formal-spaces.tex

\input{preamble}

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\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 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 abelian 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 products, so we can define the notion of a
sheaf as in Sheaves, Definition
\ref{sheaves-definition-sheaf-values-in-category}.
Moreover, products still commute with the forgetful functor
to $\textit{Sets}$ so the underlying presheaf of sets
of a sheaf of topological spaces (for example) is a sheaf of
sets. 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, groups, rings, 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, abelian groups, rings, or 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, which is the same as taking
the colimit on the level of underlying sheaves of sets, groups,
rings, or modules but comes equipped with a topology (the colimit
topology). One can sheafify presheaves of topological spaces,
abelian groups, rings, or modules and taking stalks commuts
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, abelian groups, rings, 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 topoligical 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 abelian groups, rings, 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 $\mathcal{G}$ of sets, abelian groups,
rings, modules over a ring, one can endow $\mathcal{G}$ with
the structure of a sheaf of topological spaces, abelian groups,
rings, 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, groups, rings, modules in this fashion. Let us say
that a sheaf of topological spaces, groups, rings, 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 space
(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 tranformation 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 morphsm
$i_\lambda : \Spec(A/I_\lambda) \to \mathfrak X$ 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 $\Spec(A/I_\lambda)$ occuring 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
$x \in D(r) \subset g^{-1}(\mathfrak U)$.
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 $A \to S_r$
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 defintion: an
open ideal $I$ such that any neighbourhood of $0$ contains $I^n$.
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 fuctor
$\Sh(\mathcal{C}, \textit{Top}) \to \Sh(\mathcal{C}, \textit{Top})$.
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-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.
\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}) and we win.
\end{proof}

\begin{definition}
\label{definition-toplogy-tensor-product}
Let $R$ be a topological ring. Let $M$ and $N$ be linearly
toplogized $R$-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 $J_{n + 1}$ 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. Since we have already introduced
the terminology ``taut'' for continuous ring maps, we will not introduce the
concept of an ``adic map between adic rings''.

\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, Lemmas \ref{algebra-lemma-hathat} and
\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 partially ordered 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 partially ordered set and by assumption
$X = \colim_{T \to X\text{ in }\Lambda} T$.
\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 definion 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 exlain 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 partially
ordered 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 a 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 defintion 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
Let $W \to X \times_S X$ be a morphism where $W$ is a scheme over $S$.
For each $i$ consider the diagram
$$
\xymatrix{
W \times_{(X \times_S X)} X_i \ar[d] \ar[r] &
W \times_{(X \times_S X)} (X_i \times_S X_i) \ar[d] \\
W \times_{(X \times_S X)} X \ar[r] & W
}
$$
The vertical arrows are \'etale because $X_i \to X$ is \'etale.
The top horizontal arrow is a closed immersion by
Lemma \ref{lemma-diagonal-affine-formal-algebraic-space}.
Because the maps $X_i \to X$ are jointly surjective, this certainly
implies that the lower horizontal arrow is locally of finite type, see
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-type-local}.
Since it is clearly a monomorphism, we conclude that
it is locally quasi-finite (Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}).
It is separated because any monomorphism is separated
(Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-monomorphism-separated}). This finishes
the proof.
\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 representabel 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 partially ordered 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{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 definiton}
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 $Z \to \mathfrak X$ as in
Remark \ref{remark-weak-ideals-of-definition}.
A {\it fundamental system of ideals of definiton}
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 $A$ 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.

\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.

\medskip\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 $\gamma$.
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 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
$$
J_{n - 1}^\wedge I + J_{n + 1}^\wedge \supset
J_{n - 1}^\wedge (I + J_{n + 1}^\wedge) \supset
J_{n - 1}^\wedge J_m^\wedge \supset
J_n^\wedge
$$
for all $n \geq m + 1$. Namely, the first inclusion
is trivial. The second was shown above. The third as
$J_{n - 1}J_m \supset J_{n - 1}^2 \supset J_n$,
hence $J_{n - 1}^\wedge J_m^\wedge \supset J_n^\wedge$.
Say $I = (g_1, \ldots, g_t)$. Pick $f \in J_{m + 1}^\wedge$.
Using the displayed inclusions above, 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 $f_1, \ldots, f_m$.
Case I: For some $m$ the closure of $I_m$ is open. Then $I_m$ is open by
the result of the previous paragraph. Since in $A^\wedge$ the
product of open ideals is open, we see that $I_m^k$ is open for all $k$.
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.,
$X$ 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. 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}

