Specialization for the pro-étale fundamental group

2.1 Formal schemes

We refer to [Reference Fujiwara and KatoFK18, Chapter I] for the basic terminology regarding formal schemes. We only consider formal schemes which are locally of the form $\operatorname {Spf}(A)$ where $A$ has the $I$-adic topology for a finitely generated ideal $I$, and is $I$-adically complete and separated. By an ideal of definition of $A$ we shall always mean a finitely generated ideal of definition. Recall that a continuous homomorphism of such algebras $A\to B$ is adic if $I\cdot B$ is an ideal of definition of $B$ for some (equivalently, every) ideal of definition $I$ of $A$, and a map of formal schemes $\mathfrak {Y}\to \mathfrak {X}$ is adic if it is locally of the form $\operatorname {Spf}(B)\to \operatorname {Spf}(A)$ for an adic homomorphism $A\to B$.

We first discuss a hypothesis on formal schemes over $\mathcal {O}_K$ that is more flexible than locally topologically of finite type (in the discrete case such objects were considered in [Reference BerkovichBer96a, § 1] under the terminology ‘special’).

Proof. Suppose that $A/I$ is finitely generated over $\mathcal {O}_K/\varpi$ and choose a surjection

\[ \varphi\colon (\mathcal{O}_K/\varpi)[x_1, \ldots, x_n]\to A/I. \]

Write $I = (f_1, \ldots, f_m, \varpi )$, and let $\psi \colon P \to A$ be the unique continuous $\mathcal {O}_K$-algebra homomorphism sending $x_i$ to prescribed lifts of $\varphi (x_i)$ and with $\psi (y_i)=f_i$. Such a homomorphism exists because each $f_i$ is topologically nilpotent. As $P$ is by definition the $(y_1, \ldots, y_m, \varpi )$-adic completion of $\mathcal {O}_K[x_1, \ldots, x_n, y_1, \ldots, y_m]$, the map $\psi$ is adic. Therefore, by [Reference Fujiwara and KatoFK18, Chapter I, Proposition 4.3.6] the map $\psi$ is surjective, and hence condition (b) is satisfied.

For the other direction, note that there exists a $k\geqslant 1$ such that $(\psi (y_1^{k}), \ldots, \psi (y_m^{k}), \varpi ) \subseteq I$. Then $P/(\psi (y_1^{k}), \ldots, \psi (y_m^{k}), \varpi )\simeq (\mathcal {O}_K/\varpi )[x_1,\ldots,x_n,y_1,\ldots,y_m]/(y_1^{k},\ldots,y_m^{k})$ is a finitely generated $\mathcal {O}_K/\varpi$-algebra, and hence so is its quotient $A/I$.

We denote the category of formal schemes locally formally of finite type (respectively, locally of finite type, respectively, of finite type) over $\mathcal {O}_K$ by $\operatorname {\mathbf {F Sch}}_{\mathcal {O}_K}^{\rm lfft}$ (respectively, $\operatorname {\mathbf {FSch}}_{\mathcal {O}_K}^{\rm lft}$, respectively, $\operatorname {\mathbf {FSch}}_{\mathcal {O}_K}^{\rm ft}$). The notions of (locally) formally of finite type, (locally) of finite type, and admissible are preserved under base change along $\mathcal {O}_K\to \mathcal {O}_{K'}$ for $K'$ a non-archimedean extension of $K$.

2.1 Completions

Let $\mathfrak {X}$ be a formal scheme and let $\mathcal {J}\subseteq \mathcal {O}_\mathfrak {X}$ be an admissible ideal sheaf. Consider the colimitFootnote ² $\mathfrak {X}(\mathcal {J}^{\infty })=\varinjlim \mathfrak {X}(\mathcal {J}^{n})$ of locally ringed spaces. Let us note that the underlying topological space of $\mathfrak {X}(\mathcal {J}^{n})$ is independent of $n\geqslant 1$. Let us call it $X$. We then consider the sheaves $\mathcal {O}_{\mathfrak {X}(\mathcal {J}^{n})}$ on $X$ for all $n$. We have obvious surjections $\mathcal {O}_{\mathfrak {X}(\mathcal {J}^{i})}\to \mathcal {O}_{\mathfrak {X}(\mathcal {J}^{j})}$ for $i\geqslant j\geqslant 1$. Using [Reference Fujiwara and KatoFK18, Chapter I, Proposition 1.4.2] one shows that $\mathfrak {X}(\mathcal {J}^{\infty })$ is a formal scheme. If $\mathfrak {X}=\operatorname {Spf}(A)$ and $J\subseteq A$ is the ideal corresponding to $\mathcal {J}$, then $\mathfrak {X}(\mathcal {J}^{\infty })$ is in fact equal to $\operatorname {Spf}(B)$ with $B$ the $J$-adic completion of $A$.

