Proof. Assertion (i) follows from [Reference Saïdi21, Theorem A]. (Note that the assumption in [Reference Saïdi21, Theorem A] that X is the complement in a proper rigid analytic k-curve of the disjoint union of finitely many k-rational open disks is satisfied after a finite extension of k [cf. [Reference Fresnel and Matignon6, 2.6, Théorème 6 and Corollaire 1]].) Assertion (ii) follows from [Reference Saïdi23, Theorem A].

Proof. We have a natural surjective homomorphism $\prod _{i=1}^n \operatorname {Ind} _{k(x_i)}^k \hat {\mathbb {Z}}(1)\twoheadrightarrow \widetilde {\Delta }_U$ of $G_k$ -modules mapping $\operatorname {Ind}_{k(x_i')}^k \hat {\mathbb {Z}}(1)$ onto the inertia subgroup [of $\pi _1(U,\eta )^{(\operatorname {ab})}$ ] at $x_i$ , as follows from the structure of inertia groups of Galois extensions of Henselian discrete valuation rings of residue characteristic zero. We show this map is an isomorphism. To this end, we can, without loss of generality, assume that X admits a k-compactification Y (cf. Notations). Indeed, this holds for X affinoid (cf. [Reference Saïdi21, 2.1]), and holds after possibly replacing k by a finite field extension in case X is a formal p-adic germ (cf. Proposition 5.1) which does not alter the structure of $\widetilde {\Delta }_U$ . We have a commutative diagram of exact sequences

(1.2)

where $\pi _1(Y,\eta )^{(\operatorname {ab})}\overset {\text {def}} = \pi _1(Y,\eta )/\operatorname {Ker}[\pi _1(Y_{\bar k},\bar \eta )\twoheadrightarrow \pi _1(Y_{\bar k},\bar \eta )^{\operatorname {ab}}]$ and the middle vertical map is induced by the natural homomorphism $\pi _1(X,\eta )\to \pi _1(Y,\eta )$ (cf. Notations, diagram (0.1)).

Denote by $x_i'$ the image of $x_i$ in Y, $\forall 1\le i\le n$ (note that $k(x_i)=k(x_i')$ ). Let $x_0'\in Y^{\operatorname {cl}} \setminus \{x_1',\ldots ,x_n'\}$ be a closed point which is not in the image of X (cf. Notations). Write $S'\overset {\text {def}} = \{x_0',x_1',\ldots ,x_n'\}\subset Y^{\operatorname {cl}}$ and $V\overset {\text {def}} = Y\setminus S'$ which is an affine k-curve. Let $\pi _1(V,\eta )$ be the étale fundamental group of V with geometric point $\eta $ which sits in the exact sequence $1\to \pi _1(V_{\bar k},\bar \eta )\to \pi _1(V,\eta )\to G_k \to 1,$ where $\pi _1(V_{\bar k},\bar \eta )$ is the étale fundamental group of $V_{\bar k}\overset {\text {def}} = V\times _{k}\bar k$ with geometric point $\bar \eta $ which is induced by $\eta $ . Let $\pi _1(V,\eta )^{(\operatorname {ab})}\overset {\text {def}} = \pi _1(V,\eta )/\operatorname {Ker} [\pi _1(V_{\bar k},\bar \eta )\twoheadrightarrow \pi _1(V_{\bar k},\bar \eta )^{\operatorname {ab}}]$ be the geometrically abelian fundamental group of V. We have a commutative diagram of exact sequences

(1.3)

where $\widetilde {\Delta }_V\overset {\text {def}} = \operatorname {Ker}[\pi _1(V,\eta )^{(\operatorname {ab})}\twoheadrightarrow \pi _1(Y,\eta )^{(\operatorname {ab})}]$ . The middle vertical map in diagram (1.3) is induced by the natural homomorphism $\pi _1(U,\eta )\to \pi _1(V,\eta )$ , which is induced by the scheme morphism $X\to Y$ in case X is a formal p-adic germ, and by the rigid analytic morphism $X\to Y^{\operatorname {rig}}$ and the rigid GAGA functor in case X is affinoid (here, we use the fact that $x_0'$ is not in the image of X). The right vertical map in diagram (1.3) is the middle vertical map in diagram (1.2).

One has an exact sequence of $G_k$ -modules (as follows from the well-known structure of $\pi _1(V,\eta )^{(\operatorname {ab})}$ ; see, e.g., the discussion in [Reference Saïdi24, §0])

$$ \begin{align*}0\to \hat{\mathbb{Z}}(1)\to \prod_{i=0}^n\operatorname{Ind}_{k(x_i')}^k \hat{\mathbb{Z}}(1)\to \widetilde{\Delta}_V\to 0.\end{align*} $$

Consider the composite homomorphism $\tau :\prod _{i=1}^n\operatorname {Ind}_{k(x_i')}^k \hat {\mathbb {Z}}(1)\to \widetilde {\Delta }_V$ of $G_k$ -modules:

$$ \begin{align*}\prod_{i=1}^n\operatorname{Ind}_{k(x_i')}^k \hat{\mathbb{Z}}(1)\hookrightarrow \prod_{i=0}^n\operatorname{Ind}_{k(x_i')}^k \hat{\mathbb{Z}}(1)\twoheadrightarrow \widetilde{\Delta}_V,\end{align*} $$

where the first map is the natural embedding: $(\beta _1,\ldots ,\beta _n)\mapsto (0,\beta _1,\ldots ,\beta _n)$ and the second map is as in the above exact sequence. Thus, $\tau $ is injective (cf. above exact sequence). Consider the following commutative diagram:

where the right vertical map is the one in diagram (1.3). The left vertical and lower horizontal maps are as explained above; hence, their composite is the map $\tau $ . The upper horizontal map is the natural projection $\prod _{i=1}^n\operatorname {Ind}_{k(x_i')}^k \hat {\mathbb {Z}}(1) \twoheadrightarrow \widetilde {\Delta }_U$ mentioned at the start of the proof. This map is an isomorphism since it is onto and it is injective as $\tau $ is.