\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 analgous 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 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}





\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, and
\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.
\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 partially ordered 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 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
$\mathcal{J}' \subset \mathcal{J} \subset \mathcal{O}_\mathfrak U$.
Let $\mathfrak U \cap V$ denote the open subscheme of $V$ whose
underlying topological space is that of $\mathfrak U \cap \mathfrak V$.
Similarly for $U \cap \mathfrak V$. 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 $U \subset Z_U \subset Z$ and that
$V \subset Z_V \subset Z$. 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.

\medskip\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 space by Bootstrap, Lemma
\ref{bootstrap-lemma-representable-by-spaces-permanence}
(note that the diagonal of $V \to Y$ is representable by
Lemma \ref{lemma-diagonal-morphism-formal-algebraic-spaces}).

\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. For a weak ideal of definition $I \subset A$ denote
$J(I)$ the closure of $\varphi(I)B$. By
Lemma \ref{lemma-topologically-nilpotent}
the ideal $J(I)$ is a weak ideal of definition of $B$. Using
Lemmas \ref{lemma-fibre-product-affines-over-separated} and
\ref{lemma-closure-image-ideal}
we see that the diagram displayed above is cartesian. 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} does {\bf not} mean that given a
morphism $f : X \to Y$ of affine formal algebraic spaces with $f$
representable and $Y$ McQuillan we have that $X$ is McQuillan.
\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 $f : Y \to X$ is representable by algebraic spaces and \'etale.
\end{enumerate}
Then there exists an \'etale ring map $B \to A$ such that
$$
Y = \text{Spf}(A^\wedge)
\quad\text{where}\quad
A^\wedge = \lim A/JA
$$
with $J \subset B$ running over the weak ideals of definition of $B$.
In particular, $Y$ 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 $f : X \to Y$ is surjective,
\item $B \to A$ is faithfully flat,
\item for every weak ideal of definition $J \subset B$
the ring map $B \to A/JA$ is faithfully flat, and
\item for some weak ideal of definition $J \subset B$
the ring map $B \to A/JA$ is faithfully flat.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $J \subset B$ be an ideal of definition. As every element of $J$
is topologically nilpotent, we see that every element of $1 + J$ is
a unit. It follows that $J$ is contained in the Jacobson radical of $B$
(Algebra, Lemma \ref{algebra-lemma-contained-in-radical}).
Hence a flat ring map $B \to A$ is faithfully flat if and only if
$B/J \to A/JA$ is faithfully flat
(Algebra, Lemma \ref{algebra-lemma-ff-rings}).
In this way we see that (2) -- (4) are equivalent.
If (1) holds, then for every weak ideal of definition $J \subset B$
the morphism
$\Spec(A/JA) = \Spec(B/J) \times_Y X \to \Spec(B/J)$ is surjective
which implies (3). Conversely, assume (3).
A morphism $T \to Y$ with $T$ quasi-compact
factors through $\Spec(B/J)$ for some ideal of definition $J$ of $B$
(Lemma \ref{lemma-factor-through-thickening}).
Hence $X \times_Y T = \Spec(A/JA) \times_{\Spec(B/J)} T \to T$
is surjective as a base change of the surjective morphism
$\Spec(A/JA) \to \Spec(B/J)$. Thus (1) holds.
\end{proof}





\section{Types of formal algebraic spaces}
\label{section-types}

\noindent
In this section we define Noetherian, adic*, and
countably indexed formal algebraic spaces. The types
adic, classical, and McQuillan are missing as we do not
know how to prove the analogue of the following lemmas
for those cases.

