# stacks/stacks-project

Fetching contributors…
Cannot retrieve contributors at this time
5559 lines (5022 sloc) 219 KB
 \input{preamble} % OK, start here. % \begin{document} \title{Formal Algebraic Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent Formal schemes were introduced in \cite{EGA}. A more general version of formal schemes was introduced in \cite{McQuillan} and another in \cite{Yasuda}. Formal algebraic spaces were introduced in \cite{Kn}. Related material and much besides can be found in \cite{Abbes} and \cite{Fujiwara-Kato}. This chapter introduces the notion of formal algebraic spaces we will work with. Our definition is general enough to allow most classes of formal schemes/spaces in the literature as full subcategories. \medskip\noindent Although we do discuss the comparison of some of these alternative theories with ours, we do not always give full details when it is not necessary for the logical development of the theory. \medskip\noindent Besides introducing formal algebraic spaces, we also prove a few very basic properties and we discuss a few types of morphisms. \section{Formal schemes \a la EGA} \label{section-formal-schemes-EGA} \noindent In this section we review the construction of formal schemes in \cite{EGA}. This notion, although very useful in algebraic geometry, may not always be the correct one to consider. Perhaps it is better to say that in the setup of the theory a number of choices are made, where for different purposes others might work better. And indeed in the literature one can find many different closely related theories adapted to the problem the authors may want to consider. Still, one of the major advantages of the theory as sketched here is that one gets to work with definite geometric objects. \medskip\noindent Before we start we should point out an issue with the sheaf condition for sheaves of topological rings or more generally sheaves of topological spaces. Namely, the big categories \begin{enumerate} \item category of topological spaces, \item category of topological groups, \item category of topological rings, \item category of topological modules over a given topological ring, \end{enumerate} endowed with their natural forgetful functors to $\textit{Sets}$ are not examples of types of algebraic structures as defined in Sheaves, Section \ref{sheaves-section-algebraic-structures}. Thus we cannot blithely apply to them the machinery developed in that chapter. On the other hand, each of the categories listed above has limits and equalizers and the forgetful functor to sets, groups, rings, modules commutes with them (see Topology, Lemmas \ref{topology-lemma-limits}, \ref{topology-lemma-topological-group-limits}, \ref{topology-lemma-topological-ring-limits}, and \ref{topology-lemma-topological-module-limits}). Thus we can define the notion of a sheaf as in Sheaves, Definition \ref{sheaves-definition-sheaf-values-in-category} and the underlying presheaf of sets, groups, rings, or modules is a sheaf. The key difference is that for an open covering $U = \bigcup_{i \in I} U_i$ the diagram $$\xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) }$$ has to be an equalizer diagram in the category of topological spaces, topological groups, topological rings, topological modules, i.e., that the first map identifies $\mathcal{F}(U)$ with a subspace of $\prod_{i \in I} \mathcal{F}(U_i)$ which is endowed with the product topology. \medskip\noindent The stalk $\mathcal{F}_x$ of a sheaf $\mathcal{F}$ of topological spaces, topological groups, topological rings, or topological modules at a point $x \in X$ is defined as the colimit over open neighbourhoods $$\mathcal{F}_x = \colim_{x\in U} \mathcal{F}(U)$$ in the corresponding category. This is the same as taking the colimit on the level of sets, groups, rings, or modules (see Topology, Lemmas \ref{topology-lemma-colimits}, \ref{topology-lemma-topological-group-colimits}, \ref{topology-lemma-topological-ring-colimits}, and \ref{topology-lemma-topological-module-colimits}) but comes equipped with a topology. Warning: the topology one gets depends on which category one is working with, see Examples, Section \ref{examples-section-colimit-topology}. One can sheafify presheaves of topological spaces, topological groups, topological rings, or topological modules and taking stalks commutes with this operation, see Remark \ref{remark-sheafification-of-presheaves-in-top}. \medskip\noindent Let $f : X \to Y$ be a continuous map of topological spaces. There is a functor $f_*$ from the category of sheaves of topological spaces, topological groups, topological rings, topological modules, to the corresponding category of sheaves on $Y$ which is defined by setting $f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}V)$ as usual. (We delay discussing the pullback in this setting till later.) We define the notion of an $f$-map $\xi : \mathcal{G} \to \mathcal{F}$ between a sheaf of topological spaces $\mathcal{G}$ on $Y$ and a sheaf of topological spaces $\mathcal{F}$ on $X$ in exactly the same manner as in Sheaves, Definition \ref{sheaves-definition-f-map} with the additional constraint that $\xi_V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)$ be continuous for every open $V \subset Y$. We have $$\{f\text{-maps from }\mathcal{G}\text{ to }\mathcal{F}\} = \text{Mor}_{\Sh(Y, \textit{Top})}(\mathcal{G}, f_*\mathcal{F})$$ as in Sheaves, Lemma \ref{sheaves-lemma-f-map}. Similarly for sheaves of topological groups, topological rings, topological modules. Finally, let $\xi : \mathcal{G} \to \mathcal{F}$ be an $f$-map as above. Then given $x \in X$ with image $y = f(x)$ there is a continuous map $$\xi_x : \mathcal{G}_y \longrightarrow \mathcal{F}_x$$ of stalks defined in exactly the same manner as in the discussion following Sheaves, Definition \ref{sheaves-definition-composition-f-maps}. \medskip\noindent Using the discussion above, we can define a category $LTRS$ of locally topologically ringed spaces''. An object is a pair $(X, \mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf of topological rings $\mathcal{O}_X$ whose stalks $\mathcal{O}_{X, x}$ are local rings (if one forgets about the topology). A morphism $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ of $LTRS$ is a pair $(f, f^\sharp)$ where $f : X \to Y$ is a continuous map of topological spaces and $f^\sharp : \mathcal{O}_Y \to \mathcal{O}_X$ is an $f$-map such that for every $x \in X$ the induced map $$f^\sharp_x : \mathcal{O}_{Y, f(x)} \longrightarrow \mathcal{O}_{X, x}$$ is a local homomorphism of local rings (forgetting about the topologies). The composition works in exactly the same manner as composition of morphisms of locally ringed spaces. \medskip\noindent Assume now that the topological space $X$ has a basis consisting of quasi-compact opens. Given a sheaf $\mathcal{F}$ of sets, groups, rings, modules over a ring, one can endow $\mathcal{F}$ with the structure of a sheaf of topological spaces, topological groups, topological rings, topological modules. Namely, if $U \subset X$ is quasi-compact open, we endow $\mathcal{F}(U)$ with the discrete topology. If $U \subset X$ is arbitrary, then we choose an open covering $U = \bigcup_{i \in I} U_i$ by quasi-compact opens and we endow $\mathcal{F}(U)$ with the induced topology from $\prod_{i \in I} \mathcal{F}(U_i)$ (as we should do according to our discussion above). The reader may verify (omitted) that we obtain a sheaf of topological spaces, topological groups, topological rings, topological modules in this fashion. Let us say that a sheaf of topological spaces, topological groups, topological rings, topological modules is {\it pseudo-discrete} if the topology on $\mathcal{F}(U)$ is discrete for every quasi-compact open $U \subset X$. Then the construction given above is an adjoint to the forgetful functor and induces an equivalence between the category of sheaves of sets and the category of pseudo-discrete sheaves of topological spaces (similarly for groups, rings, modules). \medskip\noindent Grothendieck and Dieudonn\'e first define formal affine schemes. These correspond to admissible topological rings $A$, see More on Algebra, Definition \ref{more-algebra-definition-topological-ring}. Namely, given $A$ one considers a fundamental system $I_\lambda$ of ideals of definition for the ring $A$. (In any admissible topological ring the family of all ideals of definition forms a fundamental system.) For each $\lambda$ we can consider the scheme $\Spec(A/I_\lambda)$. For $I_\lambda \subset I_\mu$ the induced morphism $$\Spec(A/I_\mu) \to \Spec(A/I_\lambda)$$ is a thickening because $I_\mu^n \subset I_\lambda$ for some $n$. Another way to see this, is to notice that the image of each of the maps $$\Spec(A/I_\lambda) \to \Spec(A)$$ is a homeomorphism onto the set of open prime ideals of $A$. This motivates the definition $$\text{Spf}(A) = \{\text{open prime ideals }\mathfrak p \subset A\}$$ endowed with the topology coming from $\Spec(A)$. For each $\lambda$ we can consider the structure sheaf $\mathcal{O}_{\Spec(A/I_\lambda)}$ as a sheaf on $\text{Spf}(A)$. Let $\mathcal{O}_\lambda$ be the corresponding pseudo-discrete sheaf of topological rings, see above. Then we set $$\mathcal{O}_{\text{Spf}(A)} = \lim \mathcal{O}_\lambda$$ where the limit is taken in the category of sheaves of topological rings. The pair $(\text{Spf}(A), \mathcal{O}_{\text{Spf}(A)})$ is called the {\it formal spectrum} of $A$. \medskip\noindent At this point one should check several things. The first is that the stalks $\mathcal{O}_{\text{Spf}(A), x}$ are local rings (forgetting about the topology). The second is that given $f \in A$, for the corresponding open $D(f) \cap \text{Spf}(A)$ we have $$\Gamma(D(f) \cap \text{Spf}(A), \mathcal{O}_{\text{Spf}(A)}) = A_{\{f\}} = \lim (A/I_\lambda)_f$$ as topological rings where $I_\lambda$ is a fundamental system of ideals of definition as above. Moreover, the ring $A_{\{f\}}$ is admissible too and $(\text{Spf}(A_f), \mathcal{O}_{\text{Spf}(A_{\{f\}})})$ is isomorphic to $(D(f) \cap \text{Spf}(A), \mathcal{O}_{\text{Spf}(A)}|_{D(f) \cap \text{Spf}(A)})$. Finally, given a pair of admissible topological rings $A, B$ we have \begin{equation} \label{equation-morphisms-affine-formal-schemes} \Mor_{LTRS}((\text{Spf}(B), \mathcal{O}_{\text{Spf}(B)}), (\text{Spf}(A), \mathcal{O}_{\text{Spf}(A)})) = \Hom_{cont}(A, B) \end{equation} where $LTRS$ is the category of locally topologically ringed spaces'' as defined above. \medskip\noindent Having said this, in \cite{EGA} a {\it formal scheme} is defined as a pair $(\mathfrak X, \mathcal{O}_\mathfrak X)$ where $\mathfrak X$ is a topological space and $\mathcal{O}_\mathfrak X$ is a sheaf of topological rings such that every point has an open neighbourhood isomorphic (in $LTRS$) to an affine formal scheme. A {\it morphism of formal schemes} $f : (\mathfrak X, \mathcal{O}_\mathfrak X) \to (\mathfrak Y, \mathcal{O}_\mathfrak Y)$ is a morphism in the category $LTRS$. \medskip\noindent Let $A$ be a ring endowed with the discrete topology. Then $A$ is admissible and the formal scheme $\text{Spf}(A)$ is equal to $\Spec(A)$. The structure sheaf $\mathcal{O}_{\text{Spf}(A)}$ is the pseudo-discrete sheaf of topological rings associated to $\mathcal{O}_{\Spec(A)}$, in other words, its underlying sheaf of rings is equal to $\mathcal{O}_{\Spec(A)}$ and the ring $\mathcal{O}_{\text{Spf}(A)}(U) = \mathcal{O}_{\Spec(A)}(U)$ over a quasi-compact open $U$ has the discrete topology, but not in general. Thus we can associate to every affine scheme a formal affine scheme. In exactly the same manner we can start with a general scheme $(X, \mathcal{O}_X)$ and associate to it $(X, \mathcal{O}'_X)$ where $\mathcal{O}'_X$ is the pseudo-discrete sheaf of topological rings whose underlying sheaf of rings is $\mathcal{O}_X$. This construction is compatible with morphisms and defines a functor \begin{equation} \label{equation-compare-schemes-formal-schemes} \textit{Schemes} \longrightarrow \textit{Formal Schemes} \end{equation} It follows in a straightforward manner from (\ref{equation-morphisms-affine-formal-schemes}) that this functor is fully faithful. \medskip\noindent Let $\mathfrak X$ be a formal scheme. Let us define the {\it size} of the formal scheme by the formula $\text{size}(\mathfrak X) = \max(\aleph_0, \kappa_1, \kappa_2)$ where $\kappa_1$ is the cardinality of the formal affine opens of $\mathfrak X$ and $\kappa_2$ is the supremum of the cardinalities of $\mathcal{O}_\mathfrak X(\mathfrak U)$ where $\mathfrak U \subset \mathfrak X$ is such a formal affine open. \begin{lemma} \label{lemma-fully-faithful} Choose a category of schemes $\Sch_\alpha$ as in Sets, Lemma \ref{sets-lemma-construct-category}. Given a formal scheme $\mathfrak X$ let $$h_\mathfrak X : (\Sch_\alpha)^{opp} \longrightarrow \textit{Sets},\quad h_\mathfrak X(S) = \Mor_{\textit{Formal Schemes}}(S, \mathfrak X)$$ be its functor of points. Then we have $$\Mor_{\textit{Formal Schemes}}(\mathfrak X, \mathfrak Y) = \Mor_{\textit{PSh}(\Sch_\alpha)}(h_\mathfrak X, h_\mathfrak Y)$$ provided the size of $\mathfrak X$ is not too large. \end{lemma} \begin{proof} First we observe that $h_\mathfrak X$ satisfies the sheaf property for the Zariski topology for any formal scheme $\mathfrak X$ (see Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}). This follows from the local nature of morphisms in the category of formal schemes. Also, for an open immersion $\mathfrak V \to \mathfrak W$ of formal schemes, the corresponding transformation of functors $h_\mathfrak V \to h_\mathfrak W$ is injective and representable by open immersions (see Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}). Choose an open covering $\mathfrak X = \bigcup \mathfrak U_i$ of a formal scheme by affine formal schemes $\mathfrak U_i$. Then the collection of functors $h_{\mathfrak U_i}$ covers $h_\mathfrak X$ (see Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}). Finally, note that $$h_{\mathfrak U_i} \times_{h_\mathfrak X} h_{\mathfrak U_j} = h_{\mathfrak U_i \cap \mathfrak U_j}$$ Hence in order to give a map $h_\mathfrak X \to h_\mathfrak Y$ is equivalent to giving a family of maps $h_{\mathfrak U_i} \to h_\mathfrak Y$ which agree on overlaps. Thus we can reduce the bijectivity (resp.\ injectivity) of the map of the lemma to bijectivity (resp.\ injectivity) for the pairs $(\mathfrak U_i, \mathfrak Y)$ and injectivity (resp.