Proof. This follows from Proposition 1.2 and the fact that $\widetilde {\mathcal {H}}_X\overset \sim \to \varprojlim _{U}\widetilde {\Delta } _U$ where $U=X\setminus S$ ; S runs over all finite subsets of $X^{\operatorname {cl}}$ , and $\widetilde {\Delta }_U$ is as in the proof of Proposition 1.2. (Note that $G_X^{(\operatorname {ab})}\overset \sim \to \varprojlim _{U} \pi _1(U,\eta )^{(\operatorname {ab})}$ where the limit runs over all U as above.)

Proof. The proof follows from Proposition 1.2 and the various definitions. More precisely, there exists a natural surjective homomorphism $\prod _{i=1}^n\operatorname {Ind}_{k(x_i)}^k \hat {\mathbb {Z}}(1)\twoheadrightarrow \Delta _U^{\operatorname {cn}}$ (mapping $\operatorname {Ind}_{k(x_i')}^k \hat {\mathbb {Z}}(1)$ onto the inertia subgroup of $\pi _1(U,\eta )^{(\text c-\operatorname {cn})}$ at $x_i$ , as follows from the structure of inertia groups of Galois extensions of Henselian discrete valuation rings of residue characteristic zero) which composed with the projection $\Delta _U^{\operatorname {cn}} \twoheadrightarrow \widetilde {\Delta }_U$ is the isomorphism $\prod _{i=1}^n \operatorname {Ind} _{k(x_i)}^k \hat {\mathbb {Z}}(1)\overset \sim \to \widetilde {\Delta }_U$ in Proposition 1.2 hence our assertion.

Proof. Consider the commutative diagram of exact sequences

where the right square is Cartesian. Thus, the group extension $E_U$ is the pullback of the group extension $\pi _1(U,\eta )^{(\text c-\operatorname {cn})}$ by the section s. The set of (possible) splittings of the group extension $E_U$ is in one-to-one correspondence with the set of sections of the projection $\pi _1(U,\eta )^{(\text c-\operatorname {cn})}\twoheadrightarrow G_k$ which lift the section s. We show that the group extension $E_U$ splits.

To this end, we can replace k by a finite extension over which the points $\{x_i\}_{i=1}^n$ are rational, and we can also assume $n=1$ (see the argument at the start of the proof of Lemma 2.3.1 in [Reference Saïdi19]). Further, we can replace X by a neighborhood $X'$ of the section s: that is, an étale cover $X'\to X$ corresponding to an open subgroup $H=\pi _1(X',\eta )\subset \pi _1(X,\eta )$ containing the image $s(G_k)$ of s. Indeed, if $U'\overset {\text {def}} = U\times _XX'$ , there exists a commutative diagram of natural homomorphisms

where the upper horizontal map is induced by the natural map $\pi _1(U',\eta ) \to \pi _1(U,\eta )$ (note $\Delta _{U'}=\Delta _U$ and $\pi _1(X',\eta )^{\operatorname {geo}}$ acts trivially on $\Delta _U^{\operatorname {cn}}$ ), and the various maps in this diagram commute with the projections onto $G_k$ . The section s induces a section $\tilde s:G_k\to \pi _1(X',\eta )$ of the projection $\pi _1(X',\eta )\twoheadrightarrow G_k$ , and a lifting $\tilde s_{U'}^{\text c-\operatorname {cn}}:G_k\to \pi _1(U',\eta )^{(\text c-\operatorname {cn})}$ of $\tilde s$ (as in the statement of Proposition 2.2) induces a lifting $s_U^{\text c-\operatorname {cn}}:G_k\to \pi _1(U,\eta )^{(\text c-\operatorname {cn})}$ of s as required (cf. above diagram). Now, it follows from [Reference Saïdi21, Theorem A] in case X is an affinoid, and Proposition 5.3 in this paper (cf. §5) in case X is a formal p-adic germ, that there exists (after possibly a finite extension of k) a neighborhood $X'\to X$ of s with $X'$ hyperbolic (cf. Notations). We can thus assume, without loss of generality, that X possesses a k-compactification Y with Y hyperbolic and the set $S\overset {\text {def}} = \{x\}\subset X(k)$ consists of a single k-rational point, in which case $\Delta _U^{\operatorname {cn}}\overset \sim \to \hat {\mathbb {Z}}(1)$ as a $\pi _1(X,\eta )$ -module (cf. Lemma 2.1 and Proposition 1.2).