\begin{lemma}
\label{lemma-iff-countably-indexed}
Let $S$ be a scheme. Let $X \to Y$ be a morphism of affine
formal schemes which is representable by algebraic spaces,
surjective, and flat. Then $X$ is countably indexed if and only
if $Y$ is countably indexed.
\end{lemma}

\begin{proof}
Assume $X$ is countably indexed. We write $X = \colim X_n$ as in
Lemma \ref{lemma-countable-affine-formal-algebraic-space}.
Write $Y = \colim Y_\lambda$ as in
Definition \ref{definition-affine-formal-algebraic-space}.
For every $n$ we can pick a $\lambda_n$ such that
$X_n \to Y$ factors through $Y_{\lambda_n}$, see
Lemma \ref{lemma-factor-through-thickening}.
On the other hand, for every $\lambda$ the scheme
$Y_\lambda \times_Y X$ is affine
(Lemma \ref{lemma-affine-representable-by-algebraic-spaces})
and hence $Y_\lambda \times_Y X \to X$ factors through
$X_n$ for some $n$ (Lemma \ref{lemma-factor-through-thickening}).
Picture
$$
\xymatrix{
Y_\mu \times_Y X \ar[r] \ar[d] \ar@/^2pc/[rr]_{} &
Y_\lambda \times_Y X \ar[r] \ar[d] & X_n \ar[r] \ar[d] & X \ar[d] \\
Y_\mu \ar[r]\ar@/_2pc/[rr]^{} &
Y_\lambda \ar@{..>}[r] & Y_{\lambda_n} \ar[r] & Y
}
$$
If we can show the dotted arrow exists, then we conclude that
$Y = \colim Y_{\lambda_n}$ and $Y$ is countably indexed. To do this we pick a
$\mu$ such that we have the solid arrows in the diagram. Say
$Y_\mu = \Spec(B_\mu)$, the closed subscheme $Y_\lambda$ corresponds to
$J \subset B_\mu$, and the closed subscheme $Y_{\lambda_n}$ corresponds to
$J' \subset B_\mu$. We are trying to show that $J \subset J'$.
By the diagram above we know $JA_\mu \subset J'A_\mu$
where $Y_\mu \times_Y X = \Spec(A_\mu)$.
Since $X \to Y$ is surjective and flat the morphism
$Y_\lambda \times_Y X \to Y_\lambda$ is a faithfully flat morphism
of affine schemes, hence $B_\mu \to A_\mu$ is
faithfully flat. Thus $J \subset J'$ as desired.

\medskip\noindent
Assume $Y$ is countably indexed. Then $X$ is countably indexed
by Lemma \ref{lemma-property-goes-up-affine}.
\end{proof}

\begin{lemma}
\label{lemma-iff-adic-star}
Let $S$ be a scheme. Let $X \to Y$ be a morphism of affine
formal schemes which is representable by algebraic spaces,
surjective, and flat. Then $X$ is adic* if and only if $Y$ is adic*.
\end{lemma}

\begin{proof}
Assume $Y$ is adic*. Then $X$ is adic* by
Lemma \ref{lemma-property-goes-up-affine}.

\medskip\noindent
Assume $X$ is adic*. Write $X = \text{Spf}(A)$ for some adic ring $A$
which has a finitely generated ideal $I$ such that $\{I^n\}$ is a
fundamental system of open ideals. By Lemmas \ref{lemma-iff-countably-indexed}
we see that $Y$ is countably indexed. Thus, by
Lemma \ref{lemma-countably-indexed},
we can write $Y = \text{Spf}(B)$ where $B$ is a weakly admissible
topological ring with a countable fundamental system $\{J_m\}$ of
weak ideals of definition. Set $Y_m = \Spec(B/J_m)$ so that
$Y = \colim Y_m$. The scheme $Y_m \times_Y X$ is affine
(Lemma \ref{lemma-affine-representable-by-algebraic-spaces})
and we have $X = \colim Y_m \times_Y X$. Say $Y_m \times_Y X = \Spec(A_m)$
so that $B/J_m \to A_m$ is a faithfully flat ring map.
It follows from Lemma \ref{lemma-fibre-product-affines-over-separated}
that $\Ker(A \to A_m)$ is the closure of $J_mA$.

