Proof. This follows from [Reference BourbakiBou98, Chapter III, §2 n ${}^{\circ }$ 11, Proposition 14 & Corollary 1] when M is finitely generated. See [Reference Fujiwara and KatoFK18, Chapter 0, Corollary 7.2.9 & Propositions 7.2.15 & 7.2.16] for general M.

Notation 1.1.2. If A is an adic ring and $T=(T_i)_i$ is a family of indeterminates, we denote by $A\langle T \rangle $ the algebra of restricted power series in T with coefficients in A, i.e., the I-adic completion of $A[T]$ for an ideal of definition $I\subset A$ . Unless otherwise stated, given an ideal $J\subset A\langle T\rangle $ , we denote by $A\langle T\rangle /J$ the I-adically complete quotient, i.e., the quotient of $A\langle T\rangle $ by the closure of the ideal J.

Notation 1.1.5. We denote by $\mathrm {FSch}$ the category of formal schemes and by $\mathrm {FSch}^{\mathrm {qcqs}}$ its full subcategory spanned by quasi-compact and quasi-separated formal schemes (in the sense of [Reference Fujiwara and KatoFK18, Chapter I, Definitions 1.6.1 & 1.6.5]). Note that the category $\mathrm {Sch}$ (resp., $\mathrm {Sch}^{\mathrm {qcqs}}$ ) of schemes (resp., of quasi-compact and quasi-separated schemes) can be identified with the full subcategory of $\mathrm {FSch}$ (resp., $\mathrm {FSch}^{\mathrm {qcqs}}$ ) spanned by those formal schemes for which $(0)$ is an ideal of definition.

Notation 1.1.6. The inclusion of the category of reduced schemes into $\mathrm {FSch}$ admits a right adjoint which we denote by $\mathcal {X}\mapsto \mathcal {X}_{\sigma }$ . It commutes with gluing along open immersions and satisfies $\mathcal {X}_{\sigma }=(\mathcal {X}/\mathcal {I})_{\mathrm {red}}$ whenever $\mathcal {X}$ admits an ideal of definition $\mathcal {I}\subset \mathcal {O}_{\mathcal {X}}$ . The scheme $\mathcal {X}_{\sigma }$ is called the special fiber of $\mathcal {X}$ .

Notation 1.1.8. (See [Reference Fujiwara and KatoFK18, Chapter II, §2])

We denote by $\mathrm {RigSpc}^{\mathrm {qcqs}}$ the $1$ -categorical localisation of the category $\mathrm {FSch}^{\mathrm {qcqs}}$ with respect to admissible blowups. More concretely, there is a functor $(-)^{\mathrm {rig}}:\mathrm {FSch}^{\mathrm {qcqs}}\to \mathrm {RigSpc}^{\mathrm {qcqs}}$ which is a bijection on objects and, given two quasi-compact and quasi-separated formal schemes $\mathcal {X}$ and $\mathcal {Y}$ , we have

(3) $$ \begin{align} \mathrm{Hom}_{\mathrm{RigSpc}^{\mathrm{qcqs}}}(\mathcal{Y}^{\mathrm{rig}},\mathcal{X}^{\mathrm{rig}})= \underset{\mathcal{Y}'\to \mathcal{Y} \,\in\, \mathfrak{B}(\mathcal{Y})}{\mathrm{colim}}\, \mathrm{Hom}_{\mathrm{FSch}^{\mathrm{qcqs}}}(\mathcal{Y}',\mathcal{X}). \end{align} $$

The objects of $\mathrm {RigSpc}^{\mathrm {qcqs}}$ are the quasi-compact and quasi-separated rigid analytic spaces (according to [Reference Fujiwara and KatoFK18, Chapter II, Definitions 2.1.1 & 2.1.2]). If $\mathcal {X}$ is a quasi-compact and quasi-separated formal scheme, $\mathcal {X}^{\mathrm {rig}}$ is called the Raynaud generic fiber (or simply the generic fiber) of $\mathcal {X}$ . For this reason, we sometimes write ‘ $\mathcal {X}_{\eta }$ ’ instead of ‘ $\mathcal {X}^{\mathrm {rig}}$ ’. A map in $\mathrm {RigSpc}^{\mathrm {qcqs}}$ is an open immersion if it is isomorphic to the generic fiber of an open immersion in $\mathrm {FSch}^{\mathrm {qcqs}}$ . General rigid analytic spaces are obtained by gluing along open immersions from objects in $\mathrm {RigSpc}^{\mathrm {qcqs}}$ as in [Reference Fujiwara and KatoFK18, Chapter II, §2.2. (c)]. The resulting category is denoted by $\mathrm {RigSpc}$ , and its objects are the rigid analytic spaces. There is also a generic fiber functor $(-)^{\mathrm {rig}}:\mathrm {FSch}\to \mathrm {RigSpc}$ extending the one on quasi-compact and quasi-separated formal schemes.

Notation 1.1.9. Let X be a rigid analytic space. A formal model for X is a formal scheme $\mathcal {X}$ endowed with an isomorphism $X\simeq \mathcal {X}^{\mathrm {rig}}$ (see [Reference Fujiwara and KatoFK18, Chapter II, Definition 2.1.7]). Formal models of X form a category which we denote by $\mathrm {Mdl}(X)$ . When X is quasi-compact and quasi-separated, $\mathrm {Mdl}(X)$ is cofiltered by [Reference Fujiwara and KatoFK18, Chapter II, Proposition 2.1.10]. Similarly, given a morphism $f:Y \to X$ of rigid analytic spaces, we have a category $\mathrm {Mdl}(f)$ of formal models of f whose objects are morphisms of formal schemes $\phi :\mathcal {Y} \to \mathcal {X}$ together with an isomorphism $f\simeq \phi ^{\mathrm {rig}}$ in $\mathrm {RigSpc}^{\Delta ^1}$ . When X and Y are quasi-compact and quasi-separated, the category $\mathrm {Mdl}(f)$ is cofiltered.

Notation 1.1.11. Let X be a quasi-compact and quasi-separated rigid analytic space. We define a locally ringed space $(|X|,\mathcal {O}^+_X)$ by

$$ \begin{align*}(|X|,\mathcal{O}_X^+)= \lim_{\mathcal{X}\in \mathrm{Mdl}(X)} (|\mathcal{X}|,\mathcal{O}_{\mathcal{X}}).\end{align*} $$

If $\mathcal {X}_0$ is a formal model of X and $\mathcal {I}\subset \mathcal {O}_{\mathcal {X}_0}$ is an ideal of definition, then $\mathcal {I}\mathcal {O}^+_X$ is an invertible ideal in $\mathcal {O}^+_X$ . We set $\mathcal {O}_X=\bigcup _{n\geq 0}(\mathcal {I}\mathcal {O}^+_X)^{-n}$ . Then $\mathcal {O}_X$ is a sheaf of rings which does not depend on $\mathcal {I}$ and which contains $\mathcal {O}_X^+$ . By gluing along open immersions, the assignment $X\mapsto (|X|,\mathcal {O}_X,\mathcal {O}^+_X)$ can be extended to any rigid analytic space X. For more details, we refer the reader to [Reference Fujiwara and KatoFK18, Chapter II, §3]. We say that $|X|$ is the topological space associated to X, that $\mathcal {O}_X$ is the structure sheaf of X and that $\mathcal {O}_X^+$ is the integral structure sheaf of X.