Consider the following maps (here, $X=\operatorname {Spec} A$ in case X is affinoid):

$$ \begin{align*}H^2(\pi_1(X,\eta),\hat{\mathbb{Z}}(1)) \hookrightarrow H^2(X,\hat{\mathbb{Z}}(1)) \leftarrow \operatorname{Pic} (X),\end{align*} $$

where the map $H^2(\pi _1(X,\eta ),\hat {\mathbb {Z}}(1)) \hookrightarrow H^2(X,\hat {\mathbb {Z}}(1))$ arises from the Cartan–Leray spectral sequence and is injective (cf. [Reference Serre25, Proof of Proposition 1]), and the map $\operatorname {Pic}(X)\rightarrow H^2(X,\hat {\mathbb {Z}}(1))$ is the cycle class map arising from the Kummer exact sequence in étale topology. Let $[\pi _1(U,\eta )^{(\text c-\operatorname {cn})}]\in H^2(\pi _1(X,\eta ),\hat {\mathbb {Z}}(1))$ be the class of the group extension $\pi _1(U,\eta )^{(\text c-\operatorname {cn})}$ . The image of $[\pi _1(U,\eta )^{(\text c-\operatorname {cn})}]$ in $H^2(X,\hat {\mathbb {Z}}(1))$ coincides with the image of the line bundle ${\mathcal {O}}(x)\in \operatorname {Pic}(X)$ via the Kummer map $\operatorname {Pic} (X) \rightarrow H^2(X,\hat {\mathbb {Z}}(1))$ . Indeed, this follows from the following commutative diagram:

where the right and middle vertical maps are induced by the scheme morphism $X\to Y$ if X is a formal p-adic germ, and the rigid morphism $X\to Y^{\operatorname {rig}}$ and the comparison theorems between étale cohomology and rigid analytic étale cohomology in case X is affinoid (cf. [Reference Hansen11, Theorem 1.8 and Theorem 1.9]). The right horizontal maps are the cycle class maps arising from the Kummer exact sequence in étale topology, and the left lower horizontal map is an isomorphism arising from the Cartan–Leray spectral sequence (cf. [Reference Mochizuki17, Proposition 1.1]). The pullback of the class $[\pi _1(V,\eta )^{(\text c-\operatorname {cn})}]\in H^2(\pi _1(Y,\eta ),\hat {\mathbb {Z}}(1))$ in $H^2(\pi _1(X,\eta ),\hat {\mathbb {Z}}(1))$ , where V is the complement in Y of the image of $S=\{x\}$ (cf. [Reference Saïdi19, 2.1.1] for the definition of $\pi _1(V,\eta )^{(\text c-\operatorname {cn})}$ ), coincides with the class $[\pi _1(U,\eta )^{(\text c-\operatorname {cn})}]$ (this follows from Lemma 2.1 and the various definitions). The class $[\pi _1(V,\eta )^{(\text c-\operatorname {cn})}]\in H^2(\pi _1(Y,\eta ),\hat {\mathbb {Z}}(1))\overset \sim \to H^2(Y,\hat {\mathbb {Z}}(1))$ coincides with the image of the Chern class of the line bundle ${\mathcal {O}}(y)\in \operatorname {Pic}(Y)$ where $y\in Y(k)$ is the image of x (cf. [Reference Saïdi19, Proof of Lemma 2.3.1]). Thus, the image of $[\pi _1(U,\eta )^{(\text c-\operatorname {cn})}]$ in $H^2(X,\hat {\mathbb {Z}}(1))$ coincides with the image of the line bundle ${\mathcal {O}}(x)\in \operatorname {Pic}(X)$ via the cycle class map $\operatorname {Pic} (X)\rightarrow H^2(X,\hat {\mathbb {Z}}(1))$ as claimed.

The Picard group $\operatorname {Pic}(X)$ is finite (cf. Proposition 4.1 in this paper in case X is affinoid and [Reference Saïdi22, Proposition 5.4] in case X is a formal p-adic germ). In particular, the image of $[\pi _1(U,\eta )^{(\text c-\operatorname {cn})}]$ in $H^2(X,\hat {\mathbb {Z}}(1))$ and hence the class $[\pi _1(U,\eta )^{(\text {c}-\operatorname {cn})}]$ is a torsion element of $H^2(\pi _1(X,\eta ),\hat {\mathbb {Z}}(1))$ . The class $[E_U]\in H^2(G_k,\hat {\mathbb {Z}}(1))$ of the group extension $E_U$ is the image of $[\pi _1(U,\eta )^{(\text {c}-\operatorname {cn})}]$ under the retraction map induced by s. Hence, the class $[E_U]$ is trivial since $\hat {\mathbb {Z}}$ is torsion-free, and the group extension $E_U$ splits.

Proof. Let $\{H\}_{i\in I}$ be a projective system of open subgroups of $\pi _1(X,\eta )$ containing $s(G_k)$ such that $s(G_k)=\bigcap _{i\in I}H_i$ . Thus, for $i\in I$ , the open subgroup $H_i$ corresponds to an étale finite cover $X_i\to X$ with $X_i$ geometrically connected and $H_i$ is identified with $\pi _1(X_i,\eta )$ which sits in the exact sequence $1\to \pi _1(X_{i},\eta )^{\operatorname {geo}}\to \pi _1(X_i,\eta )\to G_k\to 1$ (the geometric point; denote also $\eta $ , of $X_i$ is induced by the geometric point $\eta $ of X). Further, the section s induces a section $s_i:G_k\to \pi _1(X_i,\eta )$ of the projection $\pi _1(X_i,\eta )\twoheadrightarrow G_k$ . Let $U_i\overset {\text {def}} = U\times _XX_i$ , and let $\pi _1(U_i,\eta )^{(\text c-\operatorname {cn})}$ be the (geometrically) cuspidally central arithmetic fundamental group of $U_i$ which sits in the exact sequence $1\to \Delta _{U_i}^{\operatorname {cn}}\to \pi _1(U_i,\eta )^{(\text c-\operatorname {cn})}\to \pi _1(X_i,\eta )\to 1$ .