\medskip\noindent
Choose $n \geq 1$. There exists an $m$ such that $\Spec(A/I^n) \to Y$
factors through $Y_m$. In terms of ideals
\begin{equation}
\label{equation-first}
\forall n\ \exists m,\ J_m A \subset I^n.
\end{equation}
Choose $m \geq 1$. We can find an $n$ such that the morphism
$\Spec(A_m) \to X$ factors through $\Spec(A/I^n)$. In terms of ideals
\begin{equation}
\label{equation-second}
\forall m\ \exists n,\ I^n \subset \Ker(A \to A_m).
\end{equation}

\medskip\noindent
Fix an $m$. Pick $n$ such that $I^n \subset \Ker(A \to A_m)$
(\ref{equation-second}). Choose generators $f_1, \ldots, f_r$
of $I$. For any $E = (e_1, \ldots, e_r)$ with $|E| = \sum e_i = n$ write
$$
f_1^{e_1} \ldots f_r^{e_r} =
\sum g_{E, j} a_{E, j} + \delta_E
$$
with $g_{E, j} \in J_m$, $a_{E, j} \in A$, and $\delta_E \in I^{n + 1}$
(possible by the above). Let $J = (g_{E, j}) \subset B$.
Then we see that
$$
I^n \subset J A + I^{n + 1}
$$
As $I$ is contained in the radical of $A$ and $I^n$ is finitely generated
we see that $I^n \subset JA$ by Algebra, Lemma \ref{algebra-lemma-NAK}.

\medskip\noindent
We first apply what we just proved as follows:
since $I^n \subset J_mA$ we see that
$J_mA$ is open in $A$, hence closed, hence $\Ker(A \to A_m) = J_mA$,
in other words, $A_m = A/J_mA$. This holds for every $m$.

\medskip\noindent
Next, we pick $m$ with $J_mA \subset I$ (\ref{equation-first}).
Then choose $J \subset J_m$ with $I^n \subset JA \subset I$ as above.
For every $k \geq 1$ we define $\mathfrak b_k = \Ker(B \to A/J^kA)$.
For every $k$ there exists
an $m'$ with $J_{m'} \subset \mathfrak b_k$ as we have
$I^{nk} \subset J^kA$ and we can apply (\ref{equation-first}).
On the other hand, for every $m'$ there exists a $k$ such that
$I^k \subset J_{m'}A$ because $J_{m'}A$ is open. Then
$\mathfrak b_k$ maps to zero in $A/J_{m'}A$ which is faithfully
flat over $B/J_{m'}$. Hence $\mathfrak b_k \subset J_{m'}$.
In other words, we see that the topology on $B$ is defined by the
sequence of ideals $\mathfrak b_k$. Note that $J^k \subset \mathfrak b_k$
which implies that $\mathfrak b_k A = J^kA$.
In other words, we have reduced the problem to the
situation discussed in the following paragraph.

\medskip\noindent
We are given a ring map $B \to A$ where
\begin{enumerate}
\item $B$ is a weakly admissible topological ring with a fundamental system
$J_1 \supset J_2 \supset J_3 \supset \ldots$ of ideals of definition,
\item $A$ is a ring complete with respect to a finitely generated
ideal $I$,
\item we have $J_k A = I^k$ for all $k$, and
\item $B/J_k \to A/I^k$ is faithfully flat.
\end{enumerate}
Pick $g_1, \ldots, g_r \in J_1$
whose images in $A/I^2$ generate $I/I^2$; this is possible because
$J_1A/J_2A = I/I^2$. Then for all $k \geq 1$ we see that the elements
$g^E = g_1^{e_1} \ldots g_r^{e_r}$ with $|E| = k$ are in $J_k$
and their classes in $J_k/J_{k + 1}$ map to generators of
$I^k/I^{k + 1}$. Since $B/J_{k + 1} \to A/I^{k + 1}$
and $B/J_k \to A/I^k$ are flat we see that
$$
J_k/J_{k + 1} \otimes_{B/J_1} A/I =
J_k/J_{k + 1} \otimes_{B/J_{k + 1}} A/I^{k + 1} \to I^k/I^{k + 1}
$$
is an isomorphism (see More on Morphisms, Lemma
\ref{more-morphisms-lemma-deform}). Since $B/J_1 \to A/I$ is
faithfully flat, we conclude that the classes of the elements
$g^E$, $|E| = k$ generate $J_k/J_{k + 1}$. We claim that
$J_k = (g^E, |E| = k)$. Namely, suppose that $x_k \in J_k$.
By the above we can write
$$
x_k = \sum\nolimits_{|E| = k} b_{E, 0} g^E + x_{k + 1}
$$
with $x_{k + 1} \in J_{k + 1}$ and some $b_{E, 0} \in B$. Now we can write
$x_{k + 1}$ as follows
$$
x_{k + 1} =
\sum\nolimits_{|E| = k}
\left(\sum\nolimits_{|E'| = 1} b_{E, E'}g^{E'}\right) g^E + x_{k + 2}
$$
because every multi-index of degree $k + 1$ is a sum of a multi-index
of degree $k$ and a multi-index of degree $1$. Continuing in this
manner we can find $b_{E, E'} \in B$ such that for every $l > 1$
we have
$$
x_k = \sum\nolimits_{|E| = k}
\left(\sum\nolimits_{0 \leq |E'| < l} b_{E, E'} g^{E'}\right) g^E + x_{k + l}
$$
with some $x_{k + l} \in J_{k + l}$. Then we can finally define
$$
b_E = \sum\nolimits_{E'} b_{E, E'} g^{E'}
$$
as an element in $B$ and we see that $x_k = \sum b_E g^E$ as desired.
This finishes the proof as now $J_1$ is finitely generated and $J_k = J_1^k$
for all $k \geq 1$.
\end{proof}