The proof of the following representability result is left to the reader.

Proposition 2.3 Let $\mathfrak {X}$ be a formal scheme such that $\underline {\mathfrak {X}}$ is locally Noetherian. Then, for any closed subset $Z$ of $|\mathfrak {X}|$ the functor

\[ \widehat{\mathfrak{X}}_{Z}\colon \operatorname{\mathbf{FSch}}^{\rm op}\to\operatorname{\mathbf{Set}},\qquad \mathfrak{X}'\mapsto \{f\in\operatorname{Hom}_{\operatorname{\mathbf{FSch}}}(\mathfrak{X}',\mathfrak{X}): f(|\mathfrak{X}'|)\subseteq Z\} \]

is representable. Moreover, if $|\mathfrak {X}(\mathcal {J})|=Z$ for an admissible ideal sheaf $\mathcal {J}\subseteq \mathcal {O}_\mathfrak {X}$, then it is represented by $\mathfrak {X}(\mathcal {J}^{\infty })$.

We call the formal scheme $\widehat {\mathfrak {X}}_Z$ the completion of $\mathfrak {X}$ along $Z$. It is clear that if $\mathfrak {U}$ is an open formal subscheme of $\mathfrak {X}$, then $\widehat {\mathfrak {U}}_{U\cap Z}$ is the preimage of $\mathfrak {U}$ in $\widehat {\mathfrak {X}}_Z$. By reducing to the affine case using [Reference Fujiwara and KatoFK18, Chapter I, Proposition 3.7.11] one can show that if $\mathfrak {X}$ is moreover quasi-compact and quasi-separated, then every closed subset $Z$ is of the form $|\mathfrak {X}(\mathcal {J})|$ for some admissible ideal sheaf $\mathcal {J}\subseteq \mathcal {O}_{\mathfrak {X}}$.

The following result will play an important role in the proof of Proposition 3.5. In its statement, we use the notion of adically locally of finite presentation as in [Reference Fujiwara and KatoFK18, Chapter I, Definition 5.3.1]. Note that by [Reference Fujiwara and KatoFK18, Chapter I, Corollary 2.2.4] and [Reference Fujiwara and KatoFK18, Chapter 0, Corollary 9.2.9] this condition is automatic for a morphism of admissible formal schemes over $\mathcal {O}_K$.

Proposition 2.4 Let $\mathfrak {Y}\to \mathfrak {X}$ be a morphism adically locally of finite presentation of formal schemes and assume that the underlying reduced scheme $\underline {\mathfrak {X}}$ of $\mathfrak {X}$ is locally Noetherian. Let $Z$ be a closed subset of $\mathfrak {X}$ and $\mathcal {Z}$ any closed subscheme of $\mathfrak {X}$ with $|\mathcal {Z}|=Z$. Then if $\mathfrak {Y}_\mathcal {Z}\to \mathcal {Z}$ is finite (respectively, a disjoint union of finite morphisms), then the same is true for $\mathfrak {Y}_{\widehat {\mathfrak {X}}_Z}\to \widehat {\mathfrak {X}}_Z$.

Proof. Suppose first that the map is finite. Without loss of generality, we may assume that $\mathcal {Z}=Z$ with the reduced scheme structure. By working locally on the target, we may assume that $\mathfrak {X}=\operatorname {Spf}(A)$ and that $Z_\mathrm {red}=\operatorname {Spec}(A/J)_\mathrm {red}$ for some admissible ideal $J$ of $A$. Set $T=\operatorname {Spec}(A/J)$. Note then that $\mathfrak {Y}_T\to T$ is, by assumption, a morphism of schemes locally of finite presentation. By assumption, we have that the pullback of this map along $T_\mathrm {red}\to T$ is finite, and thus it is itself finite by [SP21, Tag 0BPG]. By [Reference Fujiwara and KatoFK18, Chapter I, Proposition 4.2.3] to verify $\mathfrak {Y}_{\widehat {\mathfrak {X}}_Z}\to \widehat {\mathfrak {X}}_Z$ is finite, it suffices to check this after base change along $T\to \widehat {\mathfrak {X}}_Z$ from where the claim follows. The second claim is clear by the previous discussion because $\mathfrak {Y}_{Z_\mathrm {red}}\to \mathfrak {Y}_{\widehat {\mathfrak {X}}_{Z}}$ is a homeomorphism, and so if $\mathfrak {Y}_{Z_\mathrm {red}}$ is a disjoint union of clopen subsets finite over $Z_\mathrm {red}$, then $\mathfrak {Y}_{\widehat {\mathfrak {X}}_Z}$ is a disjoint union of clopen subsets which, by the above, must be finite over $\widehat {\mathfrak {X}}_Z$.