Notation 1.1.13. Let X be a rigid analytic space and $x\in |X|$ a point. By [Reference Fujiwara and KatoFK18, Chapter II, Proposition 3.2.6], the local ring $\mathcal {O}_{X,x}^+$ is a prevaluative ring. (Here we use the terminology of [Reference AbbesAbb10, Définition 1.9.1].) More precisely, there is a nonzero divisor $a\in \mathcal {O}_{X,x}^+$ with the following properties:

• ○ Every finitely generated ideal of $\mathcal {O}_{X,x}^+$ containing a power of a is principal;

• ○ $\mathcal {O}_{X,x}^+[a^{-1}]=\mathcal {O}_{X,x}$ ;

• ○ $\mathfrak {m}_{X,x}=\bigcap _{n\in \mathbb {N}} a^n\mathcal {O}_{X,x}^+$ , where $\mathfrak {m}_{X,x}$ is the maximal ideal of $\mathcal {O}_{X,x}$ ;

• ○ $\mathcal {O}_{X,x}^+/\mathfrak {m}_x$ is a valuation ring of the residue field $\mathcal {O}_{X,x}/\mathfrak {m}_x$ . We denote by $\Gamma _x$ its value group (denoted multiplicatively).

We let $\kappa ^+(x)$ be the a-adic completion of $\mathcal {O}_{X,x}^+$ , $\kappa (x)$ its fraction field and $\widetilde {\kappa }(x)$ the residue field of $\kappa ^+(x)$ . We also let $\kappa ^{\circ }(x)\subset \kappa (x)$ be the subring of power bounded elements. Then $\kappa ^{\circ }(x)$ is the unique height $1$ valuation ring containing $\kappa ^+(x)$ . Moreover, $\kappa (x)$ is a non-Archimedean complete field for the norm induced by $\kappa ^{\circ }(x)$ .

Notation 1.2.1. Given a Tate ring A, we denote by $A^{\circ }\subset A$ the subring of power bounded elements and $A^{\circ \circ }\subset A^{\circ }$ the ideal of topologically nilpotent elements. We say that A is uniform if $A^{\circ }$ is bounded (which is equivalent to ask that $A^{\circ }$ is a ring of definition).

Proof. Indeed, let R and $R'$ be two Tate affinoid rings and $f:R_{\natural } \to R^{\prime }_{\natural }$ a morphism of ind-adic rings. There exists rings of definition $R_0\subset R^{\pm }$ and $R^{\prime }_0\subset R^{\prime \pm }$ contained in $R^+$ and $R^{\prime +}$ such that f restricts to a morphism of adic rings $f_0:R_0\to R^{\prime }_0$ . Then $f_0$ induces a morphism of Tate rings $f^{\pm }:R^{\pm } \to R^{\prime \pm }$ . Since $f_0$ is the restriction of f, for every ring of definition $R_1\subset R^{\pm }$ containing $R_0$ and contained in $R^+$ , there exists a ring of definition $R^{\prime }_1\subset R^{\prime \pm }$ contained in $R^{\prime +}$ and a morphism $f_1:R_1\to R^{\prime }_1$ extending $f_0$ . This shows that $f^{\pm }$ maps $R^+$ into $R^{\prime +}$ as needed.

Proof. A point $x\in |\mathrm {Spf}(A)^{\mathrm {rig}}|$ determines a morphism of adic rings $A \to \kappa ^+(x)$ , and hence a continuous valuation $v_x:A \to \Gamma _x\cup \{0\}$ landing in $\Gamma ^+_x\cup \{0\}$ . (Here $\Gamma ^+_x\subset \Gamma $ denotes the submonoid defined by the inequality $\leq 1$ .) Since the image of $\pi $ in $\kappa ^+(x)$ is nonzero, $v_x$ extends uniquely to a continuous valuation $v_x:A^{\natural \pm } \to \Gamma _x\cup \{0\}$ . Moreover, $v_x$ maps $A^{\natural +}$ into $\Gamma ^+_x\cup \{0\}$ since $A^{\natural +}$ is integral over A. Therefore, $v_x$ belongs to $\mathrm {Spa}(A^{\natural })$ . It is easy to see that $x\mapsto v_x$ is a bijection, which is continuous and open. More precisely, given elements $a_0,\ldots , a_n$ in A generating an admissible ideal of A, the open subset $|\mathrm {Spf}(A\langle \frac {a_1}{a_0},\ldots , \frac {a_n}{a_0} \rangle )^{\mathrm {rig}}| \subset |\mathrm {Spf}(A)^{\mathrm {rig}}|$ is mapped bijectively to the rational subset $|\mathrm {Spa}(A^{\natural }\langle \frac {a_1}{a_0},\ldots , \frac {a_n}{a_0}\rangle )|\subset |\mathrm {Spa}(A^{\natural })|$ . This also shows that $\mathcal {O}_{\mathrm {Spf}(A)^{\mathrm {rig}}}$ is the sheafification of $\mathcal {O}_{\mathrm {Spa}(A^{\natural })}$ .

Assertion (2) can be deduced from assertion (1) and the fact that the counit map $(R_{\natural })^{\natural } \to R$ identifies the Tate affinoid ring R with the colimit of the ind-Tate affinoid ring $(R_{\natural })^{\natural }$ .

Proof. It suffices to treat the affinoid case; the general case follows then by gluing along open immersions. Given two stably uniform Tate affinoid rings A and B, the fact that A is sheafy implies that there is a bijection $\mathrm {Hom}(A,B) \simeq \mathrm {Hom}(\mathrm {Spa}(B),\mathrm {Spa}(A))$ . It follows from Remark 1.2.3 that there is a functor $\mathrm {Spa}(A) \mapsto \mathrm {Spf}(A^+)^{\mathrm {rig}}$ , from affinoid uniform adic spaces to rigid analytic spaces, and it remains to show that the map

$$ \begin{align*}\mathrm{Hom}(A^+,B^+)\to \mathrm{Hom}(\mathrm{Spf}(B^+)^{\mathrm{rig}},\mathrm{Spf}(A^+)^{\mathrm{rig}}),\end{align*} $$

with A and B as above, is a bijection. An element of the right-hand side can be represented by a morphism $\mathcal {Y}\to \mathrm {Spf}(A^+)$ , where $\mathcal {Y}\to \mathrm {Spf}(B^+)$ is an admissible blowup. We may assume that $\mathcal {O}_{\mathcal {Y}}$ is $\pi $ -torsion-free, with $\pi $ a generator of an ideal of definition in $B^+$ . We claim that $\mathcal {O}(\mathcal {Y})=B^+$ which implies that $\mathcal {Y}\to \mathrm {Spf}(A^+)$ factors uniquely through $\mathrm {Spf}(B^+)$ , finishing the proof.

