover, $\varepsilon$ extends to a homeomorphism 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 $f \colon M \to M'$ be a homomorphism between separated torsion-free R-modules. Then $\| f(m) \|_{M'} \leqslant \| m \|_M$ for each $m \in M$ . If f is injective, then $\textrm{Coker} f$ is torsion-free if and only if $\| f(m)\|_{M'} = \| m \|_M$ for each $m \in M$ .

1. (1) the natural topology on $\Gamma(X, F)$ coincides with the $\varpi$ -adic one;

2. (2) $\Gamma(X, F)$ is complete.

Proof. We prove both statements at the same time. Let $\varpi \in R \smallsetminus \{ 0 \}$ be topologically nilpotent and for $n \in \mathbb{N}$ let $X_n$ be the closed subscheme of X defined by the equation $\varpi^n = 0$ . By definition,

\[ \Gamma(X, F) = \lim_{n \in \mathbb{N}} \Gamma(X_n, F).\]

Flatness of F over R implies that the sequence

\[ 0 \longrightarrow \varpi^n F \longrightarrow F \longrightarrow F_{\rvert X_n} \longrightarrow 0\]

is short exact. It follows that the natural map $\Gamma(X, F) / \varpi^n \Gamma(X, F) \to \Gamma(X_n, F)$ is injective, because taking global sections is a left exact functor. Since the projective limit is also left-exact, the natural map

\[ \lim_{n \in \mathbb{N}} \Gamma(X, F) / \varpi^n \Gamma(X, F) \longrightarrow \lim_{n \in \mathbb{N}} \Gamma(X_n, F)\]

is injective. Therefore, the map $\Gamma(X, F) \to \lim_{n \in \mathbb{N}} \Gamma(X, F) / \varpi^n \Gamma(X, F) $ is bijective. This proves both statements.

4.4 Proof

Let S be an R-scheme, M a quasi-coherent $\mathcal{O}_X$ -module, and $\varpi$ a non-zero 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 non-zero topologically nilpotent element $\varpi'$ such that $|\varpi'| \leqslant |\varpi|$ and a homomorphism of $\mathcal{O}_S$ -modules $\varphi \colon M \to M'$ , the previous discussion, applied with $S = S_{\varpi'}$ , yields the following commutative and exact diagram of R-modules.

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

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

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

   (4.1) \begin{equation} \label{SES:RestrictionToDivisor} 0 \longrightarrow F \stackrel{s}{\longrightarrow} F(D) \longrightarrow F(D)_{\rvert D} \longrightarrow 0.\end{equation}
   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
   \[ 0 \longrightarrow F_{\rvert X_\varpi} \stackrel{s}{\longrightarrow} F(D)_{\rvert X_\varpi} \longrightarrow F(D)_{\rvert D_\varpi} \longrightarrow 0. \]
   The statement follows by taking global sections.

2. (2) The short exact sequence of $\mathcal{O}_X$ -modules (4.1) yields the following commutative diagram.

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

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 with respect to the norms $\| \cdot \|_{\Gamma(X, F)}$ and $\| \cdot \|_{\Gamma(X\smallsetminus D, F)}$ defined in § 4.3; in particular, $\rho$ induces a homeomorphism of $\Gamma(X, F)$ with a closed subset of $\Gamma(X \smallsetminus D, F)$ .

Proof. Statements (1), (2) and (3) are 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,

(4.2) \begin{align} \Gamma(X, F(\ast D)) = M [t] / (1 - tf), \quad\ \Gamma(U, F) = M \{ t \} / (1 - tf).\end{align}

We first prove statements (2) and (3) and then statement (1) under this additional assumption. Finally, we prove statement (4) for general X.

(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 a power series $n(t) \in M \{ t \}$ 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. By Lemma 4.4 the topology on M coincides with the $\varpi$ -adic one. Therefore, 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 4.5(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,

\[ \textrm{H}^1(X, F(\ast D)) = \mathop{\operatorname{colim}}\limits_{d \in \mathbb{N}} \textrm{H}^1(X, F(dD)).\]

For $d \in \mathbb{N}$ , since X is affine and F(dD) coherent, the cohomology group $\textrm{H}^1(X, F(dD))$ vanishes for each $d \in \mathbb{N}$ , whence the claimed surjectivity. Equation (4.2) can be rewritten as the following commutative and exact diagram of A-modules.