2.2 Generic fibers of formal schemes

In this subsection we discuss the notion of the rigid generic fiber of a formal scheme locally formally of finite type over $\mathcal {O}_K$, which is a rigid $K$-space. We then note several properties of the generic fiber construction.

We first recall Raynaud's equivalence as developed by Fujiwara–Kato in [Reference Fujiwara and KatoFK18], which, in particular, gives the construction for formal schemes locally of finite type over $\mathcal {O}_K$. Let $A$ be a topologically finite type $\mathcal {O}_K$-algebra. The algebra $A_K = A[{1}/{\pi }]$ is then an affinoid $K$-algebra, i.e. an algebra topologically of finite type over $K$. The subring $A^{\circ }_K\subseteq A_K$ of powerbounded elements coincides with the integral closure of (the image of) $A$ in $A_K$ (see [Reference Fujiwara and KatoFK18, Chapter II, Corollary A.4.10]). Combining [Reference Fujiwara and KatoFK18, Chapter II] and [Reference Fujiwara and KatoFK18, Chapter II, Theorem A.5.1], there exists a unique functor

\[ (-)_\eta \colon \operatorname{\mathbf{FSch}}^{\rm ft}_{\mathcal{O}_K} \to \operatorname{\mathbf{Rig}}_K^{\rm qcqs} \]

such that $\operatorname {Spf}(A)_\eta = \operatorname {Spa}(A_K)$ for every topologically finite type $\mathcal {O}_K$-algebra $A$, and which respects open immersions and open covers. This functor naturally extends to a functor

\[ (-)_\eta \colon \operatorname{\mathbf{FSch}}^{\rm lft}_{\mathcal{O}_K} \to \operatorname{\mathbf{Rig}}_K^{\rm qs}, \]

and for $\mathfrak {X}$ locally of finite type over $\mathcal {O}_K$, the rigid $K$-space $\mathfrak {X}_\eta$ is called the rigid generic fiber of $\mathfrak {X}$. Furthermore, $(-)_\eta$ sends the class $W$ of admissible blowups to isomorphisms and induces equivalences of categories

(2.2.1)\begin{equation} \operatorname{\mathbf{FSch}}^{\rm adm,qc}_{\mathcal{O}_K}[W^{-1}] \underset{\rm incl}{\buildrel \sim \over \longrightarrow} \operatorname{\mathbf{FSch}}^{\rm ft}_{\mathcal{O}_K}[W^{-1}] \underset{(-)_\eta}{\buildrel \sim \over \longrightarrow} \operatorname{\mathbf{Rig}}_K^{\rm qcqs}. \end{equation}

Here $\operatorname {\mathbf {FSch}}^{\rm adm,qc}_{\mathcal {O}_K}$ consists of quasi-compact admissible formal schemes over $\mathcal {O}_K$, and $(-)[W^{-1}]$ denotes the localization with respect to $W$, i.e. the category obtained by formally inverting all admissible blowups. By a formal model of a rigid $K$-space $X$ we shall mean a formal scheme $\mathfrak {X}$ locally of finite type over $\mathcal {O}_K$ together with an isomorphism $\mathfrak {X}_\eta \simeq X$. By [Reference Fujiwara and KatoFK18, Chapter II, Proposition 2.1.10], for any quasi-compact and quasi-separated rigid $K$-space, the category of admissible formal models of $X$ is (equivalent to) a cofiltering poset.

If $X$ is a quasi-compact and quasi-separated rigid $K$-space, then the construction of the rigid generic fiber allows one to identify the locally topologically ringed space $(X,\mathcal {O}_X^{+})$ as $\varprojlim \, (\mathfrak {X},\mathcal {O}_\mathfrak {X})$ where $\mathfrak {X}$ runs over admissible formal models of $X$. In particular, for each admissible formal model $\mathfrak {X}$ of $X$ we have a morphism of locally topologically ringed spaces

\[ \operatorname{sp}_\mathfrak{X} \colon (X,\mathcal{O}_X^{+})\to (\mathfrak{X},\mathcal{O}_\mathfrak{X}) \]

called the specialization map for $\mathfrak {X}$. For $\mathfrak {X}=\operatorname {Spf}(A)$ affine, the underlying map of topological spaces sends a valuation $\nu \colon A_K\to \Gamma \cup \{0\}$ to the (open) prime ideal $\{x\in A: \nu (x)<1\}$ and the map on global sections is the natural map $\mathcal {O}_\mathfrak {X}(\mathfrak {X})=A\to A_K^{\circ } =\mathcal {O}_X(X)^{+}$.