Let $(\mathcal {Y}_i)_i$ be an affine open covering of $\mathcal {Y}$ and set $\mathcal {Y}_{ij}=\mathcal {Y}_i\cap \mathcal {Y}_j$ . Let $B_i$ and $B_{ij}$ be the Tate affinoid rings associated to the adic rings $\mathcal {O}(\mathcal {Y}_i)$ and $\mathcal {O}(\mathcal {Y}_{ij})$ , respectively. Then $(\mathrm {Spa}(B_i))_i$ is an open covering of $\mathrm {Spa}(B)$ , and $\mathrm {Spa}(B_{ij})=\mathrm {Spa}(B_i) \cap \mathrm {Spa}(B_j)$ . Since B is sheafy, we deduce that $B^+$ is the equaliser of the usual pair of arrows $\prod _i B_i^+ \rightrightarrows \prod _{ij}B_{ij}^+$ . Since $\mathcal {O}_{\mathcal {Y}}$ is $\pi $ -torsion-free, we have inclusions $\mathcal {O}(\mathcal {Y}_i)\subset B_i^+$ and $\mathcal {O}(\mathcal {Y}_{ij})\subset B^+_{ij}$ . This proves that $\mathcal {O}(\mathcal {Y})$ , which is the equaliser of $\prod _i \mathcal {O}(\mathcal {Y}_i) \rightrightarrows \prod _{ij}\mathcal {O}(\mathcal {Y}_{ij})$ , is contained in $B^+$ as needed.

Notation 1.3.1. Let A be an adic ring and $J\subset A$ an ideal. We denote by $J^{\mathrm {sat}}$ the ideal of A consisting of those elements $a\in A$ for which there exists an ideal of definition $I\subset A$ such that $aI\subset J$ . The ideal $J^{\mathrm {sat}}$ is called the saturation of J.

Proof. The proof of [Reference FujiwaraFuj95, Proposition 2.1.1] can be easily adapted to the situation considered in the statement. For the reader’s convenience we recall the argument.

For $m\in \mathbb {N}$ , we set $A_m=A/\pi ^m$ , $B_m=B/\pi ^m$ and $C_m=C/\pi ^m$ . Since B is rig-étale over A, there exists an integer c such that $\Omega ^1_{B_m/A_m}$ is annihilated by $\pi ^c$ independently of m. (Indeed, if B is given as in Definition 1.3.3, it suffices to take c so that $\pi ^c$ belongs to the ideal generated by $\det (\partial f_i/\partial t_j)$ .) Now let $f,f':B \to C$ be two morphisms of A-algebras inducing the same morphism $f_m:B_m \to C_m$ for some $m\geq c+1$ . We will show that $f_{m+1}=f^{\prime }_{m+1}$ , which suffices to conclude using induction.

We may consider $f_{2m}$ and $f^{\prime }_{2m}$ as deformations of $f_{m}$ . The difference between these deformations is classified by an element $\epsilon \in \mathrm {Hom}(C_m\otimes _{B_m} \Omega ^1_{B_m/A_m}, \pi ^mC/\pi ^{2m}C)$ . Since $\pi $ is a nonzero divisor of C and $\Omega ^1_{B_m/A_m}$ annihilated by $\pi ^c$ , the image of any C-linear morphism $C_m\otimes _{B_m} \Omega ^1_{B_m/A_m} \to \pi ^mC/\pi ^{2m}C$ is contained in $\pi ^{2m-c}C/\pi ^{2m}C$ . In particular, the map

$$ \begin{align*}\mathrm{Hom}(C_m\otimes_{B_m} \Omega^1_{B_m/A_m}, \pi^mC/\pi^{2m}C)\to \mathrm{Hom}(C_m\otimes_{B_m} \Omega^1_{B_m/A_m}, \pi^mC/\pi^{m+1}C)\end{align*} $$

is identically zero. Since the image of $\epsilon $ by this map classifies the difference between $f_{m+1}$ and $f^{\prime }_{m+1}$ , we get the equality $f_{m+1}=f^{\prime }_{m+1}$ .

Proof. For N sufficiently large, the injectivity of the map (5) follows from Lemma 1.3.6. The fact that there is an N which works for all $f'$ close enough to f follows from the proof of Lemma 1.3.6. (Indeed, the N depends only on the ideal generated by $\det (\partial f_i/\partial t_j)$ .)

For the surjectivity of the map (5), it is enough to solve the following problem: Given an n-tuple $c_0=(c_{0,1},\ldots , c_{0,n})$ in C such that the components of $f(c_0)$ belong to $\pi ^{2e}C$ , find an n-tuple $c=(c_1,\ldots , c_n)$ in C such that $f(c)=0$ and the components of $c-c_0$ belong to $\pi ^eC$ . (Indeed, since C is $\pi $ -torsion-free an n-tuple c such that $f(c)=0$ determines an A-morphism $B\to C$ .)

This problem can be solved using the Newton method as in the first step of the proof of [Reference AyoubAyo15, Lemme 1.1.52]. In fact, one can also remark that the argument in loc. cit. is valid more generally for non-Archimedean Banach rings, i.e., complete normed rings with a non-Archimedean norm. In particular, it applies with ‘A’, ‘C’ and ‘R’ in loc. cit. replaced with $A[\pi ^{-1}]$ , $B[\pi ^{-1}]$ and $C[\pi ^{-1}]$ endowed with the natural norms for which $A/(0)^{\mathrm {sat}}$ , $B/(0)^{\mathrm {sat}}$ and $C=C/(0)^{\mathrm {sat}}$ are the unit balls. (More precisely, for $a\in A[\pi ^{-1}]$ , we set $\|a\|=e^{-v(a)}$ , where $v(a)$ is the largest integer such that $a\in \pi ^{v(a)}(A/(0)^{\mathrm {sat}})$ and similarly for B and C.) Since $\pi $ is a nonzero divisor of C, a solution $c=(c_1,\ldots , c_n)$ in $(C[\pi ^{-1}])^n$ of the system of equations $f=0$ , close enough to $c_0$ , determines a solution in $C^n$ . We may take for N an integer which is larger than $\ln (2M^2)$ with M as in [Reference AyoubAyo15, page 46].Footnote ³ It is clear that N depends $\pi $ -adically continuously on f.

Proof. This follows by applying Proposition 1.3.7 to the rig-étale adic A-algebras B and $B'$ .

Notation 1.3.9. Let A be an adic ring. We denote by $\mathcal {E}_A$ the category of rig-étale A-algebras and $\mathcal {E}^{\prime }_A$ its full subcategory spanned by those adic A-algebras whose zero ideal is saturated. (Thus, every object $B\in \mathcal {E}^{\prime }_A$ admits a presentation $B\simeq A\langle t \rangle /(f)^{\mathrm {sat}}$ with $t=(t_1,\ldots , t_n)$ and $f=(f_1,\ldots , f_n)$ such that $\det (\partial f_i /\partial t_j)$ generates an open ideal in B.) The inclusion $\mathcal {E}_A'\to \mathcal {E}_A$ admits a left adjoint given by $B\mapsto B/(0)^{\mathrm {sat}}$ . Given a morphism of adic rings $A_1 \to A_2$ , there is are induced functors $\mathcal {E}_{A_1}\to \mathcal {E}_{A_2}$ and $\mathcal {E}^{\prime }_{A_1} \to \mathcal {E}^{\prime }_{A_2}$ given by $B\mapsto A_2\,\widehat {\otimes }_{A_1}\,B$ and $B\mapsto (A_2\,\widehat {\otimes }_{A_1}\,B)/(0)^{\mathrm {sat}}$ , respectively.