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:

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

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 $m = n_0$ , $n_i = f n_{i-1}$ for $i = 1, \ldots, d$ and $f n_d = 0$ . Since multiplication by f on M is injective, one obtains $n_0 = 0$ , thus $m = 0$ . This proves that $\alpha$ is injective.

Recall that an R-module is flat if and only if it is torsion-free. In view of this, since the natural map $\textrm{Coker} \alpha \to \textrm{Coker} \beta$ is injective by statement (2), it 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 $i \geqslant 1$ , $\varpi m_i = n_i - f n_{i-1}$ . In other words, modulo $\varpi M$ , the following congruences hold:

\begin{align*}n_0 \equiv - p , \quad fn_{i-1} \equiv n_i, \quad (i \geqslant 1).\end{align*}

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 4.5(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) Statement (3) implies $\| \rho(f) \|_{\Gamma(U, F)} = \| f \|_{\Gamma(X, F)}$ for each $f \in \Gamma(X, F)$ . According to Lemma 4.4, $\rho$ is an isometric embedding between complete metric spaces. It follows that the image is closed.

Proof of Theorem 4.3. Let $j \colon U \to X$ be the open immersion of U in X. Note that the situation here is no longer local, so X cannot be taken to be affine, contrarily to what done in the proof of the previous lemma. In particular, one needs an argument finer than simply approximating power series by polynomials. To do so, Lemma 4.6 implies that 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 an 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 4.5(1), for each $d \in \mathbb{N}$ , the map $\alpha_{d,n} \colon \Gamma(X_n, F(d D)) \to \Gamma(U_n, F)$ 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 following diagram implies $d_m \geqslant d_n$ .

The map $s^{d_m - d_n} \colon \Gamma(X_n, F(d_n D)) \to \Gamma(X_n, F(d_m D))$ is seen to be injective by applying Lemma 4.5 (1) with $d = d_n$ and $i = d_m - d_n$ . It follows that

\[s^{d_m - d_n} f_n = f_{m \rvert X_n}.\]

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 $\Gamma(X_{n}, F(d_mD))/ \Gamma(X_{m}, F(d_mD))$ is zero. However, by Lemma 4.5, the map

\[s^i \colon \frac{\Gamma(X_{n}, F(d_nD))}{\Gamma(X_{m}, F(d_nD))} \longrightarrow\frac{\Gamma(X_{n}, F(d_mD))}{\Gamma(X_{m}, F(d_mD))}\]

is injective. Therefore, the image of $f_n$ in $\Gamma(X_{n}, F(d_nD))/ \Gamma(X_{m}, F(d_nD))$ is also zero. In other words, there is $g_m \in \Gamma(X_{m}, F(d_nD))$ 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 $(\Gamma(X_m, F(d_nD)))_{m \geqslant n}$ 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 [Reference GrothendieckEGA III, Corollaire 4.1.7] (see also [FGI+05, 8.2.7, p. 191]); the general case can be found in [Reference AbbesAbb10, Corollaire 2.11.7] and [Reference Fujiwara and KatoFK18, Proposition I.11.3.3 and Lemma 0.8.8.5 (2)]. For $m \geqslant n$ let

\[\rho_m \colon \Gamma(X_m, F(d_nD)) \longrightarrow \Gamma(X_n, F(d_nD))\]

be the restriction map. According to the 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 [Sta19, Lemma 0597]. In other words, there is $g_\infty \in \lim_{m \in \mathbb{N}} \Gamma(X_m, F(d_nD))$ such that $g_{\infty \rvert X_n} = f_n$ . To conclude, let E be a flat coherent $\mathcal{O}_X$ -module. The comparison theorem in formal geometry (see [Reference GrothendieckEGA III, Corollaire 4.1.7] or [Reference Fantechi, Göttsche, Illusie, Kleiman, Nitsure and VistoliFGI⁺05, Theorem 8.2.6.1] in the Noetherian setting, or [Reference Fujiwara and KatoFK18, Theorem 0.9.2.1] in the general case) states that the natural map

\[ \varphi_E \colon \Gamma(X, E) \longrightarrow \lim_{m \geqslant n} \Gamma(X_m, E)\]

is an isomorphism. One concludes the proof by taking g to be the preimage of $g_\infty$ under the above isomorphism for $E := F(d_nD)$ .