As $|\mathfrak {X}|=|\mathfrak {X}_k|$ we shall often implicitly treat the specialization map $\operatorname {sp}_\mathfrak {X}$ as a map of topological spaces $|X|\to |\mathfrak {X}_k|$. This map is continuous, quasi-compact, closed, and surjective (see [Reference Fujiwara and KatoFK18, Chapter II, Theorem 3.1.2] and [Reference Fujiwara and KatoFK18, Chapter II, Proposition 3.1.5]). If $\varphi \colon \mathfrak {X}'\to \mathfrak {X}$ is a morphism in $\operatorname {\mathbf {FSch}}_{\mathcal {O}_K}^{\mathrm {ft}}$ then the diagram

commutes. Moreover, for any open subset $\mathfrak {U}$ of $\mathfrak {X}$ the induced map $\mathfrak {U}_\eta \to \mathfrak {X}_\eta$ is an open immersion with image $\operatorname {sp}_\mathfrak {X}^{-1}(\mathfrak {U})$ (see [Reference Fujiwara and KatoFK18, Chapter II, Proposition 3.1.3]). It follows that the definition of the specialization map may be extended to formal schemes $\mathfrak {X}$ locally of finite type over $\mathcal {O}_K$, and that this extension enjoys also the property of being continuous, quasi-compact, closed, and surjective if $\mathfrak {X}$ is admissible.

In order to extend the definition of the rigid generic fiber to the category $\operatorname {\mathbf {FSch}}^{\rm lfft}_{\mathcal {O}_K}$, we need the following result.

Proposition 2.5 (See [Reference Fujiwara and KatoFK18, Chapter II, § 9.6.(b)])

Let $\mathfrak {X}$ be a formal scheme locally formally of finite type over $\mathcal {O}_K$. Then, the functor

\[ \big(\operatorname{\mathbf{Rig}}_K^{\mathrm{qcqs}}\big)^{\rm op}\to \operatorname{\mathbf{Set}},\quad Z\mapsto \varinjlim\operatorname{Hom}_{\mathcal{O}_K}(\mathfrak{Z},\mathfrak{X}), \]

where $\mathfrak {Z}$ runs over the admissible formal models of $Z$, is representable by a rigid $K$-space $\mathfrak {X}_\eta$.

We call $\mathfrak {X}_\eta$ the rigid generic fiber of $\mathfrak {X}$. If $\mathfrak {X}$ is locally of finite type over $\mathcal {O}_K$, this agrees with the previous definition of $\mathfrak {X}_\eta$. The association $\mathfrak {X}\mapsto \mathfrak {X}_\eta$ is functorial in $\mathfrak {X}$ and commutes with finite limits and disjoint unions. Moreover, it sends admissible blowups to isomorphisms.

The following construction makes the rigid generic fiber more explicit in the affine case.

Construction 2.6 (The rigid generic fiber of an affine formal scheme)

Let $\mathfrak {X}=\operatorname {Spf}(B)$ for a topologically formally of finite type $\mathcal {O}_K$-algebra $B$, and let $J=(b_1,\ldots,b_m)$ be an ideal of definition of $B$. We denote by $B[ {J}/{\varpi }]$ the affine blowup algebra [SP21, Tag 052Q], i.e. the image of $B[x_1, \ldots, x_m]/(\pi x_i - b_i)$ in $B[ {1}/{\varpi }]$; it is independent of the choice of generators of $J$. Let $B\langle {J}/{\varpi }\rangle$ be the $J$-adic completion of $B[ {J}/\varpi ]$, which is an admissible $\mathcal {O}_K$-algebra on which the $J$-adic topology coincides with the $\varpi$-adic topology. The map $B\to B\langle J/ \varpi \rangle$ is continuous, and the morphism $\operatorname {Spf}(B\langle {J}/{\varpi }\rangle )\to \operatorname {Spf}(B)$ induces a map $\operatorname {Spf}(B\langle J/ \varpi \rangle )_\eta \to \operatorname {Spf}(B)_\eta$ of rigid $K$-spaces. Writing $B(J) = B\langle {J}/{\varpi }\rangle [ {1}/{\varpi }]$, we have $\operatorname {Spf}(B\langle {J}/{\varpi } \rangle )_\eta = \operatorname {Spa}(B(J))$.Footnote ³