Consider the following commutative diagrams:

and for $i\in I$

where the right squares are Cartesian. Thus, ${\mathcal {E}}_U$ (resp. $E_{U_i}$ ) is the pullback of the group extension $\pi _1(U,\eta )^{(\text c-\operatorname {ab})}$ (resp. $\pi _1(U_i,\eta )^{(\text c-\operatorname {cn})}$ ) by the section s (resp. $s_i$ ). There is a natural isomorphism $\Delta _U^{\operatorname {ab}}=\varprojlim _{i\in I}\Delta _{U_i}^{\operatorname {cn}}$ as follows from the facts that $\Delta _U=\Delta _{U_i}$ , $\forall i\in I$ , and given a finite quotient $\Delta _U^{\operatorname {ab}}\twoheadrightarrow H$ , there exists $i\in I$ such that $\pi _1(X_i,\eta )^{\operatorname {geo}}$ acts trivially on H. Further, there is a natural isomorphism ${\mathcal {E}}_U\overset \sim \to \varprojlim _{i\in I} E_{U_i}$ (the transition maps in the projective limit being surjective). The existence of a section $s_U^{\text c-\operatorname {ab}}:G_k\to \pi _1(U,\eta )^{(\text c-\operatorname {ab})}$ of the projection $\pi _1(U,\eta )^{(\text c-\operatorname {ab})}\twoheadrightarrow G_k$ which lifts the section s is equivalent to the splitting of the group extension ${\mathcal {E}}_U$ , and the set of those (possible) liftings $s_U^{\text c-\operatorname {ab}}$ is in one-to-one correspondence with the set of sections of the projection ${\mathcal {E}}_U\twoheadrightarrow G_k$ . The natural projection $E_{U_i}\twoheadrightarrow G_k$ splits for all $i\in I$ (see the proof of Proposition 2.2). We show that the group extension ${\mathcal {E}}_U$ splits.

Let $(P_j)_{j\in J}$ be a projective system of quotients ${\mathcal {E}}_U \twoheadrightarrow P_j$ , where $P_j$ sits in an exact sequence $1\to F_j\to P_j\to G_k\to 1$ with $F_j$ finite, and ${\mathcal {E}}_U=\varprojlim _{j\in J}P_j$ . (More precisely, write ${\mathcal {E}}_U$ as a projective limit of finite groups $\{\tilde P_j\}_{j\in J}$ where $\tilde P_j$ sits in an exact sequence $1\to F_j\to \tilde P_j\to G_j\to 1$ with $G_j$ a quotient of $G_k$ and $F_j$ a quotient of $\operatorname {Ker}({\mathcal {E}}_U\twoheadrightarrow G_k)$ . Let $1\to F_j\to P_j\to G_k\to 1$ be the pullback of the group extension $1\to F_j\to \tilde P_j\to G_j\to 1$ by $G_k\twoheadrightarrow G_j$ . Then ${\mathcal {E}}_U=\varprojlim _{j\in J}P_j$ .) The set $\operatorname {Sect}(G_k,{\mathcal {E}}_U)$ of group-theoretic sections of the projection ${\mathcal {E}}_U\twoheadrightarrow G_k$ is naturally identified with the projective limit $\varprojlim _{j\in J}\operatorname {Sect}(G_k,P_j)$ of the sets $\operatorname {Sect}(G_k,P_j)$ of group-theoretic sections of the projections $P_j\twoheadrightarrow G_k$ , $j\in J$ . The set $\operatorname {Sect}(G_k,P_j)$ is non-empty, $\forall j\in J$ . Indeed, $P_j$ (being a quotient of ${\mathcal {E}}_U$ ) is a quotient of $E_{U_i}$ for some $i\in I$ , this quotient $E_{U_i}\twoheadrightarrow P_j$ commutes with the projections onto $G_k$ , and we know the projection $E_{U_i}\twoheadrightarrow G_k$ splits, and hence the projection $P_j\twoheadrightarrow G_k$ splits. Moreover, the set $\operatorname {Sect} (G_k,P_j)$ is, up to conjugation by the elements of $F_j$ , a torsor under the group $H^1(G_k,F_j)$ which is finite since k is a p-adic local field (cf. [Reference Neukirch, Schmidt and Wingberg18, (7.1.8) Theorem (iii)]). Thus, $\operatorname {Sect} (G_k,P_j)$ is a non-empty finite set. The set $\operatorname {Sect} (G_k,{\mathcal {E}}_U)$ is non-empty being the projective limit of non-empty finite sets. This finishes the proof of Theorem 2.3.

Proof. The proof follows, using the natural identification $G_X^{\text c-\operatorname {ab}}\overset \sim \to \varprojlim _{U} \pi _1(U,\eta )^{\text c-\operatorname {ab}}$ (where U runs over all subschemes of X as in Theorem 2.3), from Theorem 2.3 and a similar argument in our context to the one used in the proof of Theorem 2.3.5 in [Reference Saïdi19]. Alternatively, one can use Theorem 2.3 and a similar argument to the one used at the end of the proof of Theorem 2.3.