\ nothing) for $(\mathfrak U_i \cap \mathfrak U_j, \mathfrak Y)$. In this way we reduce to the case where $\mathfrak X$ is an affine formal scheme. Say $\mathfrak X = \text{Spf}(A)$ for some admissible topological ring $A$. Also, choose a fundamental system of ideals of definition $I_\lambda \subset A$. \medskip\noindent We can also localize on $\mathfrak Y$. Namely, suppose that $\mathfrak V \subset \mathfrak Y$ is an open formal subscheme and $\varphi : h_\mathfrak X \to h_\mathfrak Y$. Then $$h_\mathfrak V \times_{h_\mathfrak Y, \varphi} h_\mathfrak X \to h_\mathfrak X$$ is representable by open immersions. Pulling back to $\Spec(A/I_\lambda)$ for all $\lambda$ we find an open subscheme $U_\lambda \subset \Spec(A/I_\lambda)$. However, for $I_\lambda \subset I_\mu$ the morphism $\Spec(A/I_\lambda) \to \Spec(A/I_\mu)$ pulls back $U_\mu$ to $U_\lambda$. Thus these glue to give an open formal subscheme $\mathfrak U \subset \mathfrak X$. A straightforward argument (omitted) shows that $$h_\mathfrak U = h_\mathfrak V \times_{h_\mathfrak Y} h_\mathfrak X$$ In this way we see that given an open covering $\mathfrak Y = \bigcup \mathfrak V_j$ and a transformation of functors $\varphi : h_\mathfrak X \to h_\mathfrak Y$ we obtain a corresponding open covering of $\mathfrak X$. Since $\mathfrak X$ is affine, we can refine this covering by a finite open covering $\mathfrak X = \mathfrak U_1 \cup \ldots \cup \mathfrak U_n$ by affine formal subschemes. In other words, for each $i$ there is a $j$ and a map $\varphi_i : h_{\mathfrak U_i} \to h_{\mathfrak V_j}$ such that $$\xymatrix{ h_{\mathfrak U_i} \ar[r]_{\varphi_i} \ar[d] & h_{\mathfrak V_j} \ar[d] \\ h_{\mathfrak X} \ar[r]^\varphi & h_\mathfrak Y }$$ commutes. With a few additional arguments (which we omit) this implies that it suffices to prove the bijectivity of the lemma in case both $\mathfrak X$ and $\mathfrak Y$ are affine formal schemes. \medskip\noindent Assume $\mathfrak X$ and $\mathfrak Y$ are affine formal schemes. Say $\mathfrak X = \text{Spf}(A)$ and $\mathfrak Y = \text{Spf}(B)$. Let $\varphi : h_\mathfrak X \to h_\mathfrak Y$ be a transformation of functors. Let $I_\lambda \subset A$ be a fundamental system of ideals of definition. The canonical inclusion morphism $i_\lambda : \Spec(A/I_\lambda) \to \mathfrak X$ maps to a morphism $\varphi(i_\lambda) : \Spec(A/I_\lambda) \to \mathfrak Y$. By (\ref{equation-morphisms-affine-formal-schemes}) this corresponds to a continuous map $\chi_\lambda : B \to A/I_\lambda$. Since $\varphi$ is a transformation of functors it follows that for $I_\lambda \subset I_\mu$ the composition $B \to A/I_\lambda \to A/I_\mu$ is equal to $\chi_\mu$. In other words we obtain a ring map $$\chi = \lim \chi_\lambda : B \longrightarrow \lim A/I_\lambda = A$$ This is a continuous homomorphism because the inverse image of $I_\lambda$ is open for all $\lambda$ (as $A/I_\lambda$ has the discrete topology and $\chi_\lambda$ is continuous). Thus we obtain a morphism $\text{Spf}(\chi) : \mathfrak X \to \mathfrak Y$ by (\ref{equation-morphisms-affine-formal-schemes}). We omit the verification that this construction is the inverse to the map of the lemma in this case. \medskip\noindent Set theoretic remarks. To make this work on the given category of schemes $\Sch_\alpha$ we just have to make sure all the schemes used in the proof above are isomorphic to objects of $\Sch_\alpha$. In fact, a careful analysis shows that it suffices if the schemes $\Spec(A/I_\lambda)$ occurring above are isomorphic to objects of $\Sch_\alpha$. For this it certainly suffices to assume the size of $\mathfrak X$ is at most the size of a scheme contained in $\Sch_\alpha$. \end{proof} \begin{lemma} \label{lemma-formal-scheme-sheaf-fppf} Let $\mathfrak X$ be a formal scheme. The functor of points $h_\mathfrak X$ (see Lemma \ref{lemma-fully-faithful}) satisfies the sheaf condition for fpqc coverings. \end{lemma} \begin{proof} Topologies, Lemma \ref{topologies-lemma-sheaf-property-fpqc} reduces us to the case of a Zariski covering and a covering $\{\Spec(S) \to \Spec(R)\}$ with $R \to S$ faithfully flat. We observed in the proof of Lemma \ref{lemma-fully-faithful} that $h_\mathfrak X$ satisfies the sheaf condition for Zariski coverings. \medskip\noindent Suppose that $R \to S$ is a faithfully flat ring map. Denote $\pi : \Spec(S) \to \Spec(R)$ the corresponding morphism of schemes. It is surjective and flat. Let $f : \Spec(S) \to \mathfrak X$ be a morphism such that $f \circ \text{pr}_1 = f \circ \text{pr}_2$ as maps $\Spec(S \otimes_R S) \to \mathfrak X$. By Descent, Lemma \ref{descent-lemma-equiv-fibre-product} we see that as a map on the underlying sets $f$ is of the form $f = g \circ \pi$ for some (set theoretic) map $g : \Spec(R) \to \mathfrak X$. By Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology} and the fact that $f$ is continuous we see that $g$ is continuous. \medskip\noindent Pick $y \in \Spec(R)$. Choose $\mathfrak U \subset \mathfrak X$ an affine formal open subscheme containing $g(y)$. Say $\mathfrak U = \text{Spf}(A)$ for some admissible topological ring $A$. By the above we may choose an $r \in R$ such that $y \in D(r) \subset g^{-1}(\mathfrak U)$. The restriction of $f$ to $\pi^{-1}(D(r))$ into $\mathfrak U$ corresponds to a continuous ring map $A \to S_r$ by (\ref{equation-morphisms-affine-formal-schemes}). The two induced ring maps $A \to S_r \otimes_{R_r} S_r = (S \otimes_R S)_r$ are equal by assumption on $f$. Note that $R_r \to S_r$ is faithfully flat. By Descent, Lemma \ref{descent-lemma-ff-exact} the equalizer of the two arrows $S_r \to S_r \otimes_{R_r} S_r$ is $R_r$. We conclude that $A \to S_r$ factors uniquely through a map $A \to R_r$ which is also continuous as it has the same (open) kernel as the map $A \to S_r$. This map in turn gives a morphism $D(r) \to \mathfrak U$ by (\ref{equation-morphisms-affine-formal-schemes}). \medskip\noindent What have we proved so far? We have shown that for any $y \in \Spec(R)$ there exists a standard affine open $y \in D(r) \subset \Spec(R)$ such that the morphism $f|_{\pi^{-1}(D(r))} : \pi^{-1}(D(r)) \to \mathfrak X$ factors uniquely though some morphism $D(r) \to \mathfrak X$. We omit the verification that these morphisms glue to the desired morphism $\Spec(R) \to \mathfrak X$. \end{proof} \begin{remark}[McQuillan's variant] \label{remark-mcquillan} There is a variant of the construction of formal schemes due to McQuillan, see \cite{McQuillan}. He suggests a slight weakening of the condition of admissibility. Namely, recall that an admissible topological ring is a complete (and separated by our conventions) topological ring $A$ which is linearly topologized such that there exists an ideal of definition: an open ideal $I$ such that any neighbourhood of $0$ contains $I^n$ for some $n \geq 1$. McQuillan works with what we will call {\it weakly admissible} topological rings. A weakly admissible topological ring $A$ is a complete (and separated by our conventions) topological ring which is linearly topologized such that there exists an {\it weak ideal of definition}: an open ideal $I$ such that for all $f \in I$ we have $f^n \to 0$ for $n \to \infty$. Similarly to the admissible case, if $I$ is a weak ideal of definition and $J \subset A$ is an open ideal, then $I \cap J$ is a weak ideal of definition. Thus the weak ideals of definition form a fundamental system of open neighbourhoods of $0$ and one can proceed along much the same route as above to define a larger category of formal schemes based on this notion. The analogues of Lemmas \ref{lemma-fully-faithful} and \ref{lemma-formal-scheme-sheaf-fppf} still hold in this setting (with the same proof). \end{remark} \begin{remark}[Sheafification of presheaves of topological spaces] \label{remark-sheafification-of-presheaves-in-top} \begin{reference} \cite{Gray} \end{reference} In this remark we briefly discuss sheafification of presheaves of topological spaces. The exact same arguments work for presheaves of topological abelian groups, topological rings, and topological modules (over a given topological ring). In order to do this in the correct generality let us work over a site $\mathcal{C}$. The reader who is interested in the case of (pre)sheaves over a topological space $X$ should think of objects of $\mathcal{C}$ as the opens of $X$, of morphisms of $\mathcal{C}$ as inclusions of opens, and of coverings in $\mathcal{C}$ as coverings in $X$, see Sites, Example \ref{sites-example-site-topological}. Denote $\Sh(\mathcal{C}, \textit{Top})$ the category of sheaves of topological spaces on $\mathcal{C}$ and denote $\textit{PSh}(\mathcal{C}, \textit{Top})$ the category of presheaves of topological spaces on $\mathcal{C}$. Let $\mathcal{F}$ be a presheaf of topological spaces on $\mathcal{C}$. The sheafification $\mathcal{F}^\#$ should satisfy the formula $$\Mor_{\textit{PSh}(\mathcal{C}, \textit{Top})}(\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(\mathcal{C}, \textit{Top})}(\mathcal{F}^\#, \mathcal{G})$$ functorially in $\mathcal{G}$ from $\Sh(\mathcal{C}, \textit{Top})$. In other words, we are trying to construct the left adjoint to the inclusion functor $\Sh(\mathcal{C}, \textit{Top}) \to \textit{PSh}(\mathcal{C}, \textit{Top})$. We first claim that $\Sh(\mathcal{C}, \textit{Top})$ has limits and that the inclusion functor commutes with them. Namely, given a category $\mathcal{I}$ and a functor $i \mapsto \mathcal{G}_i$ into $\Sh(\mathcal{C}, \textit{Top})$ we simply define $$(\lim \mathcal{G}_i)(U) = \lim \mathcal{G}_i(U)$$ where we take the limit in the category of topological spaces (Topology, Lemma \ref{topology-lemma-limits}). This defines a sheaf because limits commute with limits (Categories, Lemma \ref{categories-lemma-colimits-commute}) and in particular products and equalizers (which are the operations used in the sheaf axiom). Finally, a morphism of presheaves from $\mathcal{F} \to \lim \mathcal{G}_i$ is clearly the same thing as a compatible system of morphisms $\mathcal{F} \to \mathcal{G}_i$. In other words, the object $\lim \mathcal{G}_i$ is the limit in the category of presheaves of topological spaces and a fortiori in the category of sheaves of topological spaces. Our second claim is that any morphism of presheaves $\mathcal{F} \to \mathcal{G}$ with $\mathcal{G}$ an object of $\Sh(\mathcal{C}, \textit{Top})$ factors through a subsheaf $\mathcal{G}' \subset \mathcal{G}$ whose size is bounded. Here we define the {\it size} $|\mathcal{H}|$ of a sheaf of topological spaces $\mathcal{H}$ to be the cardinal $\sup_{U \in \Ob(\mathcal{C})} |\mathcal{H}(U)|$. To prove our claim we let $$\mathcal{G}'(U) = \left\{ \quad s \in \mathcal{G}(U) \quad \middle| \quad \begin{matrix} \text{there exists a covering }\{U_i \to U\}_{i \in I} \\ \text{such that } s|_{U_i} \in \Im(\mathcal{F}(U_i) \to \mathcal{G}(U_i)) \end{matrix} \quad \right\}$$ We endow $\mathcal{G}'(U)$ with the induced topology. Then $\mathcal{G}'$ is a sheaf of topological spaces (details omitted) and $\mathcal{G}' \to \mathcal{G}$ is a morphism through which the given map $\mathcal{F} \to \mathcal{G}$ factors. Moreover, the size of $\mathcal{G}'$ is bounded by some cardinal $\kappa$ depending only on $\mathcal{C}$ and the presheaf $\mathcal{F}$ (hint: use that coverings in $\mathcal{C}$ form a set by our conventions). Putting everything together we see that the assumptions of Categories, Theorem \ref{categories-theorem-adjoint-functor} are satisfied and we obtain sheafification as the left adjoint of the inclusion functor from sheaves to presheaves. Finally, let $p$ be a point of the site $\mathcal{C}$ given by a functor $u : \mathcal{C} \to \textit{Sets}$, see Sites, Definition \ref{sites-definition-point}. For a topological space $M$ the presheaf defined by the rule $$U \mapsto \text{Map}(u(U), M) = \prod\nolimits_{x \in u(U)} M$$ endowed with the product topology is a sheaf of topological spaces. Hence the exact same argument as given in the proof of Sites, Lemma \ref{sites-lemma-point-pushforward-sheaf} shows that $\mathcal{F}_p = \mathcal{F}^\#_p$, in other words, sheafification commutes with taking stalks at a point. \end{remark} \section{Conventions and notation} \label{section-conventions} \noindent The conventions from now on will be similar to the conventions in Properties of Spaces, Section \ref{spaces-properties-section-conventions}. Thus from now on the standing assumption is that all schemes are contained in a big fppf site $\Sch_{fppf}$. And all rings $A$ considered have the property that $\Spec(A)$ is (isomorphic) to an object of this big site. For topological rings $A$ we assume only that all discrete quotients have this property (but usually we assume more, compare with Remark \ref{remark-set-theoretic}). \medskip\noindent Let $S$ be a scheme and let $X$ be a space'' over $S$, i.e., a sheaf on $(\Sch/S)_{fppf}$. In this chapter we will write $X \times_S X$ for the product of $X$ with itself in the category of sheaves on $(\Sch/S)_{fppf}$ instead of $X \times X$. Moreover, if $X$ and $Y$ are spaces'' then we say "let $f : X \to Y$ be a morphism" to indicate that $f$ is a natural transformation of functors, i.e., a map of sheaves on $(\Sch/S)_{fppf}$. Similarly, if $U$ is a scheme over $S$ and $X$ is a space'' over $S$, then we say "let $f : U \to X$ be a morphism" or "let $g : X \to U$ be a morphism" to indicate that $f$ or $g$ is a map of sheaves $h_U \to X$ or $X \to h_U$ where $h_U$ is as in Categories, Example \ref{categories-example-hom-functor}. \section{Topological rings and modules} \label{section-topological-module} \noindent This section is a continuation of More on Algebra, Section \ref{more-algebra-section-topological-ring}. Let $R$ be a topological ring and let $M$ be a linearly topologized $R$-module. When we say {\it let $M_\lambda$ be a fundamental system of open submodules}'' we will mean that each $M_\lambda$ is an open submodule and that any neighbourhood of $0$ contains one of the $M_\lambda$. In other words, this means that $M_\lambda$ is a fundamental system of neighbourhoods of $0$ in $M$ consisting of submodules. Similarly, if $R$ is a linearly topologized ring, then we say {\it let $I_\lambda$ be a fundamental system of open ideals}'' to mean that $I_\lambda$ is a fundamental system of neighbourhoods of $0$ in $R$ consisting of ideals. \begin{example} \label{example-what-does-it-mean} Let $R$ be a linearly topologized ring and let $M$ be a linearly topologized $A$-module. Let $I_\lambda$ be a fundamental system of open ideals in $R$ and let $M_\mu$ be a fundamental system of open submodules of $M$. The continuity of $+ : M \times M \to M$ is automatic and the continuity of $R \times M \to M$ signifies $$\forall f, x, \mu\ \exists \lambda, \nu,\ (f + I_\lambda)(x + M_\nu) \subset fx + M_\mu$$ Since $fM_\nu + I_\lambda M_\nu \subset M_\mu$ if $M_\nu \subset M_\mu$ we see that the condition is equivalent to $$\forall x, \mu\ \exists \lambda\ I_\lambda x \subset M_\mu$$ However, it need not be the case that given $\mu$ there is a $\lambda$ such that $I_\lambda M \subset M_\mu$. For example, consider $R = k[[t]]$ with the $t$-adic topology and $M = \bigoplus_{n \in \mathbf{N}} R$ with fundamental system of open submodules given by $$M_m = \bigoplus\nolimits_{n \in \mathbf{N}} t^{nm}R$$ Since every $x \in M$ has finitely many nonzero coordinates we see that, given $m$ and $x$ there exists a $k$ such that $t^k x \in M_m$. Thus $M$ is a linearly topologized $R$-module, but it isn't true that given $m$ there is a $k$ such that $t^kM \subset M_m$. On the other hand, if $R \to S$ is a continuous map of linearly topologized rings, then the corresponding statement does hold, i.e., for every open ideal $J \subset S$ there exists an open ideal $I \subset R$ such that $IS \subset J$ (as the reader can easily deduce from continuity of the map $R \to S$). \end{example} \begin{lemma} \label{lemma-closed} Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module and let $M_\lambda$, $\lambda \in \Lambda$ be a fundamental system of open submodules. Let $N \subset M$ be a submodule. The closure of $N$ is $\bigcap_{\lambda \in \Lambda} (N + M_\lambda)$. \end{lemma} \begin{proof} Since each $N + M_\lambda$ is open, it is also closed. Hence the intersection is closed. If $x \in M$ is not in the closure of $N$, then $(x + M_\lambda) \cap N = 0$ for some $\lambda$. Hence $x \not \in N + M_\lambda$. This proves the lemma. \end{proof} \noindent Unless otherwise mentioned we endow submodules and quotient modules with the induced topology. Let $M$ be a linearly topologized module over a topological ring $R$, and let $0 \to N \to M \to Q \to 0$ is a short exact sequence of $R$-modules. If $M_\lambda$ is a fundamental system of open submodules of $M$, then $N \cap M_\lambda$ is a fundamental system of open submodules of $N$. If $\pi : M \to Q$ is the quotient map, then $\pi(M_\lambda)$ is a fundamental system of open submodules of $Q$. In particular these induced topologies are linear topologies. \begin{lemma} \label{lemma-closure} Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Then \begin{enumerate} \item $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge$ is exact, and \item $N^\wedge$ is the closure of the image of $N \to M^\wedge$. \end{enumerate} \end{lemma} \begin{proof} Let $M_\lambda$, $\lambda \in \Lambda$ be a fundamental system of open submodules. Then $N \cap M_\lambda$ is a fundamental system of open submodules of $N$ and $M_\lambda + N/N$ is a fundamental system of open submodules of $M/N$. Thus we see that (1) follows from the exactness of the sequences $$0 \to N/N \cap M_\lambda \to M/M_\lambda \to M/(M_\lambda + N) \to 0$$ and the fact that taking limits commutes with limits. The second statement follows from this and the fact that $N \to N^\wedge$ has dense image and that the kernel of $M^\wedge \to (M/N)^\wedge$ is closed. \end{proof} \begin{lemma} \label{lemma-quotient-by-closed} Let $R$ be a topological ring. Let $M$ be a complete, linearly topologized $R$-module. Let $N \subset M$ be a closed submodule. If $M$ has a countable fundamental system of neighbourhoods of $0$, then $M/N$ is complete and the map $M \to M/N$ is open. \end{lemma} \begin{proof} Let $M_n$, $n \in \mathbf{N}$ be a fundamental system of open submodules of $M$. We may assume $M_{n + 1} \subset M_n$ for all $n$. The $(M_n + N)/N$ is a fundamental system in $M/N$. Hence we have to show that $M/N = \lim M/(M_n + N)$. Consider the short exact sequences $$0 \to N/N \cap M_n \to M/M_n \to M/(M_n + N) \to 0$$ Since the transition maps of the system $\{N/N\cap M_n\}$ are surjective we see that $M = \lim M/M_n$ (by completeness of $M$) surjects onto $\lim M/(M_n + N)$ by Algebra, Lemma \ref{algebra-lemma-ML-exact-sequence}. As $N$ is closed we see that the kernel of $M \to \lim M/(M_n + N)$ is $N$ (see Lemma \ref{lemma-closed}). Finally, $M \to M/N$ is open by definition of the quotient topology. \end{proof} \begin{lemma} \label{lemma-ses} \begin{reference} \cite[Theorem 8.1]{Ma} \end{reference} Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Assume $M$ has a countable fundamental system of neighbourhoods of $0$. Then \begin{enumerate} \item $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge \to 0$ is exact, \item $N^\wedge$ is the closure of the image of $N \to M^\wedge$, \item $M^\wedge \to (M/N)^\wedge$ is open. \end{enumerate} \end{lemma} \begin{proof} We have $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge$ is exact and statement (2) by Lemma \ref{lemma-closure}. This produces a canonical map $c : M^\wedge/N^\wedge \to (M/N)^\wedge$. The module $M^\wedge/N^\wedge$ is complete and $M^\wedge \to M^\wedge/N^\wedge$ is open by Lemma \ref{lemma-quotient-by-closed}. By the universal property of completion we obtain a canonical map $b : (M/N)^\wedge \to M^\wedge/N^\wedge$. Then $b$ and $c$ are mutually inverse as they are on a dense subset. \end{proof} \begin{definition} \label{definition-toplogy-tensor-product} Let $R$ be a topological ring. Let $M$ and $N$ be linearly topologized $R$-modules. The {\it tensor product} of $M$ and $N$ is the (usual) tensor product $M \otimes_R N$ endowed with the linear topology defined by declaring $$\Im(M_\mu \otimes_R N + M \otimes_R N_\nu \longrightarrow M \otimes_R N)$$ to be a fundamental system of open submodules, where $M_\mu \subset M$ and $N_\nu \subset N$ run through fundamental systems of open submodules in $M$ and $N$. The {\it completed tensor product} $$M \widehat{\otimes}_R N = \lim M \otimes_R N/(M_\mu \otimes_R N + M \otimes_R N_\nu) = \lim M/M_\mu \otimes_R N/N_\nu$$ is the completion of the tensor product. \end{definition} \noindent Observe that the topology on $R$ is immaterial for the construction of the tensor product or the completed tensor product. If $R \to A$ and $R \to B$ are continuous maps of linearly topologized rings, then the construction above gives a tensor product $A \otimes_R B$ and a completed tensor product $A \widehat{\otimes}_R B$. \medskip\noindent We record here the notions introduced in Remark \ref{remark-mcquillan}. \begin{definition} \label{definition-weakly-admissible} Let $A$ be a linearly topologized ring. \begin{enumerate} \item An element $f \in A$ is called {\it topologically nilpotent} if $f^n \to 0$ as $n \to \infty$. \item A {\it weak ideal of definition} for $A$ is an open ideal $I \subset A$ consisting entirely of topologically nilpotent elements. \item We say $A$ is {\it weakly pre-admissible} if $A$ has a weak ideal of definition. \item We say $A$ is {\it weakly admissible} if $A$ is weakly pre-admissible and complete\footnote{By our conventions this includes separated.