5.1 Reminder on Stein spaces

We begin with a reminder on Stein spaces in the non-Archimedean framework, see also [Reference Maculan and PoineauMP21]. Recall that a K-analytic space X is said to holomorphically separable if, for $x, x' \in X$ distinct, there is $f \in \Gamma(X, \mathcal{O}_X)$ such that $|f(x)| \neq |f(x')|$ . 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 W-exhausted by affinoid domains in [Reference Maculan and PoineauMP21]. A K-analytic space Stein X is holomorphically separable and holomorphically convex by [Reference Maculan and PoineauMP21, 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 $\textrm{H}^q(X, F)$ vanishes.

Given a K-affinoid space S, closed analytic subspaces of $\mathbb{A}^{n, \textrm{an}}_K \times_K S$ are Stein. Conversely, let X be a Stein 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 $X \hookrightarrow \mathbb{A}^{n, \textrm{an}}_K$ . In particular, for a K-scheme of finite type X the K-analytic space $X^\textrm{an}$ is Stein if and only if there is a closed embedding closed embedding $X^\textrm{an} \hookrightarrow \mathbb{A}^{n, \textrm{an}}_K$ for some $n \in \mathbb{N}$ .

5.2 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 fibres, 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 fibres of $\sigma$ , there is a unique morphism of K-analytic spaces $\tilde{\tau} \colon S \to T$ such that $\tau = \tilde{\tau} \circ \sigma$ .

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

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 5.2 one obtains the following result.

In the rest of this section is devoted to the proof of Theorem 5.2.

5.3 Proofs

The proof of Theorem 5.2 follows closely that of [Reference Maculan and PoineauMP21, Theorem 6.1], in turn inspired by the proof of the complex Remmert factorization theorem. In order to control boundary, we need the following.

We recall the statement of the Stein factorization theorem [Reference BerkovichBer90, Proposition 3.3.7] with a slight improvement. Let $f \colon X \to Y$ be a proper morphism of K-analytic spaces. Then, the sheaf of $\mathcal{O}_Y$ -algebras $\mathcal{A} := f_\ast \mathcal{O}_X$ is coherent. Moreover, there is a (unique up to a unique isomorphism) K-analytic space $\tilde{Y}$ together with a finite morphism $\pi \colon \tilde{Y} \to Y$ , called the Stein factorization of f, representing the functor associating to a morphism $g \colon S \to Y$ of K-analytic spaces the set of morphisms of $\mathcal{O}_S$ -algebras $g^\ast \mathcal{A} \to \mathcal{O}_S$ . Concretely, when $Y= \mathcal{M}(B)$ is affinoid, the K-analytic space $\tilde{Y}$ is the Banach spectrum of the finite B-algebra $A:= \Gamma(X, \mathcal{O}_X)$ . Let $\tilde{f} \colon X \to \tilde{Y}$ be the morphism corresponding to the morphism $f^\ast f_\ast \mathcal{O}_X \to \mathcal{O}_X$ .

1. (1) $\tilde{f}$ is proper and surjective, $\pi$ is finite and $f = \pi \circ \tilde{f}$ ;

2. (2) the morphism $\mathcal{O}_{\tilde{Y}} \to \tilde{f}_\ast \mathcal{O}_X$ is an isomorphism;

3. (3) the fibres of $\tilde{f}$ are geometrically connected;

4. (4) given a morphism $g \colon X \to Z$ which is constant on the connected components of the fibres of f, there is a unique morphism $\tilde{g} \colon \tilde{Y} \to Z$ such that $g = \tilde{g} \circ \tilde{f}$ ;

5. (5) the construction of the Stein factorization is compatible with extension of scalars, thus given a complete valued extension K’ of K, the Stein factorization of $X \times_K K' \to Y \times_K K'$ is $\tilde{Y} \times_K K' \to Y \times_K K'$ .