Proof. Let R be the colimit of $(A_{\alpha })_{\alpha }$ taken in the category of discrete rings. We may assume that there is a smallest index o, and we fix $\pi \in A_o$ generating an ideal of definition of $A_o$ . Then $A=\lim _{n\in \mathbb {N}} R/\pi ^nR$ , and there is a map of rings $R\to A$ with kernel $J=\bigcap _n \pi ^nR$ and with dense image $\widetilde {R}\subset A$ . We split the proof into two steps.

Step 1

First, we prove that the functor (6) is essentially surjective. By Proposition 1.3.8, an object $B\in \mathcal {E}^{\prime }_A$ admits a presentation of the form $B=A\langle t\rangle /(\widetilde {f})^{\mathrm {sat}}$ , where $t=(t_1,\ldots , t_n)$ is a system of coordinates and $\widetilde {f}= (\widetilde {f}_1,\ldots , \widetilde {f}_n)$ an n-tuple in $\widetilde {R}[t]$ such that $\widetilde {g}= \det (\partial \widetilde {f}_i/\partial t_j)$ generates an open ideal in B. Using Remark 1.3.5, we can find an integer N and an element $\widetilde {h}\in A\langle t\rangle $ such that $\pi ^N-\widetilde {h}\widetilde {g}$ belongs to the closure of the ideal $(\widetilde {f})\subset \widetilde {R}[t]$ in $A\langle t\rangle $ . In particular, we may write

$$ \begin{align*}\pi^N-\widetilde{h}\widetilde{g}=\sum_{i=1}^n \widetilde{a}_i \widetilde{f}_i+\widetilde{v}\pi^{N+1}\end{align*} $$

with $\widetilde {v}\in A\langle t \rangle $ and $\widetilde {a}_1,\ldots , \widetilde {a}_n\in \widetilde {R}[t]$ . Write $\widetilde {h}=\widetilde {h}_0+\widetilde {h}_1\pi ^{N+1}$ with $\widetilde {h}_0\in \widetilde {R}[t]$ and $\widetilde {h}_1 \in A\langle t\rangle $ . Replacing $\widetilde {h}$ by $\widetilde {h}_0$ and $\widetilde {v}$ by $\widetilde {v}+\widetilde {h}_1$ , we may assume that $\widetilde {h}$ belongs to $\widetilde {R}[t]$ . It follows from Lemma 1.1.1 that the expression $\pi ^N-\widetilde {h}\widetilde {g}-\sum _{i=1}^n \widetilde {a}_i \widetilde {f}_i\in \widetilde {R}[t]$ belongs to $\pi ^{N+1}\widetilde {R}[t]$ . Said differently, we may also assume that $\widetilde {v}\in \widetilde {R}[t]$ . We now choose a lift $f=(f_1,\ldots , f_n)$ of $\widetilde {f}$ to an n-tuple in $R[t]$ and set $g=\det (\partial f_i/\partial t_j)$ . We also choose lifts $h, a_1,\ldots , a_n\in R[t]$ of $\widetilde {h},\widetilde {a}_1,\ldots , \widetilde {a}_n$ . Since the elements of J are divisible by any power of $\pi $ , we may also find a lift $v\in R[t]$ of $\widetilde {v}$ such that

$$ \begin{align*}\pi^N-hg=\sum_{i=1}^n a_if_i+v\pi^{N+1}.\end{align*} $$

For $\alpha $ sufficiently big, the previous equality can be lifted to an equality

$$ \begin{align*}\pi^N-h_{\alpha}g_{\alpha}=\sum_{i=1}^n a_{\alpha,i}f_{\alpha,i}+v_{\alpha}\pi^{N+1}\end{align*} $$

in $A_{\alpha }[t]$ with the property that $g_{\alpha }=\det (\partial f_{\alpha ,i}/\partial t_j)$ . Since $1-v_{\alpha }\pi $ is invertible in $A_{\alpha }\langle t\rangle $ , it follows that $B_{\alpha }=A_{\alpha }\langle t\rangle /(f_{\alpha })^{\mathrm {sat}}$ is a rig-étale $A_{\alpha }$ -algebra. Clearly, the functor (6) sends $B_{\alpha }$ to B.

Step 2

We now prove that the functor (6) is fully faithful. We fix two objects $B_o, C_o\in \mathcal {E}^{\prime }_{A_o}$ . For an index $\alpha $ , we set $B_{\alpha }=(B_o\widehat {\otimes }_{A_o} A_{\alpha })/(0)^{\mathrm {sat}}$ and define $C_{\alpha }$ similarly. We also set $B=(B_o\widehat {\otimes }_{A_o}A)/(0)^{\mathrm {sat}}$ and define C similarly. We want to show that

$$ \begin{align*}\underset{\alpha}{\mathrm{colim}}\, \mathrm{Hom}_{A_{\alpha}}(B_{\alpha},C_{\alpha})\to \mathrm{Hom}_A(B,C) \end{align*} $$

is a bijection. (This is enough since we are free to change the smallest index o. We also used that the colimit in equation (6) is filtered in order to describe the hom-set in the domain.) The above map can be rewritten as

$$ \begin{align*}\underset{\alpha}{\mathrm{colim}}\, \mathrm{Hom}_{A_o}(B_o,C_{\alpha})\to \mathrm{Hom}_{A_o}(B_o,C).\end{align*} $$

Since C and the $C_{\alpha }$ ’s are $\pi $ -torsion-free, we may replace $B_o$ by any rig-étale $A_o$ -algebra $B^{\prime }_o$ such that $B_o\simeq B^{\prime }_o/(0)^{\mathrm {sat}}$ . By Remark 1.3.5, we may choose $B^{\prime }_o$ topologically finitely presented. We now apply Proposition 1.3.7: There exists an integer N such that the maps

$$ \begin{align*}\mathrm{Hom}_{A_o}(B^{\prime}_o,C_{\alpha}) \to {\mathrm{im}}\left\{\mathrm{Hom}_{A_o/\pi^{2N}}(B^{\prime}_o/\pi^{2N},C_{\alpha}/\pi^{2N}) \to \mathrm{Hom}_{A_o/\pi^{N}}(B^{\prime}_o/\pi^{N},C_{\alpha}/\pi^{N})\right\}\end{align*} $$

are bijections and similarly for C (instead of $C_{\alpha }$ ). Since filtered colimits commute with taking images, we are left to show that

$$ \begin{align*}\underset{\alpha}{\mathrm{colim}}\, \mathrm{Hom}_{A_o/\pi^e}(B^{\prime}_o/\pi^e,C_{\alpha}/\pi^e)\to \mathrm{Hom}_{A_o/\pi^e}(B^{\prime}_o/\pi^e,C/\pi^e)\end{align*} $$

is a bijection for any positive integer e. This is clear since $B^{\prime }_o/\pi ^e$ is a finitely presented $A_o/\pi ^e$ -algebra and $C/\pi ^e$ is the colimit of the filtered system $(C_{\alpha }/\pi ^e)_{\alpha }$ .