}. \end{enumerate} \end{definition} \noindent Given a weak ideal of definition $I$ in a linearly topologized ring $A$ and an open ideal $J$ the intersection $I \cap J$ is a weak ideal of definition. Hence if there is one weak ideal of definition, then there is a fundamental system of open ideals consisting of weak ideals of definition. In particular, given a weakly admissible topological ring $A$ then $A = \lim A/I_\lambda$ where $\{I_\lambda\}$ is a fundamental system of weak ideals of definition. \begin{lemma} \label{lemma-topologically-nilpotent} Let $\varphi : A \to B$ be a continuous map of linearly topologized rings. \begin{enumerate} \item If $f \in A$ is topologically nilpotent, then $\varphi(f)$ is topologically nilpotent. \item If $I \subset A$ consists of topologically nilpotent elements, then the closure of $\varphi(I)B$ consists of topologically nilpotent elements. \end{enumerate} \end{lemma} \begin{proof} Part (1) is clear. Let $g$ be an element of the closure of $\varphi(I)B$. Let $J \subset B$ be an open ideal. We have to show $g^e \in J$ for some $e$. We have $g \in \varphi(I)B + J$ by Lemma \ref{lemma-closed}. Hence $g = \sum_{i = 1, \ldots, n} f_ib_i + h$ for some $f_i \in I$, $b_i \in B$ and $h \in J$. Pick $e_i$ such that $\varphi(f_i^{e_i}) \in J$. Then $g^{e_1 + \ldots + e_n + 1} \in J$. \end{proof} \begin{definition} \label{definition-taut} Let $\varphi : A \to B$ be a continuous map of linearly topologized rings. We say $\varphi$ is {\it taut}\footnote{This is nonstandard notation. The definition generalizes to modules, by saying a linearly topologized $A$-module $M$ is $A$-taut if for every open ideal $I \subset A$ the closure of $IM$ in $M$ is open and these closures form a fundamental system of neighbourhoods of $0$ in $M$.} if for every open ideal $I \subset A$ the closure of the ideal $\varphi(I)B$ is open and these closures form a fundamental system of open ideals. \end{definition} \noindent If $\varphi : A \to B$ is a continuous map of linearly topologized rings and $I_\lambda$ a fundamental system of open ideals of $A$, then $\varphi$ is taut if and only if the closures of $I_\lambda B$ are open and form a fundamental system of open ideals in $A$. \begin{lemma} \label{lemma-taut-weakly-admissible} Let $\varphi : A \to B$ be a continuous map of weakly admissible topological rings. The following are equivalent \begin{enumerate} \item $\varphi$ is taut, \item for every weak ideal of definition $I \subset A$ the closure of $\varphi(I)B$ is a weak ideal of definition of $B$ and these form a fundamental system of weak ideals of definition of $B$. \end{enumerate} \end{lemma} \begin{proof} It is clear that (2) implies (1). The other implication follows from Lemma \ref{lemma-topologically-nilpotent}. \end{proof} \begin{lemma} \label{lemma-closure-image-ideal} Let $A \to B$ be a continuous map of linearly topologized rings. Let $I \subset A$ be an ideal. The closure of $IB$ is the kernel of $B \to B \widehat{\otimes}_A A/I$. \end{lemma} \begin{proof} Let $J_\mu$ be a fundamental system of open ideals of $B$. The closure of $IB$ is $\bigcap (IB + J_\lambda)$ by Lemma \ref{lemma-closed}. Let $I_\mu$ be a fundamental system of open ideals in $A$. Then $$B \widehat{\otimes}_A A/I = \lim (B/J_\lambda \otimes_A A/(I_\mu + I)) = \lim B/(J_\lambda + I_\mu B + I B)$$ Since $A \to B$ is continuous, for every $\lambda$ there is a $\mu$ such that $I_\mu B \subset J_\lambda$, see discussion in Example \ref{example-what-does-it-mean}. Hence the limit can be written as $\lim B/(J_\lambda + IB)$ and the result is clear. \end{proof} \begin{lemma} \label{lemma-dense-image-surjective} Let $\varphi : A \to B$ be a continuous homomorphism of linearly topologized rings. If \begin{enumerate} \item $\varphi$ is taut, \item $\varphi$ has dense image, \item $A$ is complete, \item $B$ is separated, and \item $A$ has a countable fundamental system of neighbourhoods of $0$. \end{enumerate} Then $\varphi$ is surjective and open, $B$ is complete, and $B = A/K$ for some closed ideal $K \subset A$. \end{lemma} \begin{proof} We may choose a sequence of open ideals $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots$ which form a fundamental system of neighbourhoods of $0$. For each $i$ let $J_i \subset B$ be the closure of $\varphi(I_i)B$. As $\varphi$ is taut we see that these form a fundamental system of open ideals of $B$. Set $I_0 = A$ and $J_0 = B$. Let $n \geq 0$ and let $y_n \in J_n$. Since $J_n$ is the closure of $\varphi(I_n)B$ we can write $$y_n = \sum\nolimits_t \varphi(f_t)b_t + y'_{n + 1}$$ for some $f_t \in I_n$, $b_t \in B$, and $y'_{n + 1} \in J_{n + 1}$. Since $\varphi$ has dense image we can choose $a_t \in A$ with $\varphi(a_t) = b_t \bmod J_{n + 1}$. Thus $$y_n = \varphi(f_n) + y_{n + 1}$$ with $f_n = \sum f_ta_t \in I_n$ and $y_{n + 1} = y'_{n + 1} + \sum f_t(b_t - \varphi(a_t)) \in J_{n + 1}$. Thus, starting with any $y = y_0 \in B$, we can find by induction a sequence $f_m \in I_m$, $m \geq 0$ such that $$y = y_0 = \varphi(f_0 + f_1 + \ldots + f_n) + y_{n + 1}$$ with $y_{n + 1} \in J_{n + 1}$. Since $A$ is complete we see that $$x = x_0 = f_0 + f_1 + f_2 + \ldots$$ exists. Since the partial sums approximate $x$ in $A$, since $\varphi$ is continuous, and since $B$ is separated we find that $\varphi(x) = y$ because above we've shown that the images of the partial sums approximate $y$ in $B$. Thus $\varphi$ is surjective. In exactly the same manner we find that $\varphi(I_n) = J_n$ for all $n \geq 1$. This proves the lemma. \end{proof} \noindent The next lemma says $\varphi$ is taut'' if and only if $\varphi$ is adic'' for continuous maps $\varphi : A \to B$ between adic rings if $A$ has a finitely generated ideal of definition. In some sense the previously introduced notion of tautness for continuous ring maps supersedes the notion of an adic map between adic rings. See also Section \ref{section-adic}. \begin{lemma} \label{lemma-taut-is-adic} Let $\varphi : A \to B$ be a continuous map of linearly topologized rings. Let $I \subset A$ be an ideal. Assume \begin{enumerate} \item $I$ is finitely generated, \item $A$ has the $I$-adic topology, \item $B$ is complete, and \item $\varphi$ is taut. \end{enumerate} Then the topology on $B$ is the $I$-adic topology. \end{lemma} \begin{proof} Let $J_n$ be the closure of $\varphi(I^n)B$ in $B$. Since $B$ is complete we have $B = \lim B/J_n$. Let $B' = \lim B/I^nB$ be the $I$-adic completion of $B$. By Algebra, Lemma \ref{algebra-lemma-hathat-finitely-generated}, the $I$-adic topology on $B'$ is complete and $B'/I^nB' = B/I^nB$. Thus the ring map $B' \to B$ is continuous and has dense image as $B' \to B/I^nB \to B/J_n$ is surjective for all $n$. Finally, the map $B' \to B$ is taut because $(I^nB')B = I^nB$ and $A \to B$ is taut. By Lemma \ref{lemma-dense-image-surjective} we see that $B' \to B$ is open and surjective which implies the lemma. \end{proof} \section{Affine formal algebraic spaces} \label{section-affine-formal-algebraic-spaces} \noindent In this section we introduce affine formal algebraic spaces. These will in fact be the same as what are called affine formal schemes in \cite{BVGD}. However, we will call them affine formal algebraic spaces, in order to prevent confusion with the notion of an affine formal scheme as defined in \cite{EGA}. \medskip\noindent Recall that a thickening of schemes is a closed immersion which induces a surjection on underlying topological spaces, see More on Morphisms, Definition \ref{more-morphisms-definition-thickening}. \begin{definition} \label{definition-affine-formal-algebraic-space} Let $S$ be a scheme. We say a sheaf $X$ on $(\Sch/S)_{fppf}$ is an {\it affine formal algebraic space} if there exist \begin{enumerate} \item a directed set $\Lambda$, \item a system $(X_\lambda, f_{\lambda \mu})$ over $\Lambda$ in $(\Sch/S)_{fppf}$ where \begin{enumerate} \item each $X_\lambda$ is affine, \item each $f_{\lambda \mu} : X_\lambda \to X_\mu$ is a thickening, \end{enumerate} \end{enumerate} such that $$X \cong \colim_{\lambda \in \Lambda} X_\lambda$$ as fppf sheaves and $X$ satisfies a set theoretic condition (see Remark \ref{remark-set-theoretic}). A {\it morphism of affine formal algebraic spaces} over $S$ is a map of sheaves. \end{definition} \noindent Observe that the system $(X_\lambda, f_{\lambda \mu})$ is not part of the data. Suppose that $U$ is a quasi-compact scheme over $S$. Since the transition maps are monomorphisms, we see that $$X(U) = \colim X_\lambda(U)$$ by Sites, Lemma \ref{sites-lemma-directed-colimits-sections}. Thus the fppf sheafification inherent in the colimit of the definition is a Zariski sheafification which does not do anything for quasi-compact schemes. \begin{lemma} \label{lemma-diagonal-affine-formal-algebraic-space} Let $S$ be a scheme. If $X$ is an affine formal algebraic space over $S$, then the diagonal morphism $\Delta : X \to X \times_S X$ is representable and a closed immersion. \end{lemma} \begin{proof} Suppose given $U \to X$ and $V \to X$ where $U, V$ are schemes over $S$. Let us show that $U \times_X V$ is representable. Write $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. The discussion above shows that Zariski locally on $U$ and $V$ the morphisms factors through some $X_\lambda$. In this case $U \times_X V = U \times_{X_\lambda} V$ which is a scheme. Thus the diagonal is representable, see Spaces, Lemma \ref{spaces-lemma-representable-diagonal}. Given $(a, b) : W \to X \times_S X$ where $W$ is a scheme over $S$ consider the map $X \times_{\Delta, X \times_S X, (a, b)} W \to W$. As before locally on $W$ the morphisms $a$ and $b$ map into the affine scheme $X_\lambda$ for some $\lambda$ and then we get the morphism $X_\lambda \times_{\Delta_\lambda, X_\lambda \times_S X_\lambda, (a, b)} W \to W$. This is the base change of $\Delta_\lambda : X_\lambda \to X_\lambda \times_S X_\lambda$ which is a closed immersion as $X_\lambda \to S$ is separated (because $X_\lambda$ is affine). Thus $X \to X \times_S X$ is a closed immersion. \end{proof} \noindent A morphism of schemes $X \to X'$ is a thickening if it is a closed immersion and induces a surjection on underlying sets of points, see (More on Morphisms, Definition \ref{more-morphisms-definition-thickening}). Hence the property of being a thickening is preserved under arbitrary base change and fpqc local on the target, see Spaces, Section \ref{spaces-section-lists}. Thus Spaces, Definition \ref{spaces-definition-relative-representable-property} applies to thickening'' and we know what it means for a representable transformation $F \to G$ of presheaves on $(\Sch/S)_{fppf}$ to be a thickening. We observe that this does not clash with our definition (More on Morphisms of Spaces, Definition \ref{spaces-more-morphisms-definition-thickening}) of thickenings in case $F$ and $G$ are algebraic spaces. \begin{lemma} \label{lemma-covering-by-thickenings} Let $X_\lambda, \lambda \in \Lambda$ and $X = \colim X_\lambda$ be as in Definition \ref{definition-affine-formal-algebraic-space}. Then $X_\lambda \to X$ is representable and a thickening. \end{lemma} \begin{proof} The statement makes sense by the discussion in Spaces, Section \ref{spaces-section-representable} and \ref{spaces-section-representable-properties}. By Lemma \ref{lemma-diagonal-affine-formal-algebraic-space} the morphisms $X_\lambda \to X$ are representable. Given $U \to X$ where $U$ is a scheme, then the discussion following Definition \ref{definition-affine-formal-algebraic-space} shows that Zariski locally on $U$ the morphism factors through some $X_\mu$ with $\lambda \leq \mu$. In this case $U \times_X X_\lambda = U \times_{X_\mu} X_\lambda$ so that $U \times_X X_\lambda \to U$ is a base change of the thickening $X_\lambda \to X_\mu$. \end{proof} \begin{lemma} \label{lemma-factor-through-thickening} Let $X_\lambda, \lambda \in \Lambda$ and $X = \colim X_\lambda$ be as in Definition \ref{definition-affine-formal-algebraic-space}. If $Y$ is a quasi-compact algebraic space over $S$, then any morphism $Y \to X$ factors through an $X_\lambda$. \end{lemma} \begin{proof} Choose an affine scheme $V$ and a surjective \'etale morphism $V \to Y$. The composition $V \to Y \to X$ factors through $X_\lambda$ for some $\lambda$ by the discussion following Definition \ref{definition-affine-formal-algebraic-space}. Since $V \to Y$ is a surjection of sheaves, we conclude. \end{proof} \begin{lemma} \label{lemma-characterize-affine-formal-algebraic-space} Let $S$ be a scheme. Let $X$ be a sheaf on $(\Sch/S)_{fppf}$. Then $X$ is an affine formal algebraic space if and only if the following hold \begin{enumerate} \item any morphism $U \to X$ where $U$ is an affine scheme over $S$ factors through a morphism $T \to X$ which is representable and a thickening with $T$ an affine scheme over $S$, and \item a set theoretic condition as in Remark \ref{remark-set-theoretic}. \end{enumerate} \end{lemma} \begin{proof} It follows from Lemmas \ref{lemma-covering-by-thickenings} and \ref{lemma-factor-through-thickening} that an affine formal algebraic space satisfies (1) and (2). In order to prove the converse we may assume $X$ is not empty. Let $\Lambda$ be the category of representable morphisms $T \to X$ which are thickenings where $T$ is an affine scheme over $S$. This category is directed. Since $X$ is not empty, $\Lambda$ contains at least one object. If $T \to X$ and $T' \to X$ are in $\Lambda$, then we can factor $T \amalg T' \to X$ through $T'' \to X$ in $\Lambda$. Between any two objects of $\Lambda$ there is a unique arrow or none. Thus $\Lambda$ is a directed set and by assumption $X = \colim_{T \to X\text{ in }\Lambda} T$. To finish the proof we need to show that any arrow $T \to T'$ in $\Lambda$ is a thickening. This is true because $T' \to X$ is a monomorphism of sheaves, so that $T = T \times_{T'} T' = T \times_X T'$ and hence the morphism $T \to T'$ equals the projection $T \times_X T' \to T'$ which is a thickening because $T \to X$ is a thickening. \end{proof} \noindent For a general affine formal algebraic space $X$ there is no guarantee that $X$ has enough functions to separate points (for example). See Examples, Section \ref{examples-section-affine-formal-algebraic-space}. To characterize those that do we offer the following