To prove property (5), let K’ be a complete valued extension of K. Let X’, Y’ and $\tilde{Y}'$ be the K’-analytic spaces obtained from X, Y and $\tilde{Y}$ , respectively, extending scalars to K’, and let $f' \colon X' \to Y'$ , $\pi \colon \tilde{Y}' \to Y'$ and $\tilde{f} \colon X' \to \tilde{Y}'$ the morphisms induced by f, $\pi$ and $\tilde{f}$ , respectively. It suffices to show that the natural morphism $q^\ast f_\ast \mathcal{O}_X \to f'_\ast \mathcal{O}_{X'}$ is an isomorphism, where $q \colon Y' \to Y$ is the base-change morphism. This boils down to prove, when $Y = \mathcal{M}(B)$ is affinoid, that the natural homomorphism of Banach B’-algebras $\varphi \colon \Gamma(X, \mathcal{O}_X) \hat{\otimes}_B B' \to\Gamma(X', \mathcal{O}_{X'})$ is an isomorphism, where $B' = B\hat{\otimes}_K K'$ . This then follows on from the natural isomorphism $\Gamma(X', \mathcal{O}_{X'}) \simeq \Gamma(X, \mathcal{O}_X)\hat{\otimes}_K K'$ , see, for instance, [Reference Maculan and PoineauMP21, Theorem A.5]. This concludes the proof of property (5).

To prove property (3), pick a point $y \in \tilde{Y}$ and let K’ be the completion of an algebraic closure of the completed residue field $\mathcal{H}(y)$ at y. The point y defines a natural K’-point y’ of $\tilde{Y}'$ . The base-change to K’ of $\tilde{f}^{-1}(y)$ is the fibre $\tilde{f}'$ at y’, thus we need to show that this is connected. By property (5) the Stein factorization of f’ is $\pi'$ , thus the fibre of $\tilde{f}'$ at y’ is connected by [Ber90, Proposition 3.3.7].

We first prove Theorem 5.2 when the target is affinoid.

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

\[n_{-1} = 0 , \quad X_{-1} = U_{-1} = W_{-1} = \emptyset.\]

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 $\mathbb{A}^{n_i, \textrm{an}}_Y$ and K-analytic functions $f_{n_{i-1} + 1}, \ldots, f_{n_i}$ on X such that:

1. (1) the morphism $(f_1, \ldots, f_{n_i}, \pi) \colon X \to \mathbb{A}^{n_i, \textrm{an}}_Y$ induces a proper morphism $F_i \colon U_i \to W_i$ ;

2. (2) $\mathbb{D}_i := \mathbb{D}(\| f_1\|_{C_i \cup \overline{U}_{i-1} }, \ldots, \| f_{n_i}\|_{C_i \cup \overline{U}_{i-1}}) \times_K Y$ is contained in $W_i$ ; and

3. (3) $X_i := F_i^{-1}(\mathbb{D}_i) \cap U_i$ contains the holomorphically convex hull of $C_i \cup \overline{U}_{i-1}$ .

Set $C'_i := C_i \cup \overline{U}_{i - 1}$ and consider a relatively compact open subset $V_i \subset X$ containing the holomorphically convex hull of $C'_i$ 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 $f_{n_{i -1} + 1}, \ldots, f_{n_{i}}$ on X such that, for all $x \in \partial V_i$ ,

\[ \max_{j = 1, \ldots, n_i} \frac{|f_j(x)|}{\| f_j \|_{C'_i}} > 1.\]

Set $W_i := \mathbb{A}^{n_i, \textrm{an}}_Y \smallsetminus F_i(\partial V_i)$ 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 $(f_1, \ldots, f_{n_i}, \pi) \colon X \to \mathbb{A}^{n_i, \textrm{an}}_Y$ is without boundary (Lemma 5.4) 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. Let $\tilde{F}_i \colon S_i \to \mathbb{D}_i$ be the Stein factorization of the proper morphism $F_i$ and $\sigma_i \colon X_i \to S_i$ the induced morphism. The morphism $\tilde{S}_i$ is finite, hence $S_i$ is affinoid, thus Stein. It follows from Lemma 5.5 that $(S_i, \sigma_i, \tilde{F}_i)$ is the Remmert factorization of $F_i$ . In particular, 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 $\varepsilon_i \colon S_i \to S_{i+1}$ .