Proof. The question is local on X and around $s(X)$ . Thus, we may assume that $X=\mathrm {Spf}(A)^{\mathrm {rig}}$ with A an adic ring of principal ideal type, that $X'=\mathrm {Spf}(A')^{\mathrm {rig}}$ with $A'$ a rig-étale adic A-algebra and that s is induced by a morphism of A-algebras $h:A' \to A$ . Fix a generator $\pi $ of an ideal of definition of A. By Proposition 1.3.8, we may assume that $A'=A\langle t\rangle /(f)^{\mathrm {sat}}$ with $t=(t_1,\ldots , t_n)$ a system of coordinates and $f=(f_1,\ldots , f_n)$ an n-tuple in $A[t]$ such that $\det (\partial f_i/\partial t_j)$ generates an open ideal in $A'$ . Consider the A-algebra $C=A[t]/(f)$ . Then, $C[\pi ^{-1}]$ is étale over $A[\pi ^{-1}]$ and h induces a morphism of $A[\pi ^{-1}]$ -algebras $C[\pi ^{-1}]\to A[\pi ^{-1}]$ . From standard properties of ordinary étale algebras, we deduce that $\mathrm {Spec}(A[\pi ^{-1}])\to \mathrm {Spec}(C[\pi ^{-1}])$ is a clopen immersion. Passing to the analytification over A in the sense of Construction 1.1.15, we deduce a clopen immersion $\mathrm {Spf}(A)^{\mathrm {rig}} \to \mathrm {Spec}(C[\pi ^{-1}])^{\mathrm {an}}$ . But the latter factors as follows:

$$ \begin{align*}\mathrm{Spf}(A)^{\mathrm{rig}} \xrightarrow{s} \mathrm{Spf}(A')^{\mathrm{rig}} \to \mathrm{Spec}(C[\pi^{-1}])^{\mathrm{an}},\end{align*} $$

where the second map is an open immersion. This finishes the proof.

Proof. The question being local on X, we may assume that $X=\mathrm {Spf}(A)^{\mathrm {rig}}$ with A an adic ring of principal ideal type, and $Z=\mathrm {Spf}(B)^{\mathrm {rig}}$ with B a quotient of A by a closed ideal $I\subset A$ . We may also assume that $X'=\mathrm {Spf}(A')^{\mathrm {rig}}$ with $A'$ a rig-étale A-algebra and that the section s is induced by a morphism of A-adic rings $h:A'\to B$ . Let $\pi \in A$ be a generator of an ideal of definition. Without loss of generality, we may assume that B and $A'$ are $\pi $ -torsion-free.

For $N\in \mathbb {N}$ and $J\subset I$ a finitely generated ideal, consider the adic A-algebra $C_{J,\,N}=A\langle J/\pi ^N\rangle $ given as the $\pi $ -adic completion of the sub-A-algebra $A[J/\pi ^N]\subset A[\pi ^{-1}]$ generated by fractions $a/\pi ^N$ with $a\in J$ . Then B is the filtered colimit in the category of adic rings of the $C_{J,\,N}$ ’s when N and J vary. Applying Corollary 1.3.10 to this inductive system, we can find J and N such that the image of $\mathrm {Hom}_A(A',C_{J,\,N}) \to \mathrm {Hom}_A(A',B)$ contains h. This means that the section s extends to an X-morphism $\mathrm {Spf}(C_{J,\,N})^{\mathrm {rig}} \to \mathrm {Spf}(A')^{\mathrm {rig}}$ . Since $\mathrm {Spf}(C_{J,\,N})^{\mathrm {rig}}$ is an open subspace of X, this proves the existence of $\tilde {s}$ as in the proposition. That $\tilde {s}$ is an open immersion follows from Lemma 1.3.11.

Proof. According to [Reference ElkikElk73, pages 588–589], the proof of [Reference ElkikElk73, page 582, Théorème 7] can be adapted to cover the above statement. Alternatively, one can use Proposition 1.3.8 as follows. By this proposition, we may assume that the adic A-algebra B is of the form

$$ \begin{align*} B=A\langle t_1,\ldots, t_m,s_1,\ldots, s_n\rangle/ (f_1,\ldots, f_n)^{\mathrm{sat}}, \end{align*} $$

with $f_1,\ldots , f_n\in A[t_1,\ldots , t_m,s_1,\ldots , s_n]$ and such that $\det (\partial f_i/\partial s_j)$ generates an open ideal in B. (The rig-étale case corresponds to $m=0$ .) Consider the A-algebra

$$ \begin{align*} P'=A[t_1,\ldots, t_m,s_1,\ldots, s_n]/(f_1,\ldots, f_n)^{\mathrm{sat}} \end{align*} $$

whose $\pi $ -adic completion is B. Let $e\in P'$ be the image of $\det (\partial f_i/\partial s_j)$ in $P'$ . By assumption, a power of $\pi $ is a multiple of e in the $\pi $ -adic completion of $P'$ . Thus, there are elements $b,c\in B$ and an integer N such that $\pi ^N=e\cdot b+c\pi ^{N+1}$ . The A-algebra $P=P'[(1-c\pi )^{-1}]$ satisfies the properties required in the statement.

Proof. The problem being local on X and around $s(Z)$ , we may assume that $X'$ is étale over $\mathbb {B}^m_X$ and, by change of coordinates, that the composition

$$ \begin{align*}Z \xrightarrow{s} X' \to \mathbb{B}^m_X\end{align*} $$

is the zero section over Z. Applying Proposition 1.3.12 to the étale morphism $X' \to \mathbb {B}^m_X$ and the closed immersion $Z \to \mathbb {B}^m_X$ given by the zero section over Z, we find locally an open neighbourhood $U'\subset X'$ of $s(Z)$ such that $U'\to \mathbb {B}^m_X$ is also an open immersion. Letting U be the inverse image of $U'$ by the zero section $X \to \mathbb {B}^m_X$ and replacing $U'$ by $U'\times _X U$ , we may assume that $U'$ is an open neighbourhood of the zero section of $\mathbb {B}^n_U$ . Since the zero section of $\mathbb {B}^n_U$ admits a system of fundamental neighbourhoods which are m-dimensional relative balls, we may also assume that $U'$ is isomorphic to $\mathbb {B}^m_U$ as needed.