lemma. \begin{lemma} \label{lemma-mcquillan-affine-formal-algebraic-space} Let $S$ be a scheme. Let $X$ be an fppf sheaf on $(\Sch/S)_{fppf}$ which satisfies the set theoretic condition of Remark \ref{remark-set-theoretic}. The following are equivalent: \begin{enumerate} \item there exists a weakly admissible topological ring $A$ over $S$ (see Remark \ref{remark-mcquillan}) such that $X = \colim_{I \subset A\text{ weak ideal of definition}} \Spec(A/I)$, \item $X$ is an affine formal algebraic space and there exists an $S$-algebra $A$ and a map $X \to \Spec(A)$ such that for a closed immersion $T \to X$ with $T$ an affine scheme the composition $T \to \Spec(A)$ is a closed immersion, \item $X$ is an affine formal algebraic space and there exists an $S$-algebra $A$ and a map $X \to \Spec(A)$ such that for a closed immersion $T \to X$ with $T$ a scheme the composition $T \to \Spec(A)$ is a closed immersion, \item $X$ is an affine formal algebraic space and for some choice of $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} the projections $\lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda}) \to \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$ are surjective, \item $X$ is an affine formal algebraic space and for any choice of $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} the projections $\lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda}) \to \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$ are surjective. \end{enumerate} Moreover, the weakly admissible topological ring is $A = \lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$ endowed with its limit topology and the weak ideals of definition classify exactly the morphisms $T \to X$ which are representable and thickenings. \end{lemma} \begin{proof} It is clear that (5) implies (4). \medskip\noindent Assume (4) for $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. Set $A = \lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$. Let $T \to X$ be a closed immersion with $T$ a scheme (note that $T \to X$ is representable by Lemma \ref{lemma-diagonal-affine-formal-algebraic-space}). Since $X_\lambda \to X$ is a thickening, so is $X_\lambda \times_X T \to T$. On the other hand, $X_\lambda \times_X T \to X_\lambda$ is a closed immersion, hence $X_\lambda \times_X T$ is affine. Hence $T$ is affine by Limits, Proposition \ref{limits-proposition-affine}. Then $T \to X$ factors through $X_\lambda$ for some $\lambda$ by Lemma \ref{lemma-factor-through-thickening}. Thus $A \to \Gamma(X_\lambda, \mathcal{O}) \to \Gamma(T, \mathcal{O})$ is surjective. In this way we see that (3) holds. \medskip\noindent It is clear that (3) implies (2). \medskip\noindent Assume (2) for $A$ and $X \to \Spec(A)$. Write $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. Then $A_\lambda = \Gamma(X_\lambda, \mathcal{O})$ is a quotient of $A$ by assumption (2). Hence $A^\wedge = \lim A_\lambda$ is a complete topological ring, see discussion in More on Algebra, Section \ref{more-algebra-section-topological-ring}. The maps $A^\wedge \to A_\lambda$ are surjective as $A \to A_\lambda$ is. We claim that for any $\lambda$ the kernel $I_\lambda \subset A^\wedge$ of $A^\wedge \to A_\lambda$ is a weak ideal of definition. Namely, it is open by definition of the limit topology. If $f \in I_\lambda$, then for any $\mu \in \Lambda$ the image of $f$ in $A_\mu$ is zero in all the residue fields of the points of $X_\mu$. Hence it is a nilpotent element of $A_\mu$. Hence some power $f^n \in I_\mu$. Thus $f^n \to 0$ as $n \to 0$. Thus $A^\wedge$ is weakly admissible. Finally, suppose that $I \subset A^\wedge$ is a weak ideal of definition. Then $I \subset A^\wedge$ is open and hence there exists some $\lambda$ such that $I \supset I_\lambda$. Thus we obtain a morphism $\Spec(A^\wedge/I) \to \Spec(A_\lambda) \to X$. Then it follows that $X = \colim \Spec(A^\wedge/I)$ where now the colimit is over all weak ideals of definition. Thus (1) holds. \medskip\noindent Assume (1). In this case it is clear that $X$ is an affine formal algebraic space. Let $X = \colim X_\lambda$ be any presentation as in Definition \ref{definition-affine-formal-algebraic-space}. For each $\lambda$ we can find a weak ideal of definition $I \subset A$ such that $X_\lambda \to X$ factors through $\Spec(A/I) \to X$, see Lemma \ref{lemma-factor-through-thickening}. Then $X_\lambda = \Spec(A/I_\lambda)$ with $I \subset I_\lambda$. Conversely, for any weak ideal of definition $I \subset A$ the morphism $\Spec(A/I) \to X$ factors through $X_\lambda$ for some $\lambda$, i.e., $I_\lambda \subset I$. It follows that each $I_\lambda$ is a weak ideal of definition and that they form a cofinal subset of the set of weak ideals of definition. Hence $A = \lim A/I = \lim A/I_\lambda$ and we see that (5) is true and moreover that $A = \lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})$. \end{proof} \noindent With this lemma in hand we can make the following definition. \begin{definition} \label{definition-types-affine-formal-algebraic-space} Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is {\it McQuillan} if $X$ satisfies the equivalent conditions of Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space}. Let $A$ be the weakly admissible topological ring associated to $X$. We say \begin{enumerate} \item $X$ is {\it classical} if $X$ is McQuillan and $A$ is admissible, \item $X$ is {\it adic} if $X$ is McQuillan and $A$ is adic, \item $X$ is {\it adic*} if $X$ is McQuillan, $A$ is adic, and $A$ has a finitely generated ideal of definition, and \item $X$ is {\it Noetherian} if $X$ is McQuillan and $A$ is both Noetherian and adic. \end{enumerate} \end{definition} \noindent In \cite{Fujiwara-Kato} they use the terminology of finite ideal type'' for the property that an adic topological ring $A$ contains a finitely generated ideal of definition. \begin{remark} \label{remark-compare-with-affine-formal-schemes} The classical affine formal algebraic spaces correspond to the affine formal schemes considered in EGA (\cite{EGA}). To explain this we assume our base scheme is $\Spec(\mathbf{Z})$. Let $\mathfrak X = \text{Spf}(A)$ be an affine formal scheme. Let $h_\mathfrak X$ be its functor of points as in Lemma \ref{lemma-fully-faithful}. Then $h_\mathfrak X = \colim h_{\Spec(A/I)}$ where the colimit is over the collection of ideals of definition of the admissible topological ring $A$. This follows from (\ref{equation-morphisms-affine-formal-schemes}) when evaluating on affine schemes and it suffices to check on affine schemes as both sides are fppf sheaves, see Lemma \ref{lemma-formal-scheme-sheaf-fppf}. Thus $h_\mathfrak X$ is an affine formal algebraic space. In fact, it is a classical affine formal algebraic space by Definition \ref{definition-types-affine-formal-algebraic-space}. Thus Lemma \ref{lemma-fully-faithful} tells us the category of affine formal schemes is equivalent to the category of classical affine formal algebraic spaces. \end{remark} \noindent Having made the connection with affine formal schemes above, it seems natural to make the following definition. \begin{definition} \label{definition-affine-formal-spectrum} Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over $S$, see Definition \ref{definition-weakly-admissible}\footnote{See More on Algebra, Definition \ref{more-algebra-definition-topological-ring} for the classical case and see Remark \ref{remark-mcquillan} for a discussion of differences.}. The {\it formal spectrum} of $A$ is the affine formal algebraic space $$\text{Spf}(A) = \colim \Spec(A/I)$$ where the colimit is over the set of weak ideals of definition of $A$ and taken in the category $\Sh((\Sch/S)_{fppf})$. \end{definition} \noindent Such a formal spectrum is McQuillan by construction and conversely every McQuillan affine formal algebraic space is isomorphic to a formal spectrum. To be sure, in our theory there exist affine formal algebraic spaces which are not the formal spectrum of any weakly admissible topological ring. Following \cite{Yasuda} we could introduce $S$-pro-rings to be pro-objects in the category of $S$-algebras, see Categories, Remark \ref{categories-remark-pro-category}. Then every affine formal algebraic space over $S$ would be the formal spectrum of such an $S$-pro-ring. We will not do this and instead we will work directly with the corresponding affine formal algebraic spaces. \medskip\noindent The construction of the formal spectrum is functorial. To explain this let $\varphi : B \to A$ be a continuous map of weakly admissible topological rings over $S$. Then $$\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)$$ is the unique morphism of affine formal algebraic spaces such that the diagrams $$\xymatrix{ \Spec(B/J) \ar[d] \ar[r] & \Spec(A/I) \ar[d] \\ \text{Spf}(B) \ar[r] & \text{Spf}(A) }$$ commute for all weak ideals of definition $I \subset A$ and $J \subset B$ with $\varphi(I) \subset J$. Since continuity of $\varphi$ implies that for every weak ideal of definition $J \subset B$ there is a weak ideal of definition $I \subset A$ with the required property, we see that the required commutativities uniquely determine and define $\text{Spf}(\varphi)$. \begin{lemma} \label{lemma-morphism-between-formal-spectra} Let $S$ be a scheme. Let $A$, $B$ be weakly admissible topological rings over $S$. Any morphism $f : \text{Spf}(B) \to \text{Spf}(A)$ of affine formal algebraic spaces over $S$ is equal to $\text{Spf}(f^\sharp)$ for a unique continuous $S$-algebra map $f^\sharp : A \to B$. \end{lemma} \begin{proof} Let $f : \text{Spf}(B) \to \text{Spf}(A)$ be as in the lemma. Let $J \subset B$ be a weak ideal of definition. By Lemma \ref{lemma-factor-through-thickening} there exists a weak ideal of definition $I \subset A$ such that $\Spec(B/J) \to \text{Spf}(B) \to \text{Spf}(A)$ factors through $\Spec(A/I)$. By Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} we obtain an $S$-algebra map $A/I \to B/J$. These maps are compatible for varying $J$ and define the map $f^\sharp : A \to B$. This map is continuous because for every weak ideal of definition $J \subset B$ there is a weak ideal of definition $I \subset A$ such that $f^\sharp(I) \subset J$. The equality $f = \text{Spf}(f^\sharp)$ holds by our choice of the ring maps $A/I \to B/J$ which make up $f^\sharp$. \end{proof} \begin{lemma} \label{lemma-presentation-representable} Let $S$ be a scheme. Let $f : X \to Y$ be a map of presheaves on $(\Sch/S)_{fppf}$. If $X$ is an affine formal algebraic space and $f$ is representable by algebraic spaces and locally quasi-finite, then $f$ is representable (by schemes). \end{lemma} \begin{proof} Let $T$ be a scheme over $S$ and $T \to Y$ a map. We have to show that the algebraic space $X \times_Y T$ is a scheme. Write $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. Let $W \subset X \times_Y T$ be a quasi-compact open subspace. The restriction of the projection $X \times_Y T \to X$ to $W$ factors through $X_\lambda$ for some $\lambda$. Then $$W \to X_\lambda \times_S T$$ is a monomorphism (hence separated) and locally quasi-finite (because $W \to X \times_Y T \to T$ is locally quasi-finite by our assumption on $X \to Y$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-quasi-finite}). Hence $W$ is a scheme by Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}. Thus $X \times_Y T$ is a scheme by Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme}. \end{proof} \section{Countably indexed affine formal algebraic spaces} \label{section-countably-indexed} \noindent These are the affine formal algebraic spaces as in the following lemma. \begin{lemma} \label{lemma-countable-affine-formal-algebraic-space} Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item there exists a system $X_1 \to X_2 \to X_3 \to \ldots$ of thickenings of affine schemes over $S$ such that $X = \colim X_n$, \item there exists a choice $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} such that $\Lambda$ is countable. \end{enumerate} \end{lemma} \begin{proof} This follows from the observation that a countable directed set has a cofinal subset isomorphic to $(\mathbf{N}, \geq)$. See proof of Algebra, Lemma \ref{algebra-lemma-ML-limit-nonempty}. \end{proof} \begin{definition} \label{definition-countable} Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is {\it countably indexed} if the equivalent conditions of Lemma \ref{lemma-countable-affine-formal-algebraic-space} are satisfied. \end{definition} \noindent In the language of \cite{BVGD} this is expressed by saying that $X$ is an $\aleph_0$-ind scheme. \begin{lemma} \label{lemma-implications-between-types} Let $X$ be an affine formal algebraic space over a scheme $S$. \begin{enumerate} \item If $X$ is Noetherian, then $X$ is adic*. \item If $X$ is adic*, then $X$ is adic. \item If $X$ is adic, then $X$ is countably indexed. \item If $X$ is countably indexed, then $X$ is McQuillan. \end{enumerate} \end{lemma} \begin{proof} Parts (1) and (2) are immediate from the definitions. \medskip\noindent Proof of (3). By definition there exists an adic topological ring $A$ such that $X = \colim \Spec(A/I)$ where the colimit is over the ideals of definition of $A$. As $A$ is adic, there exits an ideal $I$ such that $\{I^n\}$ forms a fundamental system of neighbourhoods of $0$. Then each $I^n$ is an ideal of definition and $X = \colim \Spec(A/I^n)$. Thus $X$ is countably indexed. \medskip\noindent Proof of (4). Write $X = \lim X_n$ for some system $X_1 \to X_2 \to X_3 \to \ldots$ of thickenings of affine schemes over $S$. Then $$A = \lim \Gamma(X_n, \mathcal{O}_{X_n})$$ surjects onto each $\Gamma(X_n, \mathcal{O}_{X_n})$ because the transition maps are surjections as the morphisms $X_n \to X_{n + 1}$ are closed immersions. \end{proof} \begin{lemma} \label{lemma-countably-indexed} Let $S$ be a scheme. Let $X$ be a presheaf on $(\Sch/S)_{fppf}$. The following are equivalent \begin{enumerate} \item $X$ is a countably indexed affine formal algebraic space, \item $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a countable fundamental system of neighbourhoods of $0$, \item $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a fundamental system $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots$ of weak ideals of definition, \item $X = \text{Spf}(A)$ where $A$ is a complete topological $S$-algebra with a fundamental system of open neighbourhoods of $0$ given by a countable sequence $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots$ of ideals such that $I_n/I_{n + 1}$ is locally nilpotent, and \item $X = \text{Spf}(A)$ where $A = \lim B/J_n$ with the limit topology where $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots$ is a sequence of ideals in an $S$-algebra $B$ with $J_n/J_{n + 1}$ locally nilpotent. \end{enumerate} \end{lemma} \begin{proof} Assume (1). By Lemma \ref{lemma-implications-between-types} we can write $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra. For any presentation $X = \colim X_n$ as in Lemma \ref{lemma-countable-affine-formal-algebraic-space} part (1) we see that $A = \lim A_n$ with $X_n = \Spec(A_n)$ and $A_n = A/I_n$ for some weak ideal of definition $I_n \subset A$. This follows from the final statement of Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space} which moreover implies that $\{I_n\}$ is a fundamental system of open neighbourhoods of $0$. Thus we have a sequence $$A \supset I_1 \supset I_2 \supset I_3 \supset \ldots$$ of weak ideals of definition with $A = \lim A/I_n$. In this way we see that condition (1) implies each of the conditions (2) -- (5). \medskip\noindent Assume (5). First note that the limit topology on $A = \lim B/J_n$ is a linearly topologized, complete topology, see More on Algebra, Section \ref{more-algebra-section-topological-ring}. If $f \in A$ maps to zero in $B/J_1$, then some power maps to zero in $B/J_2$ as its image in $J_1/J_2$ is nilpotent, then a further power maps to zero in $J_2/J_3$, etc, etc. In this way we see the open ideal $\Ker(A \to B/J_1)$ is a weak ideal of definition. Thus $A$ is weakly admissible. In this way we see that (5) implies (2). \medskip\noindent It is clear that (4) is a special case of (5) by taking $B = A$. It is clear that (3) is a special case of (2). \medskip\noindent Assume $A$ is as in (2). Let $E_n$ be a countable fundamental system of neighbourhoods of $0$ in $A$. Since $A$ is a weakly admissible topological ring we can find open ideals $I_n \subset E_n$. We can also choose a weak ideal of definition $J \subset A$. Then $J \cap I_n$ is a fundamental system of weak ideals of definition of $A$ and we get $X = \text{Spf}(A) = \colim \Spec(A/(J \cap I_n))$ which shows that $X$ is a countably indexed affine formal algebraic space. \end{proof} \begin{lemma} \label{lemma-characterize-noetherian-affine} Let $S$ be a scheme. Let $X$ be an affine formal algebraic space. The following are equivalent \begin{enumerate} \item $X$ is Noetherian, \item $X$ is adic* and for some choice of $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space} the schemes $X_\lambda$ are Noetherian, \item $X$ is adic* and for any closed immersion $T \to X$ with $T$ a scheme, $T$ is Noetherian. \end{enumerate} \end{lemma} \begin{proof} This follows from the fact that if $A$ is a ring complete with respect to a finitely generated ideal $I$, then $A$ is Noetherian if and only if $A/I$ is Noetherian, see Algebra, Lemma \ref{algebra-lemma-completion-Noetherian}. Details omitted. \end{proof} \section{Formal algebraic spaces} \label{section-formal-algebraic-spaces} \noindent We take a break from our habit of introducing new concepts first for rings, then for schemes, and then for algebraic spaces, by introducing formal algebraic spaces without first introducing formal schemes. The general idea will be that a formal algebraic space is a sheaf in the fppf topology which \'etale locally is an affine formal scheme in the sense of \cite{BVGD}. Related material can be found in \cite{Yasuda}. \medskip\noindent In the definition of a formal algebraic space we are going to borrow some terminology from Bootstrap, Sections \ref{bootstrap-section-morphism-representable-by-spaces} and \ref{bootstrap-section-representable-by-spaces-properties}. \begin{definition} \label{definition-formal-algebraic-space} Let $S$ be a scheme. We say a sheaf $X$ on $(\Sch/S)_{fppf}$ is a {\it formal algebraic space} if there exist a family of maps $\{X_i \to X\}_{i \in I}$ of sheaves such that \begin{enumerate} \item $X_i$ is an affine formal algebraic space, \item $X_i \to X$ is representable by algebraic spaces and \'etale, \item $\coprod X_i \to X$ is surjective as a map of sheaves \end{enumerate} and $X$ satisfies a set theoretic condition (see Remark \ref{remark-set-theoretic}). A {\it morphism of formal algebraic spaces} over $S$ is a map of sheaves. \end{definition} \noindent Discussion. Sanity check: an affine formal algebraic space is a formal algebraic space. In the situation of the definition the morphisms $X_i \to X$ are representable (by schemes), see Lemma \ref{lemma-presentation-representable}. By Bootstrap, Lemma \ref{bootstrap-lemma-surjective-flat-locally-finite-presentation} we could instead of asking $\coprod X_i \to X$ to be surjective as a map of sheaves, require that it be surjective (which makes sense because it is representable). \medskip\noindent Our notion of a formal algebraic space is {\bf very general}. In fact, even affine formal algebraic spaces as defined above are very nasty objects. However, they do have an underlying reduced algebraic space as the following lemma demonstrates. \begin{lemma} \label{lemma-reduction-formal-algebraic-space} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. There exists a reduced algebraic space $X_{red}$ and a representable morphism $X_{red} \to X$ which is a thickening. A morphism $U \to X$ with $U$ a reduced algebraic space factors uniquely through $X_{red}$. \end{lemma} \begin{proof} First assume that $X$ is an affine formal algebraic space. Say $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. Since the transition morphisms are thickenings, the affine schemes $X_\lambda$ all have isomorphic reductions $X_{red}$. The morphism $X_{red} \to X$ is representable and a thickening by Lemma \ref{lemma-covering-by-thickenings} and the fact that compositions of thickenings are thickenings. We omit the verification of the universal property (use Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}, Schemes, Lemma \ref{schemes-lemma-map-into-reduction}, Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}, and Properties of Spaces, Lemma \ref{spaces-properties-lemma-map-into-reduction}). \medskip\noindent Let $X$ and $\{X_i \to X\}_{i \in I}$ be as in Definition \ref{definition-formal-algebraic-space}. For each $i$ let $X_{i, red} \to X_i$ be the reduction as constructed above. For $i, j \in I$ the projection $X_{i, red} \times_X X_j \to X_{i, red}$ is an \'etale (by assumption) morphism of schemes (by Lemma \ref{lemma-presentation-representable}). Hence $X_{i, red} \times_X X_j$ is reduced (see Descent, Lemma \ref{descent-lemma-reduced-local-smooth}). Thus the projection $X_{i, red} \times_X X_j \to X_j$ factors through $X_{j, red}$ by the universal property. We conclude that $$R_{ij} = X_{i, red} \times_X X_j = X_{i, red} \times_X X_{j, red} = X_i \times_X X_{j, red}$$ because the morphisms $X_{i, red} \to X_i$ are injections of sheaves. Set $U = \coprod X_{i, red}$, set $R = \coprod R_{ij}$, and denote $s, t : R \to U$ the two projections. As a sheaf $R = U \times_X U$ and $s$ and $t$ are \'etale. Then $(t, s) : R \to U$ defines an \'etale equivalence relation by our observations above. Thus $X_{red} = U/R$ is an algebraic space by Spaces, Theorem \ref{spaces-theorem-presentation}. By construction the diagram $$\xymatrix{ \coprod X_{i, red} \ar[r] \ar[d] & \coprod X_i \ar[d] \\ X_{red} \ar[r] & X }$$ is cartesian. Since the right vertical arrow is \'etale surjective and the top horizontal arrow is representable and a thickening we conclude that $X_{red} \to X$ is representable by Bootstrap, Lemma \ref{bootstrap-lemma-after-fppf-sep-lqf} (to verify the assumptions of the lemma use that a surjective \'etale morphism is surjective, flat, and locally of finite presentation and use that thickenings are separated and locally quasi-finite). Then we can use Spaces, Lemma \ref{spaces-lemma-descent-representable-transformations-property} to conclude that $X_{red} \to X$ is a thickening (use that being a thickening is equivalent to being a surjective closed immersion). \medskip\noindent Finally, suppose that $U \to X$ is a morphism with $U$ a reduced algebraic space over $S$. Then each $X_i \times_X U$ is \'etale over $U$ and therefore reduced (by our definition of reduced algebraic spaces in Properties of Spaces, Section \ref{spaces-properties-section-types-properties}). Then $X_i \times_X U \to X_i$ factors through $X_{i, red}$. Hence $U \to X$ factors through $X_{red}$ because $\{X_i \times_X U \to U\}$ is an \'etale covering. \end{proof} \begin{lemma} \label{lemma-diagonal-formal-algebraic-space} Let $S$ be a scheme. If $X$ is a formal algebraic space over $S$, then the diagonal morphism $\Delta : X \to X \times_S X$ is representable, a monomorphism, locally quasi-finite, locally of finite type, and separated. \end{lemma} \begin{proof} Suppose given $U \to X$ and $V \to X$ with $U, V$ schemes over $S$. Then $U \times_X V$ is a sheaf. Choose $\{X_i \to X\}$ as in Definition \ref{definition-formal-algebraic-space}. For every $i$ the morphism $$(U \times_X X_i) \times_{X_i} (V \times_X X_i) = (U \times_X V) \times_X X_i \to U \times_X V$$ is representable and \'etale as a base change of $X_i \to X$ and its source is a scheme (use Lemmas \ref{lemma-diagonal-affine-formal-algebraic-space} and \ref{lemma-presentation-representable}). These maps are jointly surjective hence $U \times_X V$ is an algebraic space by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}. The morphism $U \times_X V \to U \times_S V$ is a monomorphism. It is also locally quasi-finite, because on precomposing with the morphism displayed above we obtain the composition $$(U \times_X X_i) \times_{X_i} (V \times_X X_i) \to (U \times_X X_i) \times_S (V \times_X X_i) \to U \times_S V$$ which is locally quasi-finite as a composition of a closed immersion (Lemma \ref{lemma-diagonal-affine-formal-algebraic-space}) and an \'etale morphism, see Descent on Spaces, Lemma \ref{spaces-descent-lemma-locally-quasi-finite-etale-local-source}. Hence we conclude that $U \times_X V$ is a scheme by Morphisms of Spaces, Proposition \ref{spaces-morphisms-proposition-locally-quasi-finite-separated-over-scheme}. Thus $\Delta$ is representable, see Spaces, Lemma \ref{spaces-lemma-representable-diagonal}. \medskip\noindent In fact, since we've shown above that the morphisms of schemes $U \times_X V \to U \times_S V$ are aways monomorphisms and locally quasi-finite we conclude that $\Delta : X \to X \times_S X$ is a monomorphism and locally quasi-finite, see Spaces, Lemma \ref{spaces-lemma-transformation-diagonal-properties}. Then we can use the principle of Spaces, Lemma \ref{spaces-lemma-representable-transformations-property-implication} to see that $\Delta$ is separated and locally of finite type. Namely, a monomorphism of schemes is separated (Schemes, Lemma \ref{schemes-lemma-monomorphism-separated}) and a locally quasi-finite morphism of schemes is locally of finite type (follows from the definition in Morphisms, Section \ref{morphisms-section-quasi-finite}). \end{proof} \begin{lemma} \label{lemma-space-to-formal-space} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism from an algebraic space over $S$ to a formal algebraic space over $S$. Then $f$ is representable by algebraic spaces. \end{lemma} \begin{proof} Let $Z \to Y$ be a morphism where $Z$ is a scheme over $S$. We have to show that $X \times_Y Z$ is an algebraic space. Choose a scheme $U$ and a surjective \'etale morphism $U \to X$. Then $U \times_Y Z \to X \times_Y Z$ is representable surjective \'etale (Spaces, Lemma \ref{spaces-lemma-base-change-representable-transformations-property}) and $U \times_Y Z$ is a scheme by Lemma \ref{lemma-diagonal-formal-algebraic-space}. Hence the result by Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}. \end{proof} \begin{remark} \label{remark-compare-with-formal-schemes} Modulo set theoretic issues the category of formal schemes \a la EGA (see Section \ref{section-formal-schemes-EGA}) is equivalent to a full subcategory of the category of formal algebraic spaces. To explain this we assume our base scheme is $\Spec(\mathbf{Z})$. By Lemma \ref{lemma-formal-scheme-sheaf-fppf} the functor of points $h_\mathfrak X$ associated to a formal scheme $\mathfrak X$ is a sheaf in the fppf topology. By Lemma \ref{lemma-fully-faithful} the assignment $\mathfrak X \mapsto h_\mathfrak X$ is a fully faithful embedding of the category of formal schemes into the category of fppf sheaves. Given a formal scheme $\mathfrak X$ we choose an open covering $\mathfrak X = \bigcup \mathfrak X_i$ with $\mathfrak X_i$ affine formal schemes. Then $h_{\mathfrak X_i}$ is an affine formal algebraic space by Remark \ref{remark-compare-with-affine-formal-schemes}. The morphisms $h_{\mathfrak X_i} \to h_\mathfrak X$ are representable and open immersions. Thus $\{h_{\mathfrak X_i} \to h_\mathfrak X\}$ is a family as in Definition \ref{definition-formal-algebraic-space} and we see that $h_\mathfrak X$ is a formal algebraic space. \end{remark} \begin{remark} \label{remark-set-theoretic} Let $S$ be a scheme and let $(\Sch/S)_{fppf}$ be a big fppf site as in Topologies, Definition \ref{topologies-definition-big-small-fppf}. As our set theoretic condition on $X$ in Definitions \ref{definition-affine-formal-algebraic-space} and \ref{definition-formal-algebraic-space} we take: there exist objects $U, R$ of $(\Sch/S)_{fppf}$, a morphism $U \to X$ which is a surjection of fppf sheaves, and a morphism $R \to U \times_X U$ which is a surjection of fppf sheaves. In other words, we require our sheaf to be a coequalizer of two maps between representable sheaves. Here are some observations which imply this notion behaves reasonably well: \begin{enumerate} \item Suppose $X = \colim_{\lambda \in \Lambda} X_\lambda$ and the system satisfies conditions (1) and (2) of Definition \ref{definition-affine-formal-algebraic-space}. Then $U = \coprod_{\lambda \in \Lambda} X_\lambda \to X$ is a surjection of fppf sheaves. Moreover, $U \times_X U$ is a closed subscheme of $U \times_S U$ by Lemma \ref{lemma-diagonal-affine-formal-algebraic-space}. Hence if $U$ is representable by an object of $(\Sch/S)_{fppf}$ then $U \times_S U$ is too (see Sets, Lemma \ref{sets-lemma-what-is-in-it}) and the set theoretic condition is satisfied. This is always the case if $\Lambda$ is countable, see Sets, Lemma \ref{sets-lemma-what-is-in-it}. \item Sanity check. Let $\{X_i \to X\}_{i \in I}$ be as in Definition \ref{definition-formal-algebraic-space} (with the set theoretic condition as formulated above) and assume that each $X_i$ is actually an affine scheme. Then $X$ is an algebraic space. Namely, if we choose a larger big fppf site $(\Sch'/S)_{fppf}$ such that $U' = \coprod X_i$ and $R' = \coprod X_i \times_X X_j$ are representable by objects in it, then $X' = U'/R'$ will be an object of the category of algebraic spaces for this choice. Then an application of Spaces, Lemma \ref{spaces-lemma-fully-faithful} shows that $X$ is an algebraic space for $(\Sch/S)_{fppf}$. \item Let $\{X_i \to X\}_{i \in I}$ be a family of maps of sheaves satisfying conditions (1), (2), (3) of Definition \ref{definition-formal-algebraic-space}. For each $i$ we can pick $U_i \in \Ob((\Sch/S)_{fppf})$ and $U_i \to X_i$ which is a surjection of sheaves. Thus if $I$ is not too large (for example countable) then $U = \coprod U_i \to X$ is a surjection of sheaves and $U$ is representable by an object of $(\Sch/S)_{fppf}$. To get $R \in \Ob((\Sch/S)_{fppf})$ surjecting onto $U \times_X U$ it suffices to assume the diagonal $\Delta : X \to X \times_S X$ is not too wild, for example this always works if the diagonal of $X$ is quasi-compact, i.e., $X$ is quasi-separated. \end{enumerate} \end{remark} \section{Colimits of algebraic spaces along thickenings} \label{section-global-colimits} \noindent A special type of formal algebraic space is one which can globally be written as a cofiltered colimit of algebraic spaces along thickenings as in the following lemma. We will see later (in Section \ref{section-quasi-compact-quasi-separated}) that any quasi-compact and quasi-separated formal algebraic space is such a global colimit. \begin{lemma} \label{lemma-colimit-is-formal} Let $S$ be a scheme. Suppose given a directed set $\Lambda$ and a system of algebraic spaces $(X_\lambda, f_{\lambda \mu})$ over $\Lambda$ where each $f_{\lambda \mu} : X_\lambda \to X_\mu$ is a thickening. Then $X = \colim_{\lambda \in \Lambda} X_\lambda$ is a formal algebraic space over $S$. \end{lemma} \begin{proof} Since we take the colimit in the category of fppf sheaves, we see that $X$ is a sheaf. Choose and fix $\lambda \in \Lambda$. Choose an \'etale covering $\{X_{i, \lambda} \to X_\lambda\}$ where $X_i$ is an affine scheme over $S$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. For each $\mu \geq \lambda$ there exists a cartesian diagram $$\xymatrix{ X_{i, \lambda} \ar[r] \ar[d] & X_{i, \mu} \ar[d] \\ X_\lambda \ar[r] & X_\mu }$$ with \'etale vertical arrows, see More on Morphisms of Spaces, Theorem \ref{spaces-more-morphisms-theorem-topological-invariance} (this also uses that a thickening is a surjective closed immersion which satisfies the conditions of the theorem). Moreover, these diagrams are unique up to unique isomorphism and hence $X_{i, \mu} = X_\mu \times_{X_{\mu'}} X_{i, \mu'}$ for $\mu' \geq \mu$. The morphisms $X_{i, \mu} \to X_{i, \mu'}$ is a thickening as a base change of a thickening. Each $X_{i, \mu}$ is an affine scheme by Limits of Spaces, Proposition \ref{spaces-limits-proposition-affine} and the fact that $X_{i, \lambda}$ is affine. Set $X_i = \colim_{\mu \geq \lambda} X_{i, \mu}$. Then $X_i$ is an affine formal algebraic space. The morphism $X_i \to X$ is \'etale because given an affine scheme $U$ any $U \to X$ factors through $X_\mu$ for some $\mu \geq \lambda$ (details omitted). In this way we see that $X$ is a formal algebraic space. \end{proof} \noindent Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. How does one prove or check that $X$ is a global colimit as in Lemma \ref{lemma-colimit-is-formal}? To do this we look for maps $i : Z \to X$ where $Z$ is an algebraic space over $S$ and $i$ is surjective and a closed immersion, in other words, $i$ is a thickening. This makes sense as $i$ is representable by algebraic spaces (Lemma \ref{lemma-space-to-formal-space}) and we can use Bootstrap, Definition \ref{bootstrap-definition-property-transformation} as before. \begin{example} \label{example-david-hansen} Let $(A, \mathfrak m, \kappa)$ be a valuation ring, which is $(\pi)$-adically complete for some nonzero $\pi \in \mathfrak m$. Assume also that $\mathfrak m$ is not finitely generated. An example is $A = \mathcal{O}_{\mathbf{C}_p}$ and $\pi = p$ where $\mathcal{O}_{\mathbf{C}_p}$ is the ring of integers of the field of $p$-adic complex numbers $\mathbf{C}_p$ (this is the completion of the algebraic closure of $\mathbf{Q}_p$). Another example is $$A = \left\{ \sum\nolimits_{\alpha \in \mathbf{Q},\ \alpha \geq 0} a_\alpha t^\alpha \middle| \begin{matrix} a_\alpha \in \kappa \text{ and for