The proof of the claim is by any means analogous to the that of [Reference Maculan and PoineauMP21, 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 (RF $_1$ ) and (RF $_2$ ) 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 $\textrm{Hom}_K(Y, X)$ denote the set of morphisms between some K-analytic spaces X and Y. If X and Y are countable at infinity, let $\textrm{Hom}_{K\textrm{-alg}}(\mathcal{O}(X), \mathcal{O}(Y))$ 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 5.8. Let X and Y be K-analytic spaces countable at infinity with X Stein. Then the following map is bijective:

\[ \textrm{Hom}_K(Y, X) \longrightarrow \textrm{Hom}_{K\textrm{-alg}}(\mathcal{O}(X), \mathcal{O}(Y)),\quad f \colon Y \to X \longmapsto f^\sharp \colon \mathcal{O}(X) \to \mathcal{O}(Y). \]

(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. Since $Y_i$ is compact for any $i \in \mathbb{N}$ , we may replace $\{ X_i \}_{i \in \mathbb{N}}$ by a subsequence such that $f_\alpha(Y_i) \subset X_i$ for any $i \in \mathbb{N}$ and $\alpha = 1, 2$ . 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.

Here the vertical arrows are the inclusions. Therefore, the morphisms $f_i$ glue to a morphism $f \colon Y \to X$ such that $f^\sharp = \varphi$ .

1. (1) the K-analytic space X’ is Stein;

2. (2) for any affinoid Stein exhaustion $\{ X'_i\}_{i \in \mathbb{N}}$ of X’, there is an affinoid Stein exhaustion $\{ X_i \}_{i \in \mathbb{N}}$ of X such that, for each $i \in \mathbb{N}$ , $X'_i$ is a Weierstrass domain of $X_i$ .

Proof. By definition of a Weierstrass domain there are $f_1, \ldots, f_n \in \Gamma(Y,\mathcal{O}_Y)$ and real numbers $r_1, \ldots, r_n >0$ such that $Y' = \{ y \in Y : |f_j(y)| \leqslant r_j , j = 1, \ldots, n\}$ . Given an affinoid Stein exhaustion $\{ W_i \}_{i \in \mathbb{N}}$ of X define, for $i\in \mathbb{N}$ ,

\[ W'_i = \{ x \in W_i : |f_j(\pi(x))| \leqslant r_j ,\quad j = 1, \ldots, n\} . \]

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

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

Proof. By definition of Weierstrass domain, there are $f_1,\ldots, f_n \in \Gamma(Y, \mathcal{O}_Y)$ and $r_1, \ldots, r_n > 0$ such that $Y'= \{ y \in Y : |f_i(y)| \leqslant r_i \textrm{ for all } i = 1, \ldots, r\}$ . In particular,

\[ X' := X \times_Y Y' = \{ x \in X : |f_i(\pi(x))| \leqslant r_i\quad\! \textrm{ for all } i = 1, \ldots, r \} .\]

The K-analytic space X’ is holomorphically convex. Indeed, given a compact subset of $C \subset X'$ , it follows from the description that $\hat{C}_X$ is contained in X’. On on the other hand, $\hat{C}_X$ is compact because X is holomorphically convex and $\hat{C}_{X'}$ is closed in $\hat{C}_{X} \cap X' = \hat{C}_X$ , thus $\hat{C}_{X'}$ is compact. The morphism $\pi' \colon X' \to Y'$ is without boundary and Y’ is affinoid. Therefore, by Proposition 5.6, the Remmert factorization $(S', \sigma', \tilde{\pi}')$ of $\pi'$ exists. By Lemma 5.1, there is a unique morphism $\varphi \colon S' \to S$ such that $\sigma_{\rvert X'}= \varphi \circ \sigma'$ . By the open mapping theorem of Fréchet spaces [Reference SchneiderSch02, Corollary 8.7] the continuous isomorphisms $\sigma_{\rvert X'}^\sharp \colon \mathcal{O}(S \times_Y Y') \to \mathcal{O}(X')$ and $\sigma'^{\sharp} \colon \mathcal{O}(S') \to \mathcal{O}(X')$ induced by the proper morphisms $\sigma_{\rvert X'}$ and $\sigma'$ , respectively, are homeomorphisms. It follows that the homomorphism $\varphi^\sharp \colon \mathcal{O}(S \times_Y Y') \to \mathcal{O}(S')$ induced by $\varphi$ is an isomorphism of Fréchet algebras. Since $S \times_Y Y'$ and S’ are Stein (Lemma 5.9), Proposition 5.8 implies that $\varphi$ is an isomorphism.