For $J'\subseteq J$, we have an inclusion $B[ {J'}/{\varpi }]\subseteq B[ {J}/{\varpi }]$ and consequently a morphism $\operatorname {Spf}(B[ {J}/{\varpi }])\to \operatorname {Spf}(B[ {J'}/{\varpi }])$ over $\operatorname {Spf}(B)$. The induced morphism $\operatorname {Spa}(B(J))\to \operatorname {Spa}(B(J'))$ is an isomorphism onto a rational open domain of $\operatorname {Spa}(B(J'))$. Thus, the inductive system $\{\operatorname {Spa}(B(J))\}$ indexed by all finitely generated ideals of definition of $B$ gives a well-defined adic space $\varinjlim _J \operatorname {Spa}(B(J))$. As this system admits compatible maps to $\mathfrak {X}_\eta$, we obtain a morphism of rigid $K$-spaces

(2.2.2)\begin{equation} \varinjlim_J \operatorname{Spa}(B(J))\to \mathfrak{X}_\eta. \end{equation}

One then has the following concrete description of the generic fiber $\mathfrak {X}_\eta$.

Let us note that for any given ideal of definition $J_0$ of $B$, the set $\{J_0^{n}\}$ is cofinal in the set of all ideals of definition of $B$. In particular, in Lemma 2.7 we may replace $\varinjlim _J \operatorname {Spa}(B(J))$ with $\varinjlim _n \operatorname {Spa}(B(J_0^{n}))$. We often do this without comment in the following.

With this, we can give a more concrete description of the functor of points of the generic fiber in the situation dictated in Construction 2.6. It will enable us to show easily that an étale morphism of formal schemes induces an étale morphism on rigid generic fibers.

Construction 2.8 Let $\mathfrak {X}=\operatorname {Spf}(B)$ and $\mathfrak {X}'=\operatorname {Spf}(B')$ be affine objects of $\operatorname {\mathbf {FSch}}^{\mathrm {lfft}}_{\mathcal {O}_K}$ and let $\mathfrak {X}'\to \mathfrak {X}$ be a morphism over $\mathcal {O}_K$. Let $J$ be an ideal of definition of $B$ and let $J'$ be an ideal of definition of $B'$ such that $JB\subseteq J'$ (which exists because $B\to B'$ is continuous). It is then easy to see that the morphism $\mathfrak {X}'_\eta \to \mathfrak {X}_\eta$ maps $\operatorname {Spa}(B'((J')^{n}))$ into $\operatorname {Spa}(B(J^{n}))$ for all $n$.

Let $(R,R^{+})$ be a Huber $(K,\mathcal {O}_K)$-algebra. Then, any morphism $\operatorname {Spa}(R,R^{+})\to \mathfrak {X}_\eta$ of adic spaces over $(K,\mathcal {O}_K)$ must factorize through $\operatorname {Spa}(B(J^{n}))$ for some $n$ and, thus, defines a map of Huber pairs $(B(J^{n}),B(J^{n})^{\circ })\to (R,R^{+})$ which, in turn, defines a continuous map $B\to R^{+}$ of $\mathcal {O}_K$-algebras independent of the choice of $n$. Moreover, this map must send $\varpi$ to an element of $R^{\times }$. Note that one has a natural map

\[ \operatorname{Hom}_{\mathfrak{X}_\eta}(\operatorname{Spa}(R,R^{+}),\mathfrak{X}'_\eta)\to \operatorname{Hom}_B(B',R^{+}). \]

Indeed, any map $\operatorname {Spa}(R,R^{+})\to \mathfrak {X}'_\eta$ must factorize through $\operatorname {Spa}(B'((J')^{m}))$ for some $m\geqslant n$ and so defines a map of Huber pairs $(B'((J')^{m}),B'((J')^{m})^{\circ })\to (R,R^{+})$. This then defines a continuous map $B'\to R^{+}$ of $B$-algebras which is independent of $m$.

Lemma 2.9 With notation as in Construction 2.8, the map

\[ \operatorname{Hom}_{\mathfrak{X}_\eta}(\operatorname{Spa}(R,R^{+}),\mathfrak{X}'_\eta)\to\operatorname{Hom}_B(B',R^{+}) \]

is a functorial bijection. Moreover,

\[ \operatorname{Hom}_B(B',R^{+})=\varinjlim_{R_0}\operatorname{Hom}_{\mathfrak{X}}(\operatorname{Spf}(R_0),\mathfrak{X}') \]

as $R_0$ runs over the open and bounded $B$-subalgebras of $R^{+}$.Footnote ⁴

With this, we can prove the following.