all }n\text{ there are only a} \\ \text{finite number of nonzero }a_\alpha \text{ with }\alpha \leq n \end{matrix} \right\}$$ and $\pi = t$. Then $X = \text{Spf}(A)$ is an affine formal algebraic space and $\Spec(\kappa) \to X$ is a thickening which corresponds to the weak ideal of definition $\mathfrak m \subset A$ which is however not an ideal of definition. \end{example} \begin{remark}[Weak ideals of definition] \label{remark-weak-ideals-of-definition} Let $\mathfrak X$ be a formal scheme in the sense of McQuillan, see Remark \ref{remark-mcquillan}. An {\it weak ideal of definition} for $\mathfrak X$ is an ideal sheaf $\mathcal{I} \subset \mathcal{O}_\mathfrak X$ such that for all $\mathfrak U \subset \mathfrak X$ affine formal open subscheme the ideal $\mathcal{I}(\mathfrak U) \subset \mathcal{O}_\mathfrak X(\mathfrak U)$ is a weak ideal of definition of the weakly admissible topological ring $\mathcal{O}_\mathfrak X(\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. There is a one-to-one correspondence $$\{\text{weak ideals of definition for }\mathfrak X\} \leftrightarrow \{\text{thickenings }i : Z \to h_\mathfrak X\text{ as above}\}$$ This correspondence associates to $\mathcal{I}$ the scheme $Z = (\mathfrak X, \mathcal{O}_\mathfrak X/\mathcal{I})$ together with the obvious morphism to $\mathfrak X$. A {\it fundamental system of weak ideals of definition} is a collection of weak ideals of definition $\mathcal{I}_\lambda$ such that on every affine open formal subscheme $\mathfrak U \subset \mathfrak X$ the ideals $$I_\lambda = \mathcal{I}_\lambda(\mathfrak U) \subset A = \Gamma(\mathfrak U, \mathcal{O}_\mathfrak X)$$ form a fundamental system of weak ideals of definition of the weakly admissible topological ring $A$. It suffices to check on the members of an affine open covering. We conclude that the formal algebraic space $h_\mathfrak X$ associated to the McQuillan formal scheme $\mathfrak X$ is a colimit of schemes as in Lemma \ref{lemma-colimit-is-formal} if and only if there exists a fundamental system of weak ideals of definition for $\mathfrak X$. \end{remark} \begin{remark}[Ideals of definition] \label{remark-ideals-of-definition} Let $\mathfrak X$ be a formal scheme \a la EGA. An {\it ideal of definition} for $\mathfrak X$ is an ideal sheaf $\mathcal{I} \subset \mathcal{O}_\mathfrak X$ such that for all $\mathfrak U \subset \mathfrak X$ affine formal open subscheme the ideal $\mathcal{I}(\mathfrak U) \subset \mathcal{O}_\mathfrak X(\mathfrak U)$ is an ideal of definition of the admissible topological ring $\mathcal{O}_\mathfrak X(\mathfrak U)$. It suffices to check the condition on the members of an affine open covering. We do {\bf not} get the same correspondence between ideals of definition and thickenings $Z \to h_\mathfrak X$ as in Remark \ref{remark-weak-ideals-of-definition}; an example is given in Example \ref{example-david-hansen}. A {\it fundamental system of ideals of definition} is a collection of ideals of definition $\mathcal{I}_\lambda$ such that on every affine open formal subscheme $\mathfrak U \subset \mathfrak X$ the ideals $$I_\lambda = \mathcal{I}_\lambda(\mathfrak U) \subset A = \Gamma(\mathfrak U, \mathcal{O}_\mathfrak X)$$ form a fundamental system of ideals of definition of the admissible topological ring $A$. It suffices to check on the members of an affine open covering. Suppose that $\mathfrak X$ is quasi-compact and that $\{\mathcal{I}_\lambda\}_{\lambda \in \Lambda}$ is a fundamental system of weak ideals of definition. If $A$ is an admissible topological ring then all sufficiently small open ideals are ideals of definition (namely any open ideal contained in an ideal of definition is an ideal of definition). Thus since we only need to check on the finitely many members of an affine open covering we see that $\mathcal{I}_\lambda$ is an ideal of definition for $\lambda$ sufficiently large. Using the discussion in Remark \ref{remark-weak-ideals-of-definition} we conclude that the formal algebraic space $h_\mathfrak X$ associated to the quasi-compact formal scheme $\mathfrak X$ \a la EGA is a colimit of schemes as in Lemma \ref{lemma-colimit-is-formal} if and only if there exists a fundamental system of ideals of definition for $\mathfrak X$. \end{remark} \section{Completion along a closed subset} \label{section-completion} \noindent Our notion of a formal algebraic space is well adapted to taking the completion along a closed subset. \begin{lemma} \label{lemma-completion-affine-is-affine-formal-algebraic-space} Let $S$ be a scheme. Let $X$ be an affine scheme over $S$. Let $T \subset |X|$ be a closed subset. Then the functor $$(\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad U \longmapsto \{f : U \to X \mid f(|U|) \subset T\}$$ is a McQuillan affine formal algebraic space. \end{lemma} \begin{proof} Say $X = \Spec(A)$ and $T$ corresponds to the radical ideal $I \subset A$. Let $U = \Spec(B)$ be an affine scheme over $S$ and let $f : U \to X$ be an element of $F(U)$. Then $f$ corresponds to a ring map $\varphi : A \to B$ such that every prime of $B$ contains $\varphi(I) B$. Thus every element of $\varphi(I)$ is nilpotent in $B$, see Algebra, Lemma \ref{algebra-lemma-Zariski-topology}. Setting $J = \Ker(\varphi)$ we conclude that $I/J$ is a locally nilpotent ideal in $A/J$. Equivalently, $V(J) = V(I) = T$. In other words, the functor of the lemma equals $\colim \Spec(A/J)$ where the colimit is over the collection of ideals $J$ with $V(J) = T$. Thus our functor is an affine formal algebraic space. It is McQuillan (Definition \ref{definition-types-affine-formal-algebraic-space}) because the maps $A \to A/J$ are surjective and hence $A^\wedge = \lim A/J \to A/J$ is surjective, see Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space}. \end{proof} \begin{lemma} \label{lemma-completion-is-formal-algebraic-space} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. Then the functor $$(\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad U \longmapsto \{f : U \to X \mid f(|U|) \subset T\}$$ is a formal algebraic space. \end{lemma} \begin{proof} Denote $F$ the functor. Let $\{U_i \to U\}$ be an fppf covering. Then $\coprod |U_i| \to |U|$ is surjective. Since $X$ is an fppf sheaf, it follows that $F$ is an fppf sheaf. \medskip\noindent Let $\{g_i : X_i \to X\}$ be an \'etale covering such that $X_i$ is affine for all $i$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. The morphisms $F \times_X X_i \to F$ are \'etale (see Spaces, Lemma \ref{spaces-lemma-base-change-representable-transformations-property}) and the map $\coprod F \times_X X_i \to F$ is a surjection of sheaves. Thus it suffices to prove that $F \times_X X_i$ is an affine formal algebraic space. A $U$-valued point of $F \times_X X_i$ is a morphism $U \to X_i$ whose image is contained in the closed subset $g_i^{-1}(T) \subset |X_i|$. Thus this follows from Lemma \ref{lemma-completion-affine-is-affine-formal-algebraic-space}. \end{proof} \begin{definition} \label{definition-completion} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a closed subset. The formal algebraic space of Lemma \ref{lemma-completion-is-formal-algebraic-space} is called the {\it completion of $X$ along $T$}. \end{definition} \noindent In \cite[Chapter I, Section 10.8]{EGA} the notation $X_{/T}$ is used to denote the completion and we will occasionally use this notation as well. Let $f : X \to X'$ be a morphism of algebraic spaces over a scheme $S$. Suppose that $T \subset |X|$ and $T' \subset |X'|$ are closed subsets such that $|f|(T) \subset T'$. Then it is clear that $f$ defines a morphism of formal algebraic spaces $$X_{/T} \longrightarrow X'_{/T'}$$ between the completions. \begin{lemma} \label{lemma-map-completions-representable} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $T \subset |Y|$ be a closed subset and let $T' = |f|^{-1}(T) \subset |X|$. Then $X_{/T'} \to Y_{/T}$ is representable by algebraic spaces. \end{lemma} \begin{proof} Namely, suppose that $V \to Y$ is a morphism from a scheme into $Y$ such that $|V|$ maps into $T$. Then $V \times_Y X \to X$ is a morphism of algebraic spaces such that $|V \times_Y X|$ maps into $T'$. Hence the functor $V \times_{Y_{/T'}} X_{/T}$ is represented by $V \times_Y X$ and we see that the lemma holds. \end{proof} \noindent The following lemma is due to Ofer Gabber. \begin{lemma} \label{lemma-completion-countably-indexed} \begin{reference} Email by Ofer Gabber of September 11, 2014. \end{reference} Let $S$ be a scheme. Let $X = \Spec(A)$ be an affine scheme over $S$. Let $T \subset X$ be a closed subscheme. \begin{enumerate} \item If the formal completion $X_{/T}$ is countably indexed and there exist countably many $f_1, f_2, f_3, \ldots \in A$ such that $T = V(f_1, f_2, f_3, \ldots)$, then $X_{/T}$ is adic*. \item The conclusion of (1) is wrong if we omit the assumption that $T$ can be cut out by countably many functions in $X$. \end{enumerate} \end{lemma} \begin{proof} The assumption that $X_{/T}$ is countably indexed means that there exists a sequence of ideals $$A \supset J_1 \supset J_2 \supset J_3 \supset \ldots$$ with $V(J_n) = T$ such that every ideal $J \subset A$ with $V(J) = T$ there exists an $n$ such that $J \supset J_n$. \medskip\noindent To construct an example for (2) let $\omega_1$ be the first uncountable ordinal. Let $k$ be a field and let $A$ be the $k$-algebra generated by $x_\alpha$, $\alpha \in \omega_1$ and $y_{\alpha \beta}$ with $\alpha \in \beta \in \omega_1$ subject to the relations $x_\alpha = y_{\alpha \beta} x_\beta$. Let $T = V(x_\alpha)$. Let $J_n = (x_\alpha^n)$. If $J \subset A$ is an ideal such that $V(J) = T$, then $x_\alpha^{n_\alpha} \in J$ for some $n_\alpha \geq 1$. One of the sets $\{\alpha \mid n_\alpha = n\}$ must be unbounded in $\omega_1$. Then the relations imply that $J_n \subset J$. \medskip\noindent To see that (2) holds it now suffices to show that $A^\wedge = \lim A/J_n$ is not a ring complete with respect to a finitely generated ideal. For $\gamma \in \omega_1$ let $A_\gamma$ be the quotient of $A$ by the ideal generated by $x_\alpha$, $\alpha \in \gamma$ and $y_{\alpha \beta}$, $\alpha \in \gamma$. As $A/J_1$ is reduced, every topologically nilpotent element $f$ of $\lim A/J_n$ is in $J_1^\wedge = \lim J_1/J_n$. This means $f$ is an infinite series involving only a countable number of generators. Hence $f$ dies in $A_\gamma^\wedge = \lim A_\gamma/J_nA_\gamma$ for some $\gamma$. Note that $A^\wedge \to A_\gamma^\wedge$ is continuous and open by Lemma \ref{lemma-ses}. If the topology on $A^\wedge$ was $I$-adic for some finitely generated ideal $I \subset A^\wedge$, then $I$ would go to zero in some $A_\gamma^\wedge$. This would mean that $A_\gamma^\wedge$ is discrete, which is not the case as there is a surjective continuous and open (by Lemma \ref{lemma-ses}) map $A_\gamma^\wedge \to k[[t]]$ given by $x_\alpha \mapsto t$, $y_{\alpha \beta} \mapsto 1$ for $\gamma = \alpha$ or $\gamma \in \alpha$. \medskip\noindent Before we prove (1) we first prove the following: If $I \subset A^\wedge$ is a finitely generated ideal whose closure $\bar I$ is open, then $I = \bar I$. Since $V(J_n^2) = T$ there exists an $m$ such that $J_n^2 \supset J_m$. Thus, we may assume that $J_n^2 \supset J_{n + 1}$ for all $n$ by passing to a subsequence. Set $J_n^\wedge = \lim_{k \geq n} J_n/J_k \subset A^\wedge$. Since the closure $\bar I = \bigcap (I + J_n^\wedge)$ (Lemma \ref{lemma-closed}) is open we see that there exists an $m$ such that $I + J_n^\wedge \supset J_m^\wedge$ for all $n \geq m$. Fix such an $m$. We have $$J_{n - 1}^\wedge I + J_{n + 1}^\wedge \supset J_{n - 1}^\wedge (I + J_{n + 1}^\wedge) \supset J_{n - 1}^\wedge J_m^\wedge$$ for all $n \geq m + 1$. Namely, the first inclusion is trivial and the second was shown above. Because $J_{n - 1}J_m \supset J_{n - 1}^2 \supset J_n$ these inclusions show that the image of $J_n$ in $A^\wedge$ is contained in the ideal $J_{n - 1}^\wedge I + J_{n + 1}^\wedge$. Because this ideal is open we conclude that $$J_{n - 1}^\wedge I + J_{n + 1}^\wedge \supset J_n^\wedge.$$ Say $I = (g_1, \ldots, g_t)$. Pick $f \in J_{m + 1}^\wedge$. Using the last displayed inclusion, valid for all $n \geq m + 1$, we can write by induction on $c \geq 0$ $$f = \sum f_{i, c} g_i \mod J_{m + 1+ c}^\wedge$$ with $f_{i, c} \in J_m^\wedge$ and $f_{i, c} \equiv f_{i, c - 1} \bmod J_{m + c}^\wedge$. It follows that $IJ_m^\wedge \supset J_{m + 1}^\wedge$. Combined with $I + J_{m + 1}^\wedge \supset J_m^\wedge$ we conclude that $I$ is open. \medskip\noindent Proof of (1). Assume $T = V(f_1, f_2, f_3, \ldots)$. Let $I_m \subset A^\wedge$ be the ideal generated by $f_1, \ldots, f_m$. We distinguish two cases. \medskip\noindent Case I: For some $m$ the closure of $I_m$ is open. Then $I_m$ is open by the result of the previous paragraph. For any $n$ we have $(J_n)^2 \supset J_{n+1}$ by design, so the closure of $(J_n^\wedge)^2$ contains $J_{n+1}^\wedge$ and thus is open. Taking $n$ large, it follows that the closure of the product of any two open ideals in $A^\wedge$ is open. Let us prove $I_m^k$ is open for $k \ge 1$ by induction on $k$. The case $k = 1$ is our hypothesis on $m$ in Case I. For $k > 1$, suppose $I_m^{k - 1}$ is open. Then $I_m^k = I_m^{k - 1} \cdot I_m$ is the product of two open ideals and hence has open closure. But then since $I_m^k$ is finitely generated it follows that $I_m^k$ is open by the previous paragraph (applied to $I = I_m^k$), so we can continue the induction on $k$. As each element of $I_m$ is topologically nilpotent, we conclude that $I_m$ is an ideal of definition which proves that $A^\wedge$ is adic with a finitely generated ideal of definition, i.e., $X_{/T}$ is adic*. \medskip\noindent Case II. For all $m$ the closure $\bar I_m$ of $I_m$ is not open. Then the topology on $A^\wedge/\bar I_m$ is not discrete. This means we can pick $\phi(m) \geq m$ such that $$\Im(J_{\phi(m)} \to A/(f_1, \ldots, f_m)) \not = \Im(J_{\phi(m) + 1} \to A/(f_1, \ldots, f_m))$$ To see this we have used that $A^\wedge/(\bar I_m + J_n^\wedge) = A/((f_1, \ldots, f_m) + J_n)$. Choose exponents $e_i > 0$ such that $f_i^{e_i} \in J_{\phi(m) + 1}$ for $0 < m < i$. Let $J = (f_1^{e_1}, f_2^{e_2}, f_3^{e_3}, \ldots)$. Then $V(J) = T$. We claim that $J \not \supset J_n$ for all $n$ which is a contradiction proving Case II does not occur. Namely, the image of $J$ in $A/(f_1, \ldots, f_m)$ is contained in the image of $J_{\phi(m) + 1}$ which is properly contained in the image of $J_m$. \end{proof} \section{Fibre products} \label{section-fibre-products} \noindent Obligatory section about fibre products of formal algebraic spaces. \begin{lemma} \label{lemma-etale-covering-by-formal-algebraic-spaces} Let $S$ be a scheme. Let $\{X_i \to X\}_{i \in I}$ be a family of maps of sheaves on $(\Sch/S)_{fppf}$. Assume (a) $X_i$ is a formal algebraic space over $S$, (b) $X_i \to X$ is representable by algebraic spaces and \'etale, and (c) $\coprod X_i \to X$ is a surjection of sheaves. Then $X$ is a formal algebraic space over $S$. \end{lemma} \begin{proof} For each $i$ pick $\{X_{ij} \to X_i\}_{j \in J_i}$ as in Definition \ref{definition-formal-algebraic-space}. Then $\{X_{ij} \to X\}_{i \in I, j \in J_i}$ is a family as in Definition \ref{definition-formal-algebraic-space} for $X$. \end{proof} \begin{lemma} \label{lemma-fibre-products-general} Let $S$ be a scheme. Let $X, Y$ be formal algebraic spaces over $S$ and let $Z$ be a sheaf whose diagonal is representable by algebraic spaces. Let $X \to Z$ and $Y \to Z$ be maps of sheaves. Then $X \times_Z Y$ is a formal algebraic space. \end{lemma} \begin{proof} Choose $\{X_i \to X\}$ and $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space}. Then $\{X_i \times_Z Y_j \to X \times_Z Y\}$ is a family of maps which are representable by algebraic spaces and \'etale. Thus Lemma \ref{lemma-etale-covering-by-formal-algebraic-spaces} tells us it suffices to show that $X \times_Z Y$ is a formal algebraic space when $X$ and $Y$ are affine formal algebraic spaces. \medskip\noindent Assume $X$ and $Y$ are affine formal algebraic spaces. Write $X = \colim X_\lambda$ and $Y = \colim Y_\mu$ as in Definition \ref{definition-affine-formal-algebraic-space}. Then $X \times_Z Y = \colim X_\lambda \times_Z Y_\mu$. Each $X_\lambda \times_Z Y_\mu$ is an algebraic space. For $\lambda \leq \lambda'$ and $\mu \leq \mu'$ the morphism $$X_\lambda \times_Z Y_\mu \to X_\lambda \times_Z Y_{\mu'} \to X_{\lambda'} \times_Z Y_{\mu'}$$ is a thickening as a composition of base changes of thickenings. Thus we conclude by applying Lemma \ref{lemma-colimit-is-formal}. \end{proof} \begin{lemma} \label{lemma-fibre-products} Let $S$ be a scheme. The category of formal algebraic spaces over $S$ has fibre products. \end{lemma} \begin{proof} Special