Proof. It is enough to show that $f(|Y|)$ is open in $|X|$ . The question is local on X and Y. By Proposition 1.3.15, we may assume that $X=\mathrm {Spf}(A)^{\mathrm {rig}}$ , with A an adic ring of principal ideal type, and $Y=\mathrm {Spf}(B)^{\mathrm {rig}}$ , with $B=\widehat {P}$ the $\pi $ -adic completion of a finitely presented A-algebra P such that $P[\pi ^{-1}]$ is smooth over $A[\pi ^{-1}]$ . (As usual, $\pi $ is a generator of an ideal of definition of A. Also, note that finite presentation in Proposition 1.3.15 can be assumed if we don’t insist on $\pi $ -torsion-freeness.) By the Raynaud–Gruson platification theorem [Reference Raynaud and GrusonRG71, Theorem 5.2.2] and working locally over X, we may further assume that P is flat over A. By [Reference GrothendieckGro66, Chapitre IV, Théorème 2.4.6], the morphism $\mathrm {Spec}(P) \to \mathrm {Spec}(A)$ is then open, and we denote by $U\subset \mathrm {Spec}(A)$ its image. Let $(a_i)_i$ be a family in A generating the ideal defining the complement of U in $\mathrm {Spec}(A)$ . Let $A_i$ be the $\pi $ -adic completion of $A[a_i^{-1}]$ and $B_i$ the $\pi $ -adic completion of $P_i=P\otimes _A A_i$ . Set $X_i=\mathrm {Spf}(A_i)^{\mathrm {rig}}$ and $Y_i=\mathrm {Spf}(B_i)^{\mathrm {rig}}$ . By construction, $(Y_i)_i$ is an open covering of Y, and it is enough to show that $f(Y_i)$ is open in X. We will show more precisely that $f(Y_i)=X_i$ , i.e., that $Y_i \to X_i$ is surjective.

Replacing X and Y by $X_i$ and $Y_i$ , we are reduced to showing that $f:Y \to X$ is surjective, for $X=\mathrm {Spf}(A)^{\mathrm {rig}}$ and $Y=\mathrm {Spf}(B)^{\mathrm {rig}}\simeq \mathrm {Spf}(\widehat {P})^{\mathrm {rig}}$ as above, assuming furthermore that the A-algebra P is faithfully flat. To do so, it will be enough to show the following assertion. If $\mathcal {X}' \to \mathrm {Spf}(A)$ is an admissible blowup and $\mathcal {Y}'=(\mathcal {X}'\otimes _AB)/(0)^{\mathrm {sat}}$ , the induced map $\mathcal {Y}^{\prime }_{\sigma } \to \mathcal {X}^{\prime }_{\sigma }$ is surjective. (Indeed, by [Reference Fujiwara and KatoFK18, Chapter III, Proposition 3.1.5], the obvious map $|Y|\to |\mathcal {Y}^{\prime }_{\sigma }|$ is surjective.) Since P is flat over A, the formal scheme $\mathcal {X}'\times _{\mathrm {Spf}(A)}\mathrm {Spf}(B)$ is already saturated and we have an isomorphism $\mathcal {Y}'/\pi \simeq \mathcal {X}'/\pi \otimes _AP$ . In particular, we see that the map $\mathcal {Y}'/\pi \to \mathcal {X}'/\pi $ is faithfully flat and, hence, surjective as needed.

Notation 1.4.1. Given a scheme S, we denote by $\acute {\mathrm {E}}\mathrm {t}/S$ the category of étale S-schemes. Similarly, given a formal scheme $\mathcal {S}$ , we denote by $\acute {\mathrm {E}}\mathrm {t}/\mathcal {S}$ the category of étale formal $\mathcal {S}$ -schemes.

Proof. This follows immediately from [Reference GrothendieckGro67, Chapitre IV, Théorème 18.1.2].

Notation 1.4.3. Given a rigid analytic space S, we denote by $\acute {\mathrm {E}}\mathrm {t}/S$ the category of étale rigid analytic S-spaces (in the sense of Definition 1.3.3). We denote by $\acute {\mathrm {E}}\mathrm {t}^{\mathrm {gr}}/S$ the full subcategory of $\acute {\mathrm {E}}\mathrm {t}/S$ spanned by those étale rigid analytic S-spaces with good reduction (in the sense of Definition 1.1.14).

Notation 1.4.7. The étale topology is generally denoted by ‘ $\acute {\mathrm {e}}\mathrm {t}$ ’ and the Nisnevich topology is denoted by ‘ $\mathrm {nis}$ ’. Also, the Zariski topology is generally denoted by ‘ $\mathrm {zar}$ ’ and the analytic topology is denoted by ‘ $\mathrm {an}$ ’.

Notation 1.4.9.

1. (1) If S is a scheme, we denote by $\mathrm {Sch}/S$ the overcategory of S-schemes and $\mathrm {Sm}/S$ its full subcategory consisting of smooth objects. For $\tau \in \{\mathrm {nis},\acute {\mathrm {e}}\mathrm {t}\}$ , we call $(\mathrm {Sm}/S,\tau )$ the big smooth site of S.

2. (2) If $\mathcal {S}$ is a formal scheme, we denote by $\mathrm {FSch}/\mathcal {S}$ the overcategory of formal $\mathcal {S}$ -schemes and $\mathrm {FSm}/\mathcal {S}$ its full subcategory consisting of smooth objects. For $\tau \in \{\mathrm {nis},\acute {\mathrm {e}}\mathrm {t}\}$ , we call $(\mathrm {FSm}/S,\tau )$ the big smooth site of $\mathcal {S}$ .

3. (3) If S is a rigid analytic space, we denote by $\mathrm {RigSpc}/S$ the overcategory of rigid analytic S-spaces and $\mathrm {RigSm}/S$ its full subcategory consisting of smooth objects (in the sense of Definition 1.3.13). For $\tau \in \{\mathrm {nis},\acute {\mathrm {e}}\mathrm {t}\}$ , we call $(\mathrm {RigSm}/S,\tau )$ the big smooth site of S.

Proof. The statement would have been a particular case of [Reference HuberHub96, Corollary A.4], except that we don’t know a priori that the continuous functor $(-)^{\mathrm {rig}}$ defines a morphism of sites and that we do not assume that our categories have finite limits. (In fact, we are particularly interested in the case where $\underline {\mathcal {V}}$ is the category of rig-smooth formal $\mathcal {S}$ -schemes, which does not admit finite limits.) Instead of trying to modify the proof of [Reference HuberHub96, Corollary A.4], we present an independent argument. We only treat the absolute case since the relative case is similar.