Proof. We may assume that $\mathfrak {X}=\operatorname {Spa}(B)$ and $\mathfrak {X}'=\operatorname {Spa}(B')$. Suppose first that $\mathfrak {X}'\to \mathfrak {X}$ is étale. We deduce from Lemma 2.9 that for any Huber $(K,\mathcal {O}_K)$-algebra $(R,R^{+})$ and square-zero ideal $N$ of $R$ one has a commutative square

where the horizontal maps are bijections. However, the right vertical arrow is a bijection because $\mathfrak {X}'\to \mathfrak {X}$ is étale and, thus, the left vertical arrow must also be a bijection. As $f_\eta \colon \mathfrak {X}'_\eta \to \mathfrak {X}_\eta$ is locally of finite type (because both are locally of finite type over $\operatorname {Spa}(K)$) we deduce that $f_\eta$ is étale as desired.

Now suppose that $f$ is finite, and let $J$ be an ideal of definition of $B$. As $f$ is adic, the ideal $J'=J B'$ is an ideal of definition of $B'$. As $B \to B'$ is finite, $B'\langle {(J')^{n}}/{\varpi }\rangle = B\langle {J^{n}}/{\varpi }\rangle \otimes _B B'$ and it follows that $B'((J')^{n}) = B(J^{n})\otimes _B B'$. Since $\mathfrak {X}'_\eta \to \mathfrak {X}_\eta$ is the colimit of the maps $\operatorname {Spa}(B'((J')^{n})) \to \operatorname {Spa}(B(J^{n}))$, which are finite by [Reference HuberHub96, Lemma 1.4.5 iv)], it follows that this map is finite, as desired.

We end this subsection by showing that the generic fiber preserves the partial properness of the special fiber.Footnote ⁵ Recall here that a morphism of schemes is called partially proper if it is locally of finite type, quasi-separated, and satisfies the valuative criterion for properness (see [SP21, Tag 03IX]). A morphism of rigid $K$-spaces is called partially proper if it is separated and universally specializing (see [Reference HuberHub96, Definition 1.3.3]).

Before proving the lemma, we recall some facts about taut spaces. A locally spectral topological space $X$ is taut if it is quasi-separated and the closure of every quasi-compact open is quasi-compact [Reference HuberHub96, Definition 5.1.2]. It is easy to check that a locally topologically Noetherian space is taut if and only if its irreducible components are quasi-compact. Moreover, if $X$ is quasi-paracompact and quasi-separated, then $X$ is taut by [Reference HuberHub96, Lemma 5.1.3(ii)]. If $Y\to X$ is a partially proper morphism of rigid $K$-spaces and $X$ is taut, then so is $Y$, by [Reference HuberHub96, Lemma 5.1.4]. Finally, for an admissible formal scheme $\mathfrak {Y}$ over $\mathcal {O}_K$, because the specialization map $\operatorname {sp}_{\mathfrak {Y}}$ is closed, quasi-compact, and surjective, we deduce that $\mathfrak {Y}_\eta$ is taut if and only if $\mathfrak {Y}_k$ is taut.

Lemma 2.12 Let $f\colon Y\to X$ be a separated morphism of schemes locally of finite type over $k$ where $Y$ is taut. Then $f$ is partially proper if and only if the maps

\[ f_n = f\times {\rm id}_{\mathbf {A}^{n}_k} \colon Y\times \mathbf {A}^{n}_k\to X\times \mathbf {A}^{n}_k\]

are specializing for all $n\geqslant 0$.

Proof Proof of Lemma 2.11

As $f_\eta$ is separated, so is $f_k$, by [Reference Bosch and LütkebohmertBL93, Proposition 4.7]. As $\mathfrak {X}_\eta$ is quasi-paracompact and quasi-separated, it is taut, and hence so is $\mathfrak {Y}_\eta$ and therefore also $\mathfrak {Y}_k$. By Lemma 2.12, it suffices to show that $\mathfrak {Y}_k\times \mathbf {A}^{n}_k\to \mathfrak {X}_k\times \mathbf {A}^{n}_k$ is specializing for all $n\geqslant 0$. Replacing $f\colon \mathfrak {Y}\to \mathfrak {X}$ with $\mathfrak {Y}\times \widehat {\mathbf {A}}^{n}_{\mathcal {O}_K}\to \mathfrak {X}\times \widehat {\mathbf {A}}^{n}_{\mathcal {O}_K}$, we are reduced to showing that $f_k$ itself is specializing. We have the following commutative square of topological spaces.

Here, the horizontal arrows are specializing (because they are closed) and surjective. If the left arrow is specializing, then so is the diagonal composition, and then it is straightforward to check that the right arrow must be specializing as well.