case of Lemma \ref{lemma-fibre-products-general} because formal algebraic spaces have representable diagonals, see Lemma \ref{lemma-diagonal-formal-algebraic-space}. \end{proof} \begin{lemma} \label{lemma-reduction-fibre-products} Let $S$ be a scheme. Let $X \to Z$ and $Y \to Z$ be morphisms of formal algebraic spaces over $S$. Then $(X \times_Z Y)_{red} = (X_{red} \times_{Z_{red}} Y_{red})_{red}$. \end{lemma} \begin{proof} This follows from the universal property of the reduction in Lemma \ref{lemma-reduction-formal-algebraic-space}. \end{proof} \noindent We have already proved the following lemma (without knowing that fibre products exist). \begin{lemma} \label{lemma-diagonal-morphism-formal-algebraic-spaces} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The diagonal morphism $\Delta : X \to X \times_Y X$ is representable (by schemes), a monomorphism, locally quasi-finite, locally of finite type, and separated. \end{lemma} \begin{proof} Let $T$ be a scheme and let $T \to X \times_Y X$ be a morphism. Then $$T \times_{(X \times_Y X)} X = T \times_{(X \times_S X)} X$$ Hence the result follows immediately from Lemma \ref{lemma-diagonal-formal-algebraic-space}. \end{proof} \section{Separation axioms for formal algebraic spaces} \label{section-separation} \noindent This section is about absolute'' separation conditions on formal algebraic spaces. We will discuss separation conditions for morphisms of formal algebraic spaces later. \begin{lemma} \label{lemma-characterize-quasi-separated} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item the reduction of $X$ (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a quasi-separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ quasi-compact schemes the fibre product $U \times_X V$ is quasi-compact, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is quasi-compact. \end{enumerate} \end{lemma} \begin{proof} Observe that $U \times_X V$ is a scheme by Lemma \ref{lemma-diagonal-formal-algebraic-space}. Let $U_{red}, V_{red}, X_{red}$ be the reduction of $U, V, X$. Then $$U_{red} \times_{X_{red}} V_{red} = U_{red} \times_X V_{red} \to U \times_X V$$ is a thickening of schemes. From this the equivalence of (1) and (2) is clear, keeping in mind the analogous lemma for algebraic spaces, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-separated}. We omit the proof of the equivalence of (2) and (3). \end{proof} \begin{lemma} \label{lemma-characterize-separated} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item the reduction of $X$ (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a separated algebraic space, \item for $U \to X$, $V \to X$ with $U$, $V$ affine the fibre product $U \times_X V$ is affine and $$\mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}(U \times_X V)$$ is surjective. \end{enumerate} \end{lemma} \begin{proof} If (2) holds, then $X_{red}$ is a separated algebraic space by applying Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-separated} to morphisms $U \to X_{red}$ and $V \to X_{red}$ with $U, V$ affine and using that $U \times_{X_{red}} V = U \times_X V$. \medskip\noindent Assume (1). Let $U \to X$ and $V \to X$ be as in (2). Observe that $U \times_X V$ is a scheme by Lemma \ref{lemma-diagonal-formal-algebraic-space}. Let $U_{red}, V_{red}, X_{red}$ be the reduction of $U, V, X$. Then $$U_{red} \times_{X_{red}} V_{red} = U_{red} \times_X V_{red} \to U \times_X V$$ is a thickening of schemes. It follows that $(U \times_X V)_{red} = (U_{red} \times_{X_{red}} V_{red})_{red}$. In particular, we see that $(U \times_X V)_{red}$ is an affine scheme and that $$\mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}((U \times_X V)_{red})$$ is surjective, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-separated}. Then $U \times_X V$ is affine by Limits of Spaces, Proposition \ref{spaces-limits-proposition-affine}. On the other hand, the morphism $U \times_X V \to U \times V$ of affine schemes is the composition $$U \times_X V = X \times_{(X \times_S X)} (U \times_S V) \to U \times_S V \to U \times V$$ The first morphism is a monomorphism and locally of finite type (Lemma \ref{lemma-diagonal-formal-algebraic-space}). The second morphism is an immersion (Schemes, Lemma \ref{schemes-lemma-fibre-product-after-map}). Hence the composition is a monomorphism which is locally of finite type. On the other hand, the composition is integral as the map on underlying reduced affine schemes is a closed immersion by the above and hence universally closed (use Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed}). Thus the ring map $$\mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}(U \times_X V)$$ is an epimorphism which is integral of finite type hence finite hence surjective (use Morphisms, Lemma \ref{morphisms-lemma-finite-integral} and Algebra, Lemma \ref{algebra-lemma-finite-epimorphism-surjective}). \end{proof} \begin{definition} \label{definition-separated} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say \begin{enumerate} \item $X$ is {\it quasi-separated} if the equivalent conditions of Lemma \ref{lemma-characterize-quasi-separated} are satisfied. \item $X$ is {\it separated} if the equivalent conditions of Lemma \ref{lemma-characterize-separated} are satisfied. \end{enumerate} \end{definition} \noindent The following lemma implies in particular that the completed tensor product of weakly admissible topological rings is a weakly admissible topological ring. \begin{lemma} \label{lemma-fibre-product-affines-over-separated} Let $S$ be a scheme. Let $X \to Z$ and $Y \to Z$ be morphisms of formal algebraic spaces over $S$. Assume $Z$ separated. \begin{enumerate} \item If $X$ and $Y$ are affine formal algebraic spaces, then so is $X \times_Z Y$. \item If $X$ and $Y$ are McQuillan affine formal algebraic spaces, then so is $X \times_Z Y$. \item If $X$, $Y$, and $Z$ are McQuillan affine formal algebraic spaces corresponding to the weakly admissible topological $S$-algebras $A$, $B$, and $C$, then $X \times_Z Y$ corresponds to $A \widehat{\otimes}_C B$. \end{enumerate} \end{lemma} \begin{proof} Write $X = \colim X_\lambda$ and $Y = \colim Y_\mu$ as in Definition \ref{definition-affine-formal-algebraic-space}. Then $X \times_Z Y = \colim X_\lambda \times_Z Y_\mu$. Since $Z$ is separated the fibre products are affine, hence we see that (1) holds. Assume $X$ and $Y$ corresponds to the weakly admissible topological $S$-algebras $A$ and $B$ and $X_\lambda = \Spec(A/I_\lambda)$ and $Y_\mu = \Spec(B/J_\mu)$. Then $$X_\lambda \times_Z Y_\mu \to X_\lambda \times Y_\mu \to \Spec(A \otimes B)$$ is a closed immersion. Thus one of the conditions of Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space} holds and we conclude that $X \times_Z Y$ is McQuillan. If also $Z$ is McQuillan corresponding to $C$, then $$X_\lambda \times_Z Y_\mu = \Spec(A/I_\lambda \otimes_C B/J_\mu)$$ hence we see that the weakly admissible topological ring corresponding to $X \times_Z Y$ is the completed tensor product (see Definition \ref{definition-toplogy-tensor-product}). \end{proof} \begin{lemma} \label{lemma-separated-from-separated} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $U \to X$ be a morphism where $U$ is a separated algebraic space over $S$. Then $U \to X$ is separated. \end{lemma} \begin{proof} The statement makes sense because $U \to X$ is representable by algebraic spaces (Lemma \ref{lemma-space-to-formal-space}). Let $T$ be a scheme and $T \to X$ a morphism. We have to show that $U \times_X T \to T$ is separated. Since $U \times_X T \to U \times_S T$ is a monomorphism, it suffices to show that $U \times_S T \to T$ is separated. As this is the base change of $U \to S$ this follows. We used in the argument above: Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-separated}, \ref{spaces-morphisms-lemma-composition-separated}, \ref{spaces-morphisms-lemma-monomorphism-separated}, and \ref{spaces-morphisms-lemma-separated-implies-morphism-separated}. \end{proof} \section{Quasi-compact formal algebraic spaces} \label{section-quasi-compact} \noindent Here is the characterization of quasi-compact formal algebraic spaces. \begin{lemma} \label{lemma-characterize-quasi-compact} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent \begin{enumerate} \item the reduction of $X$ (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact algebraic space, \item we can find $\{X_i \to X\}_{i \in I}$ as in Definition \ref{definition-formal-algebraic-space} with $I$ finite, \item there exists a morphism $Y \to X$ representable by algebraic spaces which is \'etale and surjective and where $Y$ is an affine formal algebraic space. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{definition} \label{definition-quasi-compact} Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say $X$ is {\it quasi-compact} if the equivalent conditions of Lemma \ref{lemma-characterize-quasi-compact} are satisfied. \end{definition} \begin{lemma} \label{lemma-characterize-quasi-compact-morphism} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item the induced map $f_{red} : X_{red} \to Y_{red}$ between reductions (Lemma \ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact morphism of algebraic spaces, \item for every quasi-compact scheme $T$ and morphism $T \to Y$ the fibre product $X \times_Y T$ is a quasi-compact formal algebraic space, \item for every affine scheme $T$ and morphism $T \to Y$ the fibre product $X \times_Y T$ is a quasi-compact formal algebraic space, and \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} such that each $X \times_Y Y_j$ is a quasi-compact formal algebraic space. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{definition} \label{definition-quasi-compact-morphism} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. We say $f$ is {\it quasi-compact} if the equivalent conditions of Lemma \ref{lemma-characterize-quasi-compact-morphism} are satisfied. \end{definition} \noindent This agrees with the already existing notion when the morphism is representable by algebraic spaces (and in particular when it is representable). \begin{lemma} \label{lemma-quasi-compact-representable} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces. Then $f$ is quasi-compact in the sense of Definition \ref{definition-quasi-compact-morphism} if and only if $f$ is quasi-compact in the sense of Bootstrap, Definition \ref{bootstrap-definition-property-transformation}. \end{lemma} \begin{proof} This is immediate from the definitions and Lemma \ref{lemma-characterize-quasi-compact-morphism}. \end{proof} \section{Quasi-compact and quasi-separated formal algebraic spaces} \label{section-quasi-compact-quasi-separated} \noindent The following result is due to Yasuda, see \cite[Proposition 3.32]{Yasuda}. \begin{lemma} \label{lemma-structure-quasi-compact-quasi-separated} \begin{reference} \cite[Proposition 3.32]{Yasuda} \end{reference} Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated formal algebraic space over $S$. Then $X = \colim X_\lambda$ for a system of algebraic spaces $(X_\lambda, f_{\lambda \mu})$ over a directed set $\Lambda$ where each $f_{\lambda \mu} : X_\lambda \to X_\mu$ is a thickening. \end{lemma} \begin{proof} By Lemma \ref{lemma-characterize-quasi-compact} we may choose an affine formal algebraic space $Y$ and a representable surjective \'etale morphism $Y \to X$. Write $Y = \colim Y_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. \medskip\noindent Pick $\lambda \in \Lambda$. Then $Y_\lambda \times_X Y$ is a scheme by Lemma \ref{lemma-presentation-representable}. The reduction (Lemma \ref{lemma-reduction-formal-algebraic-space}) of $Y_\lambda \times_X Y$ is equal to the reduction of $Y_{red} \times_{X_{red}} Y_{red}$ which is quasi-compact as $X$ is quasi-separated and $Y_{red}$ is affine. Therefore $Y_\lambda \times_X Y$ is a quasi-compact scheme. Hence there exists a $\mu \geq \lambda$ such that $\text{pr}_2 : Y_\lambda \times_X Y \to Y$ factors through $Y_\mu$, see Lemma \ref{lemma-factor-through-thickening}. Let $Z_\lambda$ be the scheme theoretic image of the morphism $\text{pr}_2 : Y_\lambda \times_X Y \to Y_\mu$. This is independent of the choice of $\mu$ and we can and will think of $Z_\lambda \subset Y$ as the scheme theoretic image of the morphism $\text{pr}_2 : Y_\lambda \times_X Y \to Y$. Observe that $Z_\lambda$ is also equal to the scheme theoretic image of the morphism $\text{pr}_1 : Y \times_X Y_\lambda \to Y$ since this is isomorphic to the morphism used to define $Z_\lambda$. We claim that $Z_\lambda \times_X Y = Y \times_X Z_\lambda$ as subfunctors of $Y \times_X Y$. Namely, since $Y \to X$ is \'etale we see that $Z_\lambda \times_X Y$ is the scheme theoretic image of the morphism $$\text{pr}_{13} = \text{pr}_1 \times \text{id}_Y : Y \times_X Y_\lambda \times_X Y \longrightarrow Y \times_X Y$$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}. By the same token, $Y \times_X Z_\lambda$ is the scheme theoretic image of the morphism $$\text{pr}_{13} = \text{id}_Y \times \text{pr}_2 : Y \times_X Y_\lambda \times_X Y \longrightarrow Y \times_X Y$$ The claim follows. Then $R_\lambda = Z_\lambda \times_X Y = Y \times_X Z_\lambda$ together with the morphism $R_\lambda \to Z_\lambda \times_S Z_\lambda$ defines an \'etale equivalence relation. In this way we obtain an algebraic space $X_\lambda = Z_\lambda/R_\lambda$. By construction the diagram $$\xymatrix{ Z_\lambda \ar[r] \ar[d] & Y \ar[d] \\ X_\lambda \ar[r] & X }$$ is cartesian (because $X$ is the coequalizer of the two projections $R = Y \times_X Y \to Y$, because $Z_\lambda \subset Y$ is $R$-invariant, and because $R_\lambda$ is the restriction of $R$ to $Z_\lambda$). Hence $X_\lambda \to X$ is representable and a closed immersion, see Spaces, Lemma \ref{spaces-lemma-morphism-sheaves-with-P-effective-descent-etale}. On the other hand, since $Y_\lambda \subset Z_\lambda$ we see that $(X_\lambda)_{red} = X_{red}$, in other words, $X_\lambda \to X$ is a thickening. Finally, we claim that $$X = \colim X_\lambda$$ We have $Y \times_X X_\lambda = Z_\lambda \supset Y_\lambda$. Every morphism $T \to X$ where $T$ is a scheme over $S$ lifts \'etale locally to a morphism into $Y$ which lifts \'etale locally into a morphism into some $Y_\lambda$. Hence $T \to X$ lifts \'etale locally on $T$ to a morphism into $X_\lambda$. This finishes the proof. \end{proof} \begin{remark} \label{remark-structure-quasi-compact-quasi-separated} In this remark we translate the statement and proof of Lemma \ref{lemma-structure-quasi-compact-quasi-separated} into the language of formal schemes \`a la EGA. Looking at Remark \ref{remark-ideals-of-definition} we see that the lemma can be translated as follows \begin{itemize} \item[$(*)$] Every quasi-compact and quasi-separated formal scheme has a fundamental system of ideals of definition. \end{itemize} To prove this we first use the induction principle (reformulated for quasi-compact and quasi-separated formal schemes) of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle} to reduce to the following situation: $\mathfrak X = \mathfrak U \cup \mathfrak V$ with $\mathfrak U$, $\mathfrak V$ open formal subschemes, with $\mathfrak V$ affine, and the result is true for $\mathfrak U$, $\mathfrak V$, and $\mathfrak U \cap \mathfrak V$. Pick any ideals of definition $\mathcal{I} \subset \mathcal{O}_\mathfrak U$ and $\mathcal{J} \subset \mathcal{O}_\mathfrak V$. By our assumption that we have a fundamental system of ideals of definition on $\mathfrak U$ and $\mathfrak V$ and because $\mathfrak U \cap \mathfrak V$ is quasi-compact, we can find ideals of definition $\mathcal{I}' \subset \mathcal{I}$ and $\mathcal{J}' \subset \mathcal{J}$ such that $$\mathcal{I}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{J}|_{\mathfrak U \cap \mathfrak V} \quad\text{and}\quad \mathcal{J}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{I}|_{\mathfrak U \cap \mathfrak V}$$ Let $U \to U' \to \mathfrak U$ and $V \to V' \to \mathfrak V$ be the closed immersions determined by the ideals of definition $\mathcal{I}' \subset \mathcal{I} \subset \mathcal{O}_\mathfrak U$ and $\mathcal{J}' \subset \mathcal{J} \subset \mathcal{O}_\mathfrak V$. Let $\mathfrak U \cap V$ denote the open subscheme of $V$ whose underlying topological space is that of $\mathfrak U \cap \mathfrak V$. By our choice of $\mathcal{I}'$ there is a factorization $\mathfrak U \cap V \to U'$. We define similarly $U \cap \mathfrak V$ which factors through $V'$. Then we consider $$Z_U = \text{scheme theoretic image of } U \amalg (\mathfrak U \cap V) \longrightarrow U'$$ and $$Z_V = \text{scheme theoretic image of } (U \cap \mathfrak V) \amalg V \longrightarrow V'$$ Since taking scheme theoretic images of quasi-compact morphisms commutes with restriction to opens (Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}) we see that $Z_U \cap \mathfrak V = \mathfrak U \cap Z_V$. Thus $Z_U$ and $Z_V$ glue to a scheme $Z$ which comes equipped with a morphism $Z \to \mathfrak X$. Analogous to the discussion in Remark \ref{remark-weak-ideals-of-definition} we see that $Z$ corresponds to a weak ideal of definition $\mathcal{I}_Z \subset \mathcal{O}_\mathfrak X$. Note that $Z_U \subset U'$ and that $Z_V \subset V'$. Thus the collection of all $\mathcal{I}_Z$ constructed in this manner forms a fundamental system of weak ideals of definition. Hence a subfamily gives a fundamental system of ideals of definition, see Remark \ref{remark-ideals-of-definition}. \end{remark} \section{Morphisms representable by algebraic spaces} \label{section-representable} \noindent Let $f : X \to Y$ be a morphism of formal algebraic spaces which is representable by algebraic spaces. For these types of morphisms we have a lot of theory at our disposal, thanks to the work done in the chapters on algebraic spaces. \begin{lemma} \label{lemma-composition-representable} The composition of morphisms representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes). \end{lemma} \begin{proof} See Bootstrap, Lemma \ref{bootstrap-lemma-composition-transformation}. \end{proof} \begin{lemma} \label{lemma-base-change-representable} A base change of a morphism representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes). \end{lemma} \begin{proof} See Bootstrap, Lemma \ref{bootstrap-lemma-base-change-transformation}. \end{proof} \begin{lemma} \label{lemma-permanence-representable} Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces over $S$. If $g \circ f : X \to Z$ is representable by algebraic spaces, then $f : X \to Y$ is representable by algebraic spaces. \end{lemma} \begin{proof} Note that the diagonal of $Y \to Z$ is representable by Lemma \ref{lemma-diagonal-morphism-formal-algebraic-spaces}. Thus $X \to Y$ is representable by algebraic spaces by Bootstrap, Lemma \ref{bootstrap-lemma-representable-by-spaces-permanence}. \end{proof} \noindent The property of being representable by algebraic spaces is local on the source and the target. \begin{lemma} \label{lemma-representable-by-algebraic-spaces-local} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item the morphism $f$ is representable by algebraic spaces, \item there exists a commutative diagram $$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$$ where $U$, $V$ are formal algebraic spaces, the vertical arrows are representable by algebraic spaces, $U \to X$ is surjective \'etale, and $U \to V$ is representable by algebraic spaces, \item for any commutative diagram $$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$$ where $U$, $V$ are formal algebraic spaces and the vertical arrows are representable by algebraic spaces, the morphism $U \to V$ is representable by algebraic spaces, \item there exists a covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $j$ a covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Definition \ref{definition-formal-algebraic-space} such that $X_{ji} \to Y_j$ is representable by algebraic spaces for each $j$ and $i$, \item there exist a covering $\{X_i \to X\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $i$ a factorization $X_i \to Y_i \to Y$ where $Y_i$ is an affine formal algebraic space, $Y_i \to Y$ is representable by algebraic spaces, such that $X_i \to Y_i$ is representable by algebraic spaces, and \item add more here. \end{enumerate} \end{lemma} \begin{proof} It is clear that (1) implies (2) because we can take $U = X$ and $V = Y$. Conversely, (2) implies (1) by Bootstrap, Lemma \ref{bootstrap-lemma-representable-by-spaces-cover} applied to $U \to X \to Y$. \medskip\noindent Assume (1) is true and consider a diagram as in (3). Then $U \to Y$ is representable by algebraic spaces (as the composition $U \to X \to Y$, see Bootstrap, Lemma \ref{bootstrap-lemma-composition-transformation}) and factors through $V$. Thus $U \to V$ is representable by algebraic spaces by Lemma \ref{lemma-permanence-representable}. \medskip\noindent It is clear that (3) implies (2). Thus now (1) -- (3) are equivalent. \medskip\noindent Observe that the condition in (4) makes sense as the fibre product $Y_j \times_Y X$ is a formal algebraic space by Lemma \ref{lemma-fibre-products}. It is clear that (4) implies (5). \medskip\noindent Assume $X_i \to Y_i \to Y$ as in (5). Then we set $V = \coprod Y_i$ and $U = \coprod X_i$ to see that (5) implies (2). \medskip\noindent Finally, assume (1) -- (3) are true. Thus we can choose any covering $\{Y_j \to Y\}$ as in Definition \ref{definition-formal-algebraic-space} and for each $j$ any covering $\{X_{ji} \to Y_j \times_Y X\}$ as in Definition \ref{definition-formal-algebraic-space}. Then $X_{ij} \to Y_j$ is representable by algebraic spaces by (3) and we see that (4) is true. This concludes the proof. \end{proof} \begin{lemma} \label{lemma-algebraic-space-over-affine-formal} Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space over $S$. Let $f : X \to Y$ be a map of sheaves on $(\Sch/S)_{fppf}$ which is representable by algebraic spaces. Then $X$ is a formal algebraic space. \end{lemma} \begin{proof} Write $Y = \colim Y_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. For each $\lambda$ the fibre product $X \times_Y Y_\lambda$ is an algebraic space. Hence $X = \colim X \times_Y Y_\lambda$ is a formal algebraic space by Lemma \ref{lemma-colimit-is-formal}. \end{proof} \begin{lemma} \label{lemma-representable-by-algebraic-spaces} Let $S$ be a scheme. Let $Y$ be a formal algebraic space over $S$. Let $f : X \to Y$ be a map of sheaves on $(\Sch/S)_{fppf}$ which is representable by algebraic spaces. Then $X$ is a formal algebraic space. \end{lemma} \begin{proof} Let $\{Y_i \to Y\}$ be as in Definition \ref{definition-formal-algebraic-space}. Then $X \times_Y Y_i \to X$ is a family of morphisms representable by algebraic spaces, \'etale, and jointly surjective. Thus it suffices to show that $X \times_Y Y_i$ is a formal algebraic space, see Lemma \ref{lemma-etale-covering-by-formal-algebraic-spaces}. This follows from Lemma \ref{lemma-algebraic-space-over-affine-formal}. \end{proof} \begin{lemma} \label{lemma-affine-representable-by-algebraic-spaces} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then $f$ is representable (by schemes) and affine. \end{lemma} \begin{proof} The first assertion follows from Lemma \ref{lemma-presentation-representable}. Write $Y = \colim Y_\mu$ and $X = \colim X_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. For the second, let $T \to Y$ be a morphism where $T$ is a scheme over $S$. We have to show that $X \times_Y T \to T$ is affine, see Spaces, Definition \ref{spaces-definition-relative-representable-property}. To do this we may assume that $T$ is affine and we have to prove that $X \times_Y T$ is affine. In this case $T \to Y$ factors through $Y_\mu \to Y$ for some $\mu$, see Lemma \ref{lemma-factor-through-thickening}. Since $f$ is quasi-compact we see that $X \times_Y T$ is quasi-compact (Lemma \ref{lemma-characterize-quasi-compact-morphism}). Hence $X \times_Y T \to X$ factors through $X_\lambda$ for some $\lambda$. Similarly $X_\lambda \to Y$ factors through $Y_\mu$ after increasing $\mu$. Then $X \times_Y T = X_\lambda \times_{Y_\mu} T$. We conclude as fibre products of affine schemes are affine. \end{proof} \begin{lemma} \label{lemma-property-goes-up-affine-morphism} Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space. Let $f : X \to Y$ be a map of sheaves on $(\Sch/S)_{fppf}$ which is representable and affine. Then \begin{enumerate} \item $X$ is an affine formal algebraic space. \item if $Y$ is countably indexed, then $X$ is countably indexed. \item if $Y$ is adic*, then $X$ is adic*, \item if $Y$ is Noetherian and $f$ is (locally) of finite type, then $X$ is Noetherian. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). Write $Y = \colim_{\lambda \in \Lambda} Y_\lambda$ as in Definition \ref{definition-affine-formal-algebraic-space}. Since $f$ is representable and affine, the fibre products $X_\lambda = Y_\lambda \times_Y X$ are affine. And $X = \colim Y_\lambda \times_Y X$. Thus $X$ is an affine formal algebraic space. \medskip\noindent Proof of (2). If $Y$ is countably indexed, then in the argument above we may assume $\Lambda$ is countable. Then we immediately see that $X$ is countably indexed too. \medskip\noindent Proof of (3). Assume $Y$ is adic*. Then $Y = \text{Spf}(B)$ for some adic topological ring $B$ which has a finitely generated ideal $J$ such that $\{J^n\}$ is a fundamental system of open ideals. Of course, then $Y = \colim \Spec(B/J^n)$. The schemes $X \times_Y \Spec(B/J^n)$ are affine and we can write $X \times_Y \Spec(B/J^n) = \Spec(A_n)$. Then $X = \colim \Spec(A_n)$. The $B$-algebra maps $A_{n + 1} \to A_n$ are surjective and induce isomorphisms $A_{n + 1}/J^nA_{n + 1} \to A_n$. By Algebra, Lemma \ref{algebra-lemma-limit-complete} the ring $A = \lim A_n$ is $J$-adically complete and $A/J^n A = A_n$. Hence $X = \text{Spf}(A^\wedge)$ is adic*. \medskip\noindent Proof of (4). Combining (3) with Lemma \ref{lemma-implications-between-types} we see that $X$ is adic*. Thus we can use the criterion of Lemma \ref{lemma-characterize-noetherian-affine}. First, it tells us the affine schemes $Y_\lambda$ are Noetherian. Then $X_\lambda \to Y_\lambda$ is of finite type, hence $X_\lambda$ is Noetherian too (Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}). Then the criterion tells us $X$ is Noetherian and the proof is complete. \end{proof} \begin{lemma} \label{lemma-property-goes-up-affine} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then \begin{enumerate} \item if $Y$ is countably indexed, then $X$ is countably indexed. \item if $Y$ is adic*, then $X$ is adic*, \item if $Y$ is Noetherian and $f$ is (locally) of finite type, then $X$ is Noetherian. \end{enumerate} \end{lemma} \begin{proof} Combine Lemmas \ref{lemma-affine-representable-by-algebraic-spaces} and \ref{lemma-property-goes-up-affine-morphism}. \end{proof} \begin{lemma} \label{lemma-representable-affine} Let $S$ be a scheme. Let $\varphi : A \to B$ be a continuous map of weakly admissible topological rings over $S$. The following are equivalent \begin{enumerate} \item $\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)$ is representable by algebraic spaces, \item $\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)$ is representable (by schemes), \item $\varphi$ is taut, see Definition \ref{definition-taut}. \end{enumerate} \end{lemma} \begin{proof} Parts (1) and (2) are equivalent by Lemma \ref{lemma-affine-representable-by-algebraic-spaces}. \medskip\noindent Assume the equivalent conditions (1) and (2) hold. If $I \subset A$ is a weak ideal of definition, then $\Spec(A/I) \to \text{Spf}(A)$ is representable and a thickening (this is clear from the construction of the formal spectrum but it also follows from Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space}). Then $\Spec(A/I) \times_{\text{Spf}(A)} \text{Spf}(B) \to \text{Spf}(B)$ is representable and a thickening as a base change. Hence by Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space} there is a weak ideal of definition $J(I) \subset B$ such that $\Spec(A/I) \times_{\text{Spf}(A)} \text{Spf}(B) = \Spec(B/J(I))$ as subfunctors of $\text{Spf}(B)$. We obtain a cartesian diagram $$\xymatrix{ \Spec(B/J(I)) \ar[d] \ar[r] & \Spec(A/I) \ar[d] \\ \text{Spf}(B) \ar[r] & \text{Spf}(A) }$$ By Lemma \ref{lemma-fibre-product-affines-over-separated} we see that $B/J(I) = B \widehat{\otimes}_A A/I$. It follows that $J(I)$ is the closure of the ideal $\varphi(I)B$, see Lemma \ref{lemma-closure-image-ideal}. Since $\text{Spf}(A) = \colim \Spec(A/I)$ with $I$ as above, we find that $\text{Spf}(B) = \colim \Spec(B/J(I))$. Thus the ideals $J(I)$ form a fundamental system of weak ideals of definition (see Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space}). Hence (3) holds. \medskip\noindent Assume (3) holds. We are essentially just going to reverse the arguments given in the previous paragraph. Let $I \subset A$ be a weak ideal of definition. By Lemma \ref{lemma-fibre-product-affines-over-separated} we get a cartesian diagram $$\xymatrix{ \text{Spf}(B \widehat{\otimes}_A A/I) \ar[d] \ar[r] & \Spec(A/I) \ar[d] \\ \text{Spf}(B) \ar[r] & \text{Spf}(A) }$$ If $J(I)$ is the closure of $IB$, then $J(I)$ is open in $B$ by tautness of $\varphi$. Hence if $J$ is open in $B$ and $J \subset J(B)$, then $B/J \otimes_A A/I = B/(IB + J) = B/J(I)$ because $J(I) = \bigcap_{J \subset B\text{ open}} (IB + J)$ by Lemma \ref{lemma-closed}. Hence the limit defining the completed tensor product collapses to give $B \widehat{\otimes}_A A/I = B/J(I)$. Thus $\text{Spf}(B \widehat{\otimes}_A A/I) = \Spec(B/J(I))$. This proves that $\text{Spf}(B) \times_{\text{Spf}(A)} \Spec(A/I)$ is representable for every weak ideal of definition $I \subset A$. Since every morphism $T \to \text{Spf}(A)$ with $T$ quasi-compact factors through $\Spec(A/I)$ for some weak ideal of definition $I$ (Lemma \ref{lemma-factor-through-thickening}) we conclude that $\text{Spf}(\varphi)$ is representable, i.e., (2) holds. This finishes the proof. \end{proof} \begin{example} \label{example-representable-morphism-from-completion} Let $B$ be a weakly admissible topological ring. Let $B \to A$ be a ring map (no topology). Then we can can consider $$A^\wedge = \lim A/JA$$ where the limit is over all weak ideals of definition $J$ of $B$. Then $A^\wedge$ (endowed with the limit topology) is a complete linearly topologized ring. The (open) kernel $I$ of the surjection $A^\wedge \to A/JA$ is the closure of $JA^\wedge$, see Lemma \ref{lemma-closed}. By Lemma \ref{lemma-topologically-nilpotent} we see that $I$ consists of topologically nilpotent elements. Thus $I$ is a weak ideal of definition of $A^\wedge$ and we conclude $A^\wedge$ is a weakly admissible topological ring. Thus $\varphi : B \to A^\wedge$ is taut map of weakly admissible topological rings and $$\text{Spf}(A^\wedge) \longrightarrow \text{Spf}(B)$$ is a special case of the phenomenon studied in Lemma \ref{lemma-representable-affine}. \end{example} \begin{remark}[Warning] \label{remark-warning} Lemma \ref{lemma-representable-affine} is sharp in the following two senses: \begin{enumerate} \item If $A$ and $B$ are weakly admissible rings and $\varphi : A \to B$ is a continuous map, then $\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)$ is in general not representable. \item If $f : Y \to X$ is a representable morphism of affine formal algebraic spaces and $X = \text{Spf}(A)$ is McQuillan, then it does not follow that $Y$ is McQuillan. \end{enumerate} An example for (1) is to take $A = k$ a field (with discrete topology) and $B = k[[t]]$ with the $t$-adic topology. An example for (2) is given in Examples, Section \ref{examples-section-affine-formal-algebraic-space}. \end{remark} \noindent The warning above notwithstanding, we do have the following result. \begin{lemma} \label{lemma-etale} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces over $S$. Assume \begin{enumerate} \item $Y$ is McQuillan, i.e., equal to $\text{Spf}(B)$ for some weakly admissible topological $S$-algebra $B$, and \item $f : X \to Y$ is representable by algebraic spaces and \'etale. \end{enumerate} Then there exists an \'etale ring map $B \to A$ such that $$X = \text{Spf}(A^\wedge) \quad\text{where}\quad A^\wedge = \lim A/JA$$ with $J \subset B$ running over the weak ideals of definition of $B$. In particular, $X$ is McQuillan. \end{lemma} \begin{proof} Choose a weak ideal of definition $J_0 \subset B$. Set $Y_0 = \Spec(B/J_0)$ and $X_0 = Y_0 \times_Y X$. Then $X_0 \to Y_0$ is an \'etale morphism of affine schemes (see Lemma \ref{lemma-affine-representable-by-algebraic-spaces}). Say $X_0 = \Spec(A_0)$. By Algebra, Lemma \ref{algebra-lemma-lift-etale} we can find an \'etale algebra map $B \to A$ such that $A_0 \cong A/J_0A$. Consider an ideal of definition $J \subset J_0$. As above we may write $\Spec(B/J) \times_Y X = \Spec(\bar A)$ for some \'etale ring map $B/J \to \bar A$. Then both $B/J \to \bar A$ and $B/J \to A/JA$ are \'etale ring maps lifting the \'etale ring map $B/J_0 \to A_0$. By More on Algebra, Lemma \ref{more-algebra-lemma-locally-nilpotent-henselian} there is a unique $B/J$-algebra isomorphism $\varphi_J : A/JA \to \bar A$ lifting the identification modulo $J_0$. Since the maps $\varphi_J$ are unique they are compatible for varying $J$. Thus $$X = \colim \Spec(B/J) \times_Y X = \colim \Spec(A/JA)$$ and we see that the lemma holds. \end{proof} \begin{lemma} \label{lemma-etale-surjective} With notation and assumptions as in Lemma \ref{lemma-etale}. The following are equivalent \begin{enumerate} \item