Proof. Assume that the projection $\pi _1(X,\eta )\twoheadrightarrow G_k$ splits and let $s:G_k\to \pi _1(X,\eta )$ be a section of this projection. By Theorem 2.4, there exists a section $s^{\text c-\operatorname {ab}}:G_k\to G_X^{(\text c-\operatorname {ab})}$ of the projection $G_X^{(\text c-\operatorname {ab})}\twoheadrightarrow G_k$ which lifts the section s. The section $s^{\text c-\operatorname {ab}}$ induces naturally a section $\tilde s:G_k\to G_X^{(\operatorname {ab})}$ of the projection $G_X^{(\operatorname {ab})}\twoheadrightarrow G_k$ (see §1 for the definition of $G_X^{(\operatorname {ab})}$ and note that $G_X^{(\operatorname {ab})}$ is a quotient of $G_X^{(\text c-\operatorname {ab})}$ ). Let $G_Y\overset {\text {def}} = \operatorname {Gal} (\overline K/K)$ be the absolute Galois group of the function field K of Y, and let $G_Y^{(\operatorname {ab})}\overset {\text {def}} = G_Y/\operatorname {Ker} [ \operatorname {Gal} (\overline K/K\bar k)\twoheadrightarrow \operatorname {Gal} (\overline K/K\bar k)^{\operatorname {ab}} ]$ be its geometrically abelian quotient. We have a commutative diagram

where the left vertical map is induced by the natural map $G_X\to G_Y$ , which is induced by the scheme morphism $X\to Y$ in case X is a formal p-adic germ, and by the rigid analytic morphism $X\to Y^{\operatorname {rig}}$ and the rigid GAGA functor in case X is affinoid. The section $\tilde s:G_k\to G_X^{(\operatorname {ab})}$ induces a section $s^{\dagger }:G_k\to G_Y^{(\operatorname {ab})}$ of the projection $G_Y^{(\operatorname {ab})}\twoheadrightarrow G_k$ (cf. above diagram). The existence of the section $s^{\dagger }$ implies that $\operatorname {index} (Y)=1$ as was observed by Esnault and Wittenberg (see [Reference Esnault and Wittenberg5, Remark 2.3(ii)] and [Reference Saïdi24, Theorem A] for a more general result).

Proof. Assertion (i) follows from Theorem 2.4 and the fact that there exists a natural homomorphism $G_X^{(\text c-\operatorname {ab})}\to G_Y^{(\text c-\operatorname {ab})}$ , induced by the natural homomorphism $G_X\to G_Y$ , which commutes with the projections to $G_k$ . Assertion (ii) follows from assertion (i) and Theorem 2.3.5 in [Reference Saïdi19].

Proof. Assertion (i) follows from Theorem 2.4 and the fact that there exists a natural projection $G_X^{(\text c-\operatorname {ab})}\twoheadrightarrow G_X^{(1/p^2-\operatorname {sol})}$ which commutes with the projections onto $G_k$ . Assertion (ii) follows from (i) and the fact that there exists a natural homomorphism $G_X^{(1/p^2-\operatorname {sol})}\to G_Y^{(1/p^2-\operatorname {sol})}$ , induced by the homomorphism $G_X\to G_Y$ , which commutes with the projections onto $G_k$ (cf. diagram (3.1) and [Reference Saïdi22, Theorem 3.4.4]).

Proof of Lemma 4.2

Follows from the above discussion and the fact that we have an exact sequence

$$ \begin{align*}M \to \varprojlim _{n\ge 1}\operatorname{Pic} ({\mathcal{S}}_n) \to \varprojlim _{n\ge 1} \text {coker}(\phi_n)\to 0,\end{align*} $$

where the first map is induced by the maps $\phi _n:M\to \operatorname {Pic} ({\mathcal {S}}_n)$ , $n\ge 1$ , and $\varprojlim _{n\ge 1} \text {coker}(\phi _n)$ is finite if assertions (A) and (B) are satisfied.

This finishes the proof of Lemma 4.2.

Proof of assertion (A)

Let $\{\eta _r\}_{r=1}^s$ be the generic points of ${\mathcal {X}}_0$ , let $\rho :{\mathcal {S}}_0^{\text {nor}}\to {\mathcal {S}}_0$ be the morphism of normalization, let $\widetilde E_i\overset {\text {def}} = \rho ^{-1}(E_i)$ , $1\le i\le t$ , and let $H_r=\overline {\{\eta _r\}}$ be the closure in ${\mathcal {S}}_0^{\text {nor}}$ of the (inverse image in ${\mathcal {S}}_0$ of the) generic point $\eta _r$ of ${\mathcal {X}}_0$ , $1\le r\le s$ . Thus, $H_r$ is a connected affine normal one-dimensional scheme over F. Let

be the composite map where the first map is the pullback of line bundles via the normalization morphism $\rho :{\mathcal {S}}_0^{\text {nor}}\to {\mathcal {S}}_0$ , and the map $\text {deg}$ is obtained by taking the degree of a line bundle on each irreducible component $D_{i,j}$ of $E_i$ .

Proof of Claim 1

We have a commutative diagram of exact sequences

where $A_1$ and $A_2$ are defined so that the above sequences are exact, and $A_2$ is finite as follows from the facts that the sheaf $\rho _{*}({\mathcal {O}}_{S_0^{\text {nor}}} ^{\times })/{\mathcal {O}}_{S_0}^{\times }$ is a skyscraper sheaf and the residue fields at closed points of ${\mathcal {S}}_0$ are finite fields. The kernel $\text {ker}(\text {deg})=\oplus _{r=1}^s \operatorname {Pic} (H_r) \oplus (\oplus _{i=1}^t \operatorname {Pic}^0 (\widetilde E_i))$ of the right lower vertical map is finite: $\operatorname {Pic}^0 (\widetilde E_i)$ is finite since $\widetilde E_i$ is a proper and non-singular curve over a finite field, and for $1\le r\le s$ it holds $\operatorname {Pic}(H_r)$ is finite since $H_r$ is an affine and normal one-dimensional scheme of finite type over the finite field F. Indeed, assume for simplicity that $H_r$ is geometrically connected over F. Let $\ell /F$ be a finite extension such that $U_r\overset {\text {def}} = H_r\times _{\operatorname {Spec} F}\operatorname {Spec} \ell $ admits a smooth and connected compactification $C_r$ with $(C_r\setminus U_r) (\ell )\neq \emptyset $ . Let $U_r\to H_r$ be the canonical morphism, and let $\operatorname {Pic}(H_r)\to \operatorname {Pic} (U_r)$ be the induced map of pullback of line bundles. Then $\operatorname {Ker} [ \operatorname {Pic}(H_r)\to \operatorname {Pic} (U_r)]$ is finite (cf. [Reference Guralnick, Jaffe and Raskind10, Theorem 1.8]). Further, the map $\operatorname {Pic}^0(C_r)\to \operatorname {Pic} (U_r)$ obtained by restricting a degree $0$ line bundle on $C_r$ to $U_r$ is surjective (if $x\in (C_r \setminus U_r)(\ell )$ and $D\in \operatorname {Pic} (U_r)$ has degree m then $D-mx\in \operatorname {Pic}^0(C_r)$ restricts to D on $U_r$ ); hence, $\operatorname {Pic} (U_r)$ is finite since $\operatorname {Pic} ^0(C_r)$ is finite. From the above, it follows that $\operatorname {Pic}(H_r)$ is finite.