2.3 Tube open subsets

Let $\mathfrak {X}$ be a formal scheme of finite type over $\mathcal {O}_K$ and let $Z\subseteq |\mathfrak {X}|$ be a closed subset. We then define the tube open subset of $\mathfrak {X}_\eta$ associated with $Z$, denoted $T(\mathfrak {X}\,|\,Z)$, to be the open subset $\operatorname {sp}_\mathfrak {X}^{-1}(Z)^{\circ }$, i.e. the topological interior of $\operatorname {sp}_\mathfrak {X}^{-1}(Z)$. With the notation $]Z[$, such opens were first considered by Berthelot in the context of rigid cohomology [Reference BerthelotBer96b]. The following topological properties of tube open subsets of $\mathfrak {X}_\eta$ are important for the proof of Theorem 3.7.

We now relate the tube open subset $T(\mathfrak {X}\,|\,Z)$ with the generic fiber $(\widehat {\mathfrak {X}}_{Z})_\eta$ of the formal completion of $\mathfrak {X}$ along $Z$.

3.1 Geometric coverings

We briefly recall the theory of the pro-étale fundamental group [Reference Bhatt and ScholzeBS15, § 7], which is based on the following notion of a covering space.

The following lemma plays a pivotal role in the construction of the specialization map.

By [Reference Bhatt and ScholzeBS15, Lemma 7.4.1], for a connected and locally topologically Noetherian scheme $X$ with a geometric point $\overline {x}$, the pair $(\operatorname {\mathbf {Cov}}_X,F_{\overline {x}})$ where $F_{\overline {x}}\colon \operatorname {\mathbf {Cov}}_X\to \operatorname {\mathbf {Set}}$ is the fiber functor at $\overline {x}$, is a tame infinite Galois category (see [Reference Bhatt and ScholzeBS15, § 7.2]). Following [Reference Bhatt and ScholzeBS15, Definition 7.4.2] we denote the fundamental group of $(\operatorname {\mathbf {Cov}}_X,F_{\overline {x}})$ by $\pi _1^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}(X,\overline {x})$ and call it the pro-étale fundamental group of $X$. By definition, it is a Noohi topological group (see [Reference Bhatt and ScholzeBS15, § 7.1]) for which the induced functor

\[ F_{\overline{x}} \colon \operatorname{\mathbf{Cov}}_X {\buildrel \sim \over \longrightarrow} \pi_1^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}(X, \overline{x})\text{-}\operatorname{\mathbf{Sets}} \]

is an equivalence, where the target is the category of (discrete) sets with a continuous action of $\pi _1^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}(X, \overline {x})$.

3.2 Coverings of rigid-analytic spaces

Next, we recall various notions of ‘covering spaces’ of a rigid space, as defined in [Reference Achinger, Lara and YoucisALY21b]. Here, we only use de Jong covering spaces, whereas the more general $\mathrm {adm}$-covering spaces and ${\unicode{x00E9}}{\text{t}}$-covering spaces are needed in the next section.

Let $X$ be a rigid $K$-space, and let $\tau \in \{\mathrm {oc},\mathrm {adm},{\unicode{x00E9}}{\text{t}}\}$. By a $\tau$-cover of $X$ we shall mean a surjective morphism $U\to X$ such that:

1. (i) if $\tau =\mathrm {oc}$, then $U\to X$ is the disjoint union of overconvergent open immersions into $X$;

2. (ii) if $\tau =\mathrm {adm}$, then $U\to X$ is the disjoint union of open immersions into $X$;

3. (iii) if $\tau ={\unicode{x00E9}}{\text{t}}$, then $U\to X$ is étale.

When $X$ is a taut rigid $K$-space (see [Reference HuberHub96, Definition 5.1.2]), then $\operatorname {\mathbf {Cov}}_X^{\mathrm {oc}}$ is naturally equivalent (via the functor in [Reference HuberHub96, § 8.3]) to the category of étale covering spaces of the associated Berkovich space $X^{\mathrm {Berk}}$ considered in [Reference de JongdJ95b]. Thus, we refer to $\text {oc}$-covering spaces as de Jong covering spaces.

For a geometric point $\overline {x}$ let $F_{\overline {x}}\colon \operatorname {\mathbf {\acute {E}t}}_X\to \operatorname {\mathbf {Set}}$ be the fiber functor $F_{\overline {x}}(Y)=\pi _0(Y_{\overline {x}})$. For a connected rigid $K$-space $X$ and $\tau =\mathrm {oc}$, de Jong essentially showed that the pair $(\operatorname {\mathbf {UCov}}^{\mathrm {oc}}_X,F_{\overline {x}})$ is a tame infinite Galois category (see [Reference de JongdJ95b, Theorem 2.9]). More generally, for a connected rigid $K$-space $X$ one knows from [Reference Achinger, Lara and YoucisALY21b, Theorem 2] that the pair $(\operatorname {\mathbf {UCov}}^{\tau }_X,F_{\overline {x}})$ is a tame infinite Galois category for each $\tau \in \{\mathrm {oc},\mathrm {adm},{\unicode{x00E9}}{\text{t}}\}$. In particular, we have an equivalence of categories

\[ F_{\overline{x}}\colon \operatorname{\mathbf{UCov}}_X^{\tau} {\buildrel \sim \over \longrightarrow} \pi_1(\operatorname{\mathbf{UCov}}^{\tau}_X, F_{\overline{x}})\text{-}\operatorname{\mathbf{Sets}}. \]