By [SGAIV1, Exposé III, Théorème 4.1], we may assume that $\underline {\mathcal {V}}\subset \mathrm {FSch}^{\mathrm {qcqs}}$ and that $\underline {V}$ is the full subcategory of $\mathrm {RigSpc}^{\mathrm {qcqs}}$ spanned by objects of the form $\mathcal {X}^{\mathrm {rig}}$ for $\mathcal {X}\in \underline {\mathcal {V}}$ . The rig topology on $\underline {\mathcal {V}}$ is not subcanonical (except for very special choices of $\underline {\mathcal {V}}$ ). We denote by $\underline {\mathcal {V}}'$ the full subcategory of the category of sheaves of sets on $(\underline {\mathcal {V}},\mathrm {rig})$ spanned by sheafifications of representable presheaves. The obvious functor $a:\underline {\mathcal {V}}\to \underline {\mathcal {V}}'$ , sending a formal scheme $\mathcal {X}$ to the sheaf associated of the presheaf represented by $\mathcal {X}$ , induces an equivalence of sites $(\underline {\mathcal {V}}',\mathrm {rig})\simeq (\underline {\mathcal {V}},\mathrm {rig})$ , where the topology of $(\underline {\mathcal {V}}',\mathrm {rig})$ is the one induced from the canonical topology on the topos of sheaves on $(\underline {\mathcal {V}},\mathrm {rig})$ . (This is a well-known fact which follows, for example, from [SGAIV1, Exposé IV, Corollaire 1.2.1]; see also [Reference AyoubAyo07b, Corollaire 4.4.52].) To prove the lemma, we remark that there is an equivalence of categories $\underline {\mathcal {V}}'\simeq \underline {V}$ which identifies the rig topology on $\underline {\mathcal {V}}'$ with the analytic topology on $\underline {V}$ . Indeed, for an admissible blowup $\mathcal {Y}'\to \mathcal {Y}$ in $\underline {\mathcal {V}}$ , the diagonal map $\mathcal {Y}'\to \mathcal {Y}'\times _{\mathcal {Y}}\mathcal {Y}'$ is a rig cover, which implies that $a\mathcal {Y}'\to a\mathcal {Y}$ is an isomorphism. Using that the Zariski topology is subcanonical on $\underline {\mathcal {V}}$ , we deduce that

$$ \begin{align*}\mathrm{Hom}_{\underline{\mathcal{V}}'}(a\mathcal{Y},a\mathcal{X})= \underset{\mathcal{Y}'\to \mathcal{Y}\,\in\, \mathfrak{B}(\mathcal{Y})}{\mathrm{colim}}\, \mathrm{Hom}_{\underline{\mathcal{V}}}(\mathcal{Y}',\mathcal{X})\end{align*} $$

for any $\mathcal {X},\mathcal {Y}\in \underline {\mathcal {V}}$ . The result follows then by comparison with equation (3).

Proof. The morphism (7) is finite étale if and only if $\mathrm {Spf}(B) \to \mathrm {Spf}(A)$ is finite rig-étale. So we need to show that the morphism (7) is surjective if and only if $|\mathrm {Spf}(B)^{\mathrm {rig}}|\to |\mathrm {Spf}(A)^{\mathrm {rig}}|$ is surjective. This follows easily from the description of $|\mathrm {Spf}(A)^{\mathrm {rig}}|$ in terms of valuation rings of residue fields of points of $\mathrm {Spec}(A)\smallsetminus \mathrm {Spec}(A/I)$ and [Reference BourbakiBou98, Chapter VI, §8, n ${}^{\circ }$ 6, Proposition 6].

Proof. Let $(\mathcal {U}_j\to \mathcal {X})_{j\in J}$ be a rig-étale cover. We may assume that J is finite (see Remark 1.4.6) and that $\mathcal {X}=\mathrm {Spf}(A)$ is affine with A an adic ring of principal ideal type. We fix a generator $\pi \in A$ of an ideal of definition of A. By Proposition 1.3.15, we may refine the rig-étale cover and assume that each $\mathcal {U}_j$ is the adic completion of a finite presentation A-scheme $U_j$ which is étale over $A[\pi ^{-1}]$ . (Note that finite presentation in Proposition 1.3.15 can be assumed if we don’t insist on $\pi $ -torsion-freeness.) By the Raynaud–Gruson platification theorem [Reference Raynaud and GrusonRG71, Theorem 5.2.2], there exists an admissible blowup $X'\to X=\mathrm {Spec}(A)$ such that the strict transform $U^{\prime }_j \to X'$ of $U_j \to X$ is flat for every j. In particular, the morphism $U^{\prime }_j \to X'$ is also quasi-finite.

Let $\mathcal {U}^{\prime }_j$ and $\mathcal {X}'$ be the adic completions of $U^{\prime }_j$ and $X'$ . By construction, we have $\mathcal {X}^{\prime \mathrm {rig}}\simeq \mathcal {X}^{\mathrm {rig}}$ and $\mathcal {U}^{\prime \mathrm {rig}}_j\simeq \mathcal {U}_j^{\mathrm {rig}}$ . Thus, $(\mathcal {U}^{\prime }_j \to \mathcal {X}')_j$ is also a rig-étale cover. Since $\mathcal {O}_{\mathcal {X}'}$ and the $\mathcal {O}_{\mathcal {U}_j'}$ ’s are $\pi $ -torsion-free, we deduce that the family $(\mathcal {U}^{\prime }_j \to \mathcal {X}')_j$ is jointly surjective. Equivalently, the family of quasi-finite morphisms $(U^{\prime }_j/\pi \to X'/\pi )_j$ is jointly surjective. Using standard properties of the Nisnevich topology, we can find a family of étale morphisms $(Y^{\prime }_i \to X')_i$ such that:

1. (1) $(Y^{\prime }_i/\pi \to X'/\pi )_i$ is a Nisnevich cover of $X'/\pi $ ;

2. (2) for every index i there is a index j and a clopen immersion $Z^{\prime }_i \to U^{\prime }_j\times _{X'} Y^{\prime }_i$ such that $Z^{\prime }_i \to Y^{\prime }_i$ is finite and $Z^{\prime }_i/\pi \to Y^{\prime }_i/\pi $ is surjective.

In addition to being finite, the morphism $Z^{\prime }_i \to Y^{\prime }_i$ is flat and étale over $Y^{\prime }_i[\pi ^{-1}]$ . Since $Z^{\prime }_i/\pi \to Y^{\prime }_i/\pi $ is surjective, we may replace $Y^{\prime }_i$ by an open neighbourhood of $Y^{\prime }_i/\pi $ and assume that $Z^{\prime }_i \to Y^{\prime }_i$ is also surjective. In particular, we see that $Z^{\prime }_i[\pi ^{-1}]\to Y^{\prime }_i[\pi ^{-1}]$ is a finite étale covering. If $\mathcal {Y}^{\prime }_i$ and $\mathcal {Z}^{\prime }_i$ denote the adic completions of $Y^{\prime }_i$ and $Z_i$ , Lemma 1.4.17 implies that the morphisms $\mathcal {Z}^{\prime }_i \to \mathcal {Y}^{\prime }_i$ are finite rig-étale coverings. Moreover, the family $(\mathcal {Y}_i'\to \mathcal {X}')_i$ is a Nisnevich cover by point (1) above. Finally, the family $(\mathcal {Z}^{\prime }_i\to \mathcal {X})_i$ refines the initial rig-étale cover as needed.