This finishes the proof of Claim 1.

Consider the composite map

Proof of Claim 2

First, we prove that the kernel of the map $\operatorname {Pic} ({\mathcal {S}}_n)\to \operatorname {Pic} ({\mathcal {S}}_{n-1})$ is finite for $n\ge 2$ . Write ${\mathcal {I}}_n$ for the sheaf of ideals of ${\mathcal {O}}_{{\mathcal {S}}}$ defining ${\mathcal {S}}_n$ . We have an exact sequence of sheaves on ${\mathcal {S}}_n$ :

$$ \begin{align*}1\to 1+({\mathcal{I}}_{n-1}/{\mathcal{I}}_n)\to {\mathcal{O}}_{S_n}^{\times}\to {\mathcal{O}}_{S_{n-1}}^{\times}\to 1\end{align*} $$

which induces an exact sequence in cohomology

$$ \begin{align*}H^1({\mathcal{S}}_n,1+({\mathcal{I}}_{n-1}/{\mathcal{I}}_n))\to \operatorname{Pic}({\mathcal{S}}_n)\to \operatorname{Pic}({\mathcal{S}}_{n-1})\to H^2({\mathcal{S}}_n,1+({\mathcal{I}}_{n-1}/{\mathcal{I}}_n)).\end{align*} $$

Further, the truncated exponential map $\alpha \mapsto 1+\alpha $ induces an isomorphism of sheaves ${\mathcal {I}}_{n-1}/{\mathcal {I}}_n\overset \sim \to 1+({\mathcal {I}}_{n-1}/{\mathcal {I}}_n)$ [ $({\mathcal {I}}_{n-1}/{\mathcal {I}}_n)^2=0$ ]; hence, $H^2({\mathcal {S}}_n,1+{\mathcal {I}}_{n-1}/{\mathcal {I}}_n)=0$ and the map $\operatorname {Pic}({\mathcal {S}}_n)\to \operatorname {Pic}({\mathcal {S}}_{n-1})$ is surjective. Moreover, $H^1({\mathcal {S}}_n,{\mathcal {I}}_{n-1}/{\mathcal {I}}_n)$ is finite. Indeed, $H^1({\mathcal {S}}_n,{\mathcal {I}}_{n-1}/{\mathcal {I}}_n)$ is a finitely generated $B_n$ -module with finite support since the morphism $\lambda _n^{-1}({\mathcal {Z}}_n\setminus \{z_1,\ldots ,z_t\})\to {\mathcal {Z}}_n\setminus \{z_1,\ldots ,z_t\}$ is affine and $R^1(\pi _n)_{*}({\mathcal {I}}_{n-1}/{\mathcal {I}}_n)$ is the sheaf associated with the $B_n$ -module $H^1({\mathcal {S}}_n,{\mathcal {I}}_{n-1}/{\mathcal {I}}_n)$ ; here, $\lambda _n:{\mathcal {S}}_n\to {\mathcal {Z}}_n$ is the proper morphism induced by $\lambda $ . This shows that the kernel of the map $\operatorname {Pic} ({\mathcal {S}}_n)\to \operatorname {Pic} ({\mathcal {S}}_{n-1})$ is finite for all $n\ge 2$ . A similar argument shows that the kernel of the map $\operatorname {Pic} ({\mathcal {S}}_1)\to \operatorname {Pic} ({\mathcal {S}}_{0})$ is finite. Hence, using Claim 1, $\text {ker}(\psi _n)$ is finite.

This finishes the proof of Claim 2.

In light of Claim 2, and in order to prove assertion (A), it suffices to prove that the cokernel of the composite map

is finite. The latter follows from the nondegeneracy of the intersection pairing $(\oplus _{j=1}^{n_i}\mathbb {Z})\times (\oplus _{j=1}^{n_i}\mathbb {Z})\to \mathbb {Z}$ on each fiber $E_i$ (cf. [Reference Shafarevich26, Lemma on page 69 and the discussion on page 71 after this lemma]), $1\le i\le t$ .

This finishes the proof of assertion (A).