We denote the fundamental group of $(\operatorname {\mathbf {UCov}}^{\mathrm {oc}}_X,F_{\overline {x}})$ by $\pi _1^{\mathrm {dJ}}(X,\overline {x})$ and call it the de Jong fundamental group.

One of the upshots of the material discussed and developed in § 2 is the ability to prove the following criterion for when the rigid generic fiber of an étale map of formal schemes is a de Jong covering space.

3.3 The étale site of a formal scheme

We now recall the topological invariance of the étale site of a formal scheme.

3.4 Statement of the main result

Let $\mathfrak {X}$ be a formal scheme locally of finite type over $\mathcal {O}_K$ and let $X=\mathfrak {X}_\eta$ be its rigid generic fiber. We have the functors between the categories of étale objects

where $(-)_\eta$ (respectively, $(-)_k$) is the functor which maps $\mathfrak {Y}\to \mathfrak {X}$ to $\mathfrak {Y}_{\eta }\to \mathfrak {X}_{\eta }=X$ (respectively, $\mathfrak {Y}_k\to \mathfrak {X}_k$). The functor $(-)_k$ is an equivalence by Proposition 3.6. Let us denote by

\[ u\colon \operatorname{\mathbf{\acute{E}t}}_{\mathfrak{X}_k} \to \operatorname{\mathbf{\acute{E}t}}_{X} \]

the composition $u=(-)_\eta \circ (-)_k^{-1}$. The main result of this section can then be stated as follows.

Let us interpret this result in terms of fundamental groups. To this end, we first need to discuss compatible choices of base points for $\mathfrak {X}_k$ and $X$. Let $L$ be an algebraically closed non-archimedean field and let $\overline {\xi }\colon \operatorname {Spf}(\mathcal {O}_L)\to \mathfrak {X}$ be a morphism of formal schemes, inducing geometric points $\overline {\xi }_\eta \colon \operatorname {Spa}(L)\to X$ and $\overline {\xi }_0\colon \operatorname {Spec}(\ell )\to \mathfrak {X}_k$, where $\ell = \mathcal {O}_L/\mathfrak {m}_L$ is the residue field of $L$. In this situation, if $\mathfrak {Y}\to \mathfrak {X}$ is an étale morphism, then the induced maps

\[ \operatorname{Hom}_X(\operatorname{Spa}(L), \mathfrak{Y}_\eta) \leftarrow \operatorname{Hom}_{\mathfrak{X}}(\operatorname{Spf}(\mathcal{O}_L), \mathfrak{Y}) \to \operatorname{Hom}_{\mathfrak{X}_k}(\operatorname{Spec}(\ell), \mathfrak{Y}_k) \]

are bijective. To see this, we may replace $\mathfrak {X}$ with $\operatorname {Spf}(\mathcal {O}_L)$, and then it suffices to show that every étale morphism $\mathfrak {Y}\to \mathfrak {X}$ is a disjoint union of copies of $\mathfrak {X}$. This follows from Proposition 3.6 because $\underline {\mathfrak {X}}=\operatorname {Spec}(\ell )$.

Corollary 3.8 Let $\mathfrak {X}$ be a quasi-paracompact formal scheme locally of finite type over $\mathcal {O}_K$ whose special fiber $\mathfrak {X}_k$ and rigid generic fiber $X=\mathfrak {X}_\eta$ are both connected. Then for a compatible choice of base points as previously discussed, the functor $u$ induces a continuous homomorphism of Noohi groups

\[ \pi_1^{\mathrm{dJ}}(X, \overline{\xi}_\eta) \to \pi_1^{{\textrm{pro}}\unicode{x00E9}{\textrm{t}}}(\mathfrak{X}_k, \overline{\xi}_0). \]

Its image is dense if $\mathfrak {X}$ is $\eta$-normal (see Appendix A).

The natural converse to Theorem 3.7 is relatively straightforward given our setup, and we obtain the following (cf. [Reference Fujiwara and KatoFK18, Chapter II, Theorem 7.5.17]).