Proof. Without loss of generality, we may assume that the indexing category of the inverse system $(\mathcal {S}_{\alpha })_{\alpha }$ admits a final object o. We may replace $\mathcal {S}_o$ by the blowup of a finitely generated ideal of definition and each $\mathcal {S}_{\alpha }$ by its strict transform, and assume that the $\mathcal {S}_{\alpha }$ ’s are locally of principal ideal type. The question being local for the Zariski topology on $\mathcal {S}_o$ , we may assume that the formal schemes $\mathcal {S}_{\alpha }$ ’s are affine of principal ideal type. We set $A_{\alpha }=\mathcal {O}(\mathcal {S}_{\alpha })$ and $A=\mathcal {O}(S)$ , and we employ Notation 1.3.9. Using Corollary 1.4.13, it is enough to show that the morphism of sites

$$ \begin{align*}(\mathcal{E}^{\prime}_A,{\mathrm{rig}}\acute{\mathrm{e}}\mathrm{t}) \to \lim_{\alpha}(\mathcal{E}^{\prime}_{A_{\alpha}},{\mathrm{rig}}\acute{\mathrm{e}}\mathrm{t})\end{align*} $$

is an equivalence. Corollary 1.3.10 gives an equivalence on the underlying categories, and it remains to show that the topologies match. For this, we need to show that every rig-étale cover in $\mathcal {E}^{\prime }_A$ can be refined by the image of a rig-étale cover in $\mathcal {E}^{\prime }_{A_{\alpha }}$ for $\alpha $ small enough. This follows readily from Proposition 1.4.19.

Proof. Assertion (1) follows immediately from [Reference Fujiwara and KatoFK18, Chapter II, Proposition 3.2.6] and the definition of $\kappa ^+(s)$ ; see Notation 1.1.13. To prove assertion (2), we may assume that $\mathcal {S}=\mathrm {Spf}(A)$ is affine and prove that the A-algebra $\kappa ^+(\overline {s})$ is a filtered colimit of rig-étale adic A-algebras in the category of adic rings. Let $s\in S$ be the image of $\overline {s}$ . Using assertion (1), we may write

$$ \begin{align*}\kappa^+(s)=\underset{\alpha}{\mathrm{colim}}\, A_{\alpha},\end{align*} $$

in the category of adic rings, where $A_{\alpha }$ are adic A-algebras such that the $\mathrm {Spf}(A_{\alpha })^{\mathrm {rig}}$ are open neighbourhoods of s in $S=\mathrm {Spf}(A)^{\mathrm {rig}}$ . Applying Corollary 1.3.10 to the inductive system $(A_{\alpha })_{\alpha }$ , we see that every rig-étale $\kappa ^+(s)$ -algebra whose zero ideal is saturated is a filtered colimit in the category of adic rings of rig-étale adic A-algebras. Thus, it is enough to show that $\kappa ^+(\overline {s})$ is a filtered colimit of adic rig-étale $\kappa ^+(s)$ -algebras. This follows immediately from Remark 1.4.23 and the following fact. If $L/\kappa (s)$ is a finite separable extension and $R\subset L$ is a sub- $\kappa ^+(s)$ -algebra of finite type with fraction field L, then R is a rig-étale $\kappa ^+(s)$ -algebra. (We leave it to the reader to find a presentation of R as in Definition 1.3.3(1).)

Proof. We only treat the second part. By a standard argument, one reduces to prove the following two assertions.

1. (1) Every étale cover of a geometric rigid point $\overline {s}$ splits.

2. (2) A family $(Y_i \to X)_i$ in $\acute {\mathrm {E}}\mathrm {t}/S$ is an étale cover if, for every geometric rigid point $\overline {s} \to S$ and every S-morphism $\overline {s} \to X$ , there exists i and an X-morphism $\overline {s}\to Y_i$ .

The first assertion follows from Proposition 1.4.19 (and Corollary 1.4.13). The second assertion follows from Definition 1.4.5.

Proof. Fix an étale cover $(U_i \to U)_i$ of U with $U_i$ quasi-compact and quasi-separated. Given an algebraic geometric rigid point $\overline {s} \to S$ , we consider $\overline {u}=\overline {s}\times _SU$ . This is a quasi-compact open rigid analytic subspace of $\overline {s}$ . Thus, $\overline {u}$ is either empty or $\overline {u}\to U$ is an algebraic geometric rigid point of U. In both cases, the morphism $\overline {u}\to U$ factors through $U_i$ for some i. Using Corollary 1.4.20 and Lemma 1.4.26, there exists an étale neighbourhood $V_{\overline {s}}\to S$ of $\overline {s}$ such that $V_{\overline {s}}\times _S U$ factors through $U_i$ . This shows that the base change of the étale cover $(V_{\overline {s}}\to S)_{\overline {s}}$ refines $(U_i \to U)_i$ as needed.

Notation 2.1.1. Given an $\infty $ -category $\mathcal {C}$ , we denote by $\mathcal {P}(\mathcal {C})$ the $\infty $ -category of presheaves on $\mathcal {C}$ with values in the $\infty $ -category $\mathcal {S}$ of Kan complexes. If $\mathcal {C}$ is endowed with a Grothendieck topology $\tau $ , we denote by $\mathrm {Shv}^{(\wedge )}_{\tau }(\mathcal {C})$ the full sub- $\infty $ -category of $\mathcal {P}(\mathcal {C})$ spanned by the $\tau $ -(hyper)sheaves. Thus, $\mathrm {Shv}_{\tau }(\mathcal {C})$ is the $\infty $ -topos associated to the site $(\mathcal {C},\tau )$ as in [Reference LurieLur09, Definition 6.2.2.6] and $\mathrm {Shv}_{\tau }^{\wedge }(\mathcal {C})$ is its hypercompletion in the sense of [Reference LurieLur09, §6.5.2].

Notation 2.1.2. Given an $\infty $ -category $\mathcal {C}$ , we denote by $\mathrm {PSh}(\mathcal {C};\Lambda )$ the $\infty $ -category of presheaves of $\Lambda $ -modules on $\mathcal {C}$ , i.e., contravariant functors from $\mathcal {C}$ to $\mathrm {Mod}_{\Lambda }$ . If $\mathcal {C}$ is endowed with a Grothendieck topology $\tau $ , we denote by $\mathrm {Shv}^{(\wedge )}_{\tau }(\mathcal {C};\Lambda )$ the full sub- $\infty $ -category of $\mathrm {PSh}(\mathcal {C};\Lambda )$ spanned by the $\tau $ -(hyper)sheaves. (For the precise meaning, see Definition 2.3.1 below.) We denote by

(8) $$ \begin{align} \mathrm{L}_{\tau}:\mathrm{PSh}(\mathcal{C};\Lambda) \to \mathrm{Shv}^{(\wedge)}_{\tau}(\mathcal{C};\Lambda) \end{align} $$

the left adjoint to the obvious inclusion. This functor is called $\tau $ -(hyper)sheafification. We also denote by

(9) $$ \begin{align} (-)^{\wedge}:\mathrm{Shv}_{\tau}(\mathcal{C};\Lambda) \to \mathrm{Shv}^{\wedge}_{\tau}(\mathcal{C};\Lambda) \end{align} $$

the left adjoint to the obvious inclusion. This functor is called hypercompletion.