Proof of assertion (B)

Let ${\mathcal {J}}$ be an ample invertible ${\mathcal {O}}_S$ -ideal such that $\text {Supp} ({\mathcal {O}}_S/{\mathcal {J}})={\mathcal {S}}_0$ . The existence of such ${\mathcal {J}}$ follows from the facts that $H_r$ is affine (cf. Proof of Assertion A), $1\le r\le s$ , the intersection pairing $(\oplus _{j=1}^{n_i}\mathbb {Z})\times (\oplus _{j=1}^{n_i}\mathbb {Z})\to \mathbb {Z}$ on each fiber $E_i$ is negative definite (cf. [Reference Shafarevich26, Lemma on page 69 and the discussion on page 71 after this lemma]), and the numerical criterion of ampleness on curves. More precisely, $\forall \ 1\le i\le t$ , one can find a divisor $D=\sum _{ j=1}^{n_i}m_{ij} D_{i,j}$ with $m_{i,j}<0$ and $D.D_{i,j}> 0$ for all $1\le j \le n_j$ .

For $m\ge 1$ , let ${\mathcal {S}}^{\prime }_m$ be the closed subscheme of ${\mathcal {S}}$ defined by the sheaf of ideals ${\mathcal {J}}^m$ . To prove Assertion B, it suffices to prove that there exists $m_0>0$ such that the map

$$ \begin{align*}\operatorname{Pic} ({\mathcal{S}}^{\prime}_{m+1})\to \operatorname{Pic} ({\mathcal{S}}^{\prime}_m)\end{align*} $$

is an isomorphism for any $m>m_0$ . We have an exact sequence of shaves on ${\mathcal {S}}^{\prime }_{m+1}$ :

$$ \begin{align*}1\to {\mathcal{J}}^{m}/{\mathcal{J}}^{m+1}\to {\mathcal{O}}_{S^{\prime}_{m+1}}^{\times}\to {\mathcal{O}}_{S^{\prime}_{m}}^{\times}\to 1,\end{align*} $$

where the map ${\mathcal {J}}^{m}/{\mathcal {J}}^{m+1}\to {\mathcal {O}}_{S^{\prime }_{m+1}}^{\times }$ maps a local section $\alpha $ to $1+\alpha $ , which induces an exact sequence in cohomology

$$ \begin{align*}H^1({\mathcal{S}}^{\prime}_{m+1},{\mathcal{J}}^{m}/{\mathcal{J}}^{m+1})\to \operatorname{Pic}({\mathcal{S}}^{\prime}_{m+1})\to \operatorname{Pic}({\mathcal{S}}^{\prime}_{m})\to 0.\end{align*} $$

Now, there exists $m_0>0$ such that $H^1({\mathcal {S}}^{\prime }_{m+1},{\mathcal {J}}^{m}/{\mathcal {J}}^{m+1})=0$ if $m\ge m_0$ by [Reference Grothendieck and Dieudonné9, première partie, Proposition 2.2.1].

This finishes the proof of assertion (B).

This finishes the proof of Proposition 4.1.

Proof. By the main result in [Reference Epp4, Introduction], there exists a finite extension $L/K$ with uniformizing parameter $\pi _L$ such that $A_{{\mathcal {O}}_L}^{\operatorname {nor}}/\pi _LA_{{\mathcal {O}}_L}^{\operatorname {nor}}$ is reduced. Note that $A_{{\mathcal {O}}_L}^{\operatorname {nor}}$ is a normal two-dimensional complete local ring with perfect residue field (cf. [Reference Bourbaki2, Chap. IX, §4, Lemma 1] and our assumption that X is geometrically connected). Without loss of generality, we will assume that $A/\pi A$ is reduced. We show that there exist a proper, flat, connected, and normal relative R-curve ${\mathcal {Y}}\to \operatorname {Spec} R$ , a closed point $y\in {\mathcal {Y}}$ , and an isomorphism $\hat {\mathcal {O}}_{{\mathcal {Y}},y}\overset \sim \to A$ .

First, $A/\pi A$ is a (reduced) one-dimensional complete local ring with residue field $\ell $ , hence is isomorphic to a quotient $\ell [[x_1,\ldots ,x_t]]/{\mathfrak {a}}$ of a formal power series ring $\ell [[x_1,\ldots ,x_t]]$ over $\ell $ (cf. [Reference Bourbaki2, chapitre IX, §3]). It then follows from [Reference Artin1, Theorem 3.8] and basic facts on the theory of algebraic curves, that there exist a proper and reduced connected (but not necessarily irreducible) $\ell $ -curve Z, a closed point $y\in Z$ , and an isomorphism $\hat {\mathcal {O}}_{Z,y}\overset \sim \to A/\pi A$ where $\hat {\mathcal {O}}_{Z,y}$ is the completion of the local ring ${\mathcal {O}}_{Z,y}$ of Z at y. Moreover, Z is non-singular outside y. There exists a rational function f on Z which defines a finite generically separable morphism $f:Z\to \Bbb P^1_{\ell }$ such that $y=f^{-1}(\infty )$ (cf. [Reference Harbater and Stevenson12, Proof of Theorem 3]). Thus, by considering the completion of the morphism f above $\infty $ , we obtain a finite generically separable morphism $\bar g:\operatorname {Spec} (A/\pi A)\to \operatorname {Spec} (\ell [[t]])$ where t is a local parameter at $\infty $ . This morphism lifts to a finite morphism $g:\operatorname {Spf} A\to \operatorname {Spf} (R[[T]])$ of formal schemes (cf. [Reference Harbater and Stevenson12, Lemma 2]). Let $\widetilde Z\to Z$ be the morphism of normalization, and let $\{x_1,\ldots ,x_m\}\subset \widetilde Z$ be the pre-image of y. There is a one-to-one correspondence between the set $\{\mathfrak {p}_1,\ldots ,\mathfrak {p}_m\}\subset \operatorname {Spec} A$ of prime ideals of height $1$ containing $\pi $ and the set $\{x_1,\ldots ,x_m\}$ , $\mathfrak {p}_i$ corresponds to $x_i$ , $1\le i\le m$ . The composite morphism $\widetilde Z\to Z\to \Bbb P^1_{\ell }$ induces, by completion above $\infty $ , finite separable morphisms $\bar g_i:\operatorname {Spec} \operatorname {Fr} (\hat {\mathcal {O}}_{\widetilde Z,x_i})\to \operatorname {Spec} \ell ((t))$ where $\operatorname {Fr} (\hat {\mathcal {O}}_{\widetilde Z,x_i})$ is the fraction field of the completion $\hat {\mathcal {O}}_{\widetilde Z,x_i}$ of the local ring ${\mathcal {O}}_{\widetilde Z,x_i}$ of $\widetilde Z$ at $x_i$ , $1\le i\le m$ (with the above notations $t=T \quad\mod \pi $ ).