Proof of Theorem 5.2. 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 Proposition 5.6, the Remmert factorization $(S_i, \sigma_i, \tilde{\pi}_i)$ of $\pi \colon X_i \to Y_i$ exists. Moreover, by Proposition 5.10, the Remmert factorization $(S_i, \sigma_i, \tilde{\pi}_i)$ is canonically isomorphic to $(S'_i, \sigma_{i + 1 \rvert S'_i}, \tilde{\pi}_{i + 1 \rvert S'_i})$ , where $S'_i := \tilde{\pi}^{-1}(Y_{i+1})$ . In what follows, we identify $S_i$ with $S_i'$ through the preceding canonical isomorphism. Thanks to Lemma 5.9, 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 $i' \leqslant i$ and $j \in \mathbb{N}$ , the affinoid space $S_{i'j}$ 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 (RF $_1$ ) and (RF $_2$ ) are deduced from those of $\sigma_i$ , for $i \in \mathbb{N}$ . Analogously, the morphism $\pi$ is seen to be without boundary.

6.1 Statements

Given a coherent sheaf F on a K-analytic space X the topology on its global sections 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 $\mathcal{V}$ is an affinoid G-cover of X, the restriction map induces a homeomorphism

\[ \Gamma(X, F) \stackrel{\sim}{\longrightarrow} \lim_{V \in \mathcal{V}} \Gamma(V, F),\]

the target being endowed with the usual topology on the projective limit. 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. The main result of this section is the analogue of Theorem 4.3 in the context of analytic spaces. In order to state it, we need to construct the sheaf of functions with polar singularities. We begin with the following general lemma, which will be used repeatedly.

Lemma 6.1. Let X be a compact analytic space and $(F_i, \varphi_{ij})_{i, j \in I}$ a directed system of sheaves of sets for the G-topology on X. Then, the canonical map is bijective

\[ \mathop{\operatorname{colim}}\limits_{i \in I} \Gamma(X, F_i)\stackrel{\sim}{\longrightarrow} \Gamma(X, \mathop{\operatorname{colim}}\limits_{i \in I} F_i).\]

We are now in position to define the sheaf of functions with polar singularities.

1. (1) If X is compact, then the maps $F(dD) \to F(\ast D)$ induce an isomorphism

   \[ \mathop{\operatorname{colim}}\limits_{d \in \mathbb{N}} \Gamma(X, F(dD)) \stackrel{\sim}{\longrightarrow} \Gamma(X, F(\ast D)).\]

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

We are now in position to state the main result. Recall that an $\mathcal{O}_X$ -module over a ringed space X is said to be semi-reflexive if the homomorphism $F \to F^{\vee\vee}$ is injective.

Theorem 6.3. 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.

It is sometimes useful to combine Theorem 6.3 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. If this is the case, for any coherent $\mathcal{O}_X$ -module F, the natural morphism $F^\vee \to j_\ast j^\ast F^\vee$ is injective. In particular, if F is semi-reflexive, then $F \to j_\ast j^\ast F$ is injective.

Proposition 6.4. Let Z be a closed analytic subspace of a Hausdorff K-analytic space X. 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.

6.2 Extension of semi-reflexive modules

When X is proper Theorem 6.3 will be deduced from Theorem 4.3 by passing to the generic fibre. However, to do so we need to extend modules from the generic fibre to the formal scheme preserving the hypotheses needed to apply Theorem 4.3. To start with recall that coherent sheaves can be extended from the generic fibre 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 [Reference Bosch and LütkebohmertBL93, Proposition 5.6], [Reference AbbesAbb10, Proposition 5.6] or [Reference Fujiwara and KatoFK18, Corollary II.5.3.3] in the general case. Note that, for 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 to preserve the semi-reflexivity and the flatness on the Cartier divisor. We start with the following.

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

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