Consider the formal closed unit disk $D=\operatorname {Spf} R<\frac {1}{T}>$ with parameter $\frac {1}{T}$ and its special fiber $D_{\ell }=\operatorname {Spec} \ell [\frac {1}{t}]$ ( $D_{\ell } \overset \sim \to \Bbb A^1_{\ell }$ ). By a result of Gabber and Katz (cf. [Reference Katz14, Main Theorem 1.4.1]), there exists, for $1\le i\le m$ , a finite cover $\bar h_i:C_i\to D_{\ell }$ with $C_i$ connected, which only (tamely) ramifies above the point $\frac {1}{t}=0$ and such that the completion of $\bar h_i$ above $t=0$ is generically isomorphic to the cover $\bar g_i:\operatorname {Spec} \operatorname {Fr} (\hat {\mathcal {O}}_{\widetilde Z,x_i})\to \operatorname {Spec} \ell ((t))$ . Using formal patching techniques (cf. [Reference Saïdi23, 1.2]), one can lift the covers $\bar h_i$ to finite covers $h_i:Y_i\to D$ which only ramify above the point $\frac {1}{T}=0$ , $1\le i\le m$ . (Outside $\frac {1}{T}=0$ , the existence of such a lifting follows from the theorems of lifting of étale covers [cf. [Reference Grothendieck8, Exposé I, Corollaire 8.4]]. In a formal neighborhood of $\frac {1}{T}=0$ , such a lifting is possible under the tameness condition: étale locally near $\frac {1}{t}$ the cover $\bar h_i$ is defined by an equation $y^s=\frac {1}{t^e}$ , where $s\ge 1$ is an integer prime to the characteristic of $\ell $ , and one lifts to the cover defined by $Y^s=\frac {1}{T^e}$ .) For $1\le i\le m$ , let $\hat A_{\mathfrak {p}_i}$ be the completion of the localization $A_{\mathfrak {p}_i}$ of A at ${\mathfrak {p}}_i$ . Thus, $\hat A_{\mathfrak {p}_i}$ is a complete discrete valuation ring with uniformizing parameter $\pi $ (recall $A/\pi A$ is reduced) and residue field $\operatorname {Fr} (\hat {\mathcal {O}}_{\widetilde Z,x_i})$ . Let B be the completion of the localization of $R[[T]]$ at $\pi $ . Thus, B is a complete discrete valuation ring with residue field $\ell ((t))$ . The finite cover $g:\operatorname {Spf} A\to \operatorname {Spf} (R[[T]])$ induces, by pullback to $\operatorname {Spf} B$ , finite covers $g_i:\operatorname {Spf} \hat A_{\mathfrak {p}_i}\to \operatorname {Spf} B$ which (by construction) lift the covers $\bar g_i:\operatorname {Spec} \operatorname {Fr} (\hat {\mathcal {O}}_{\widetilde Z,x_i})\to \operatorname {Spec} \ell ((t))$ , $1\le i\le m$ . Further, the cover $h_i:Y_i\to D$ induces, by pullback to $\operatorname {Spf} B$ , a finite cover $\tilde h_i:\operatorname {Spf} B_i\to \operatorname {Spf} B$ which by construction lifts the cover $\bar g_i:\operatorname {Spec} \operatorname {Fr} (\hat {\mathcal {O}}_{\widetilde Z,x_i})\to \operatorname {Spec} \ell ((t))$ . Thus, the covers $\tilde h_i:\operatorname {Spf} B_i\to \operatorname {Spf} B$ and $g_i:\operatorname {Spf} \hat A_{\mathfrak {p}_i}\to \operatorname {Spf} B$ are isomorphic since $\bar g_i$ is generically separable. Using formal patching techniques (cf. [Reference Grothendieck8, Exposé I, Corollaire 8.4]), one can patch the covers $g:\operatorname {Spf} A\to \operatorname {Spf} (R[[T]])$ and $h_i:Y_i\to D$ , $1\le i\le m$ , to construct a finite cover ${\mathcal {Y}}\to \Bbb P^1_{R}$ in the category of formal schemes with ${\mathcal {Y}}$ normal, connected, proper, and flat over $\operatorname {Spf} R$ . The special fiber ${\mathcal {Y}}_{\ell }\overset {\text {def}} = {\mathcal {Y}}\times _{\operatorname {Spec} R}\operatorname {Spec} \ell $ of ${\mathcal {Y}}$ consists of m irreducible components which intersect at the point y and is (by construction) non-singular outside y. The formal curve ${\mathcal {Y}}$ is algebraic by formal GAGA and (by construction) $\hat {\mathcal {O}}_{{\mathcal {Y}},y}\overset \sim \to A$ as required.

Proof. This follows easily from Proposition 5.1, Remark 5.2, and Theorem 3 in [Reference Saïdi23].