\begin{lemma}
\label{lemma-iff-noetherian}
Let $S$ be a scheme. Let $X \to Y$ be a morphism of affine
formal schemes which is representable by algebraic spaces,
surjective, flat, and (locally) of finite type. Then $X$ is Noetherian
if and only if $Y$ is Noetherian.
\end{lemma}

\begin{proof}
Observe that a Noetherian affine formal algebraic space is adic*, see
Lemma \ref{lemma-implications-between-types}. Thus by
Lemma \ref{lemma-iff-adic-star} we may assume that both $X$ and $Y$
are adic*. We will use the criterion of
Lemma \ref{lemma-characterize-noetherian-affine}
to see that the lemma holds. Namely, write $Y = \colim Y_n$
as in Lemma \ref{lemma-countable-affine-formal-algebraic-space}.
For each $n$ set $X_n = Y_n \times_Y X$. Then $X_n$ is an
affine scheme (Lemma \ref{lemma-affine-representable-by-algebraic-spaces})
and $X = \colim X_n$. Each of the morphisms $X_n \to Y_n$ is
faithfully flat and of finite type. Thus the lemma follows from the
fact that in this situation $X_n$ is Noetherian if and only if $Y_n$
is Noetherian, see
Algebra, Lemma \ref{algebra-lemma-descent-Noetherian} (to go down)
and
Algebra, Lemma \ref{algebra-lemma-Noetherian-permanence} (to go up).
\end{proof}

\begin{lemma}
\label{lemma-type-local}
Let $S$ be a scheme. Let $P \in \{countably\ indexed, adic*, Noetherian\}$.
Let $X$ be a formal algebraic space over $S$.
The following are equivalent
\begin{enumerate}
\item if $Y$ is an affine formal algebraic space and
$f : Y \to X$ is representable by algebraic spaces and \'etale,
then $Y$ has property $P$,
\item for some $\{X_i \to X\}_{i \in I}$ as in
Definition \ref{definition-formal-algebraic-space}
each $X_i$ has property $P$.
\end{enumerate}
\end{lemma}

\begin{proof}
It is clear that (1) implies (2). Assume (2) and let
$Y \to X$ be as in (1). Since the fibre products $X_i \times_Y X$
are formal algebraic spaces (Lemma \ref{lemma-fibre-products-general})
we can pick coverings $\{X_{ij} \to X_i \times_X Y\}$ as in
Definition \ref{definition-formal-algebraic-space}.
Since $Y$ is quasi-compact, there exist
$(i_1, j_1), \ldots, (i_n, j_n)$ such that
$$
X_{i_1 j_1} \amalg \ldots \amalg X_{i_n j_n} \longrightarrow Y
$$
is surjective. Then $X_{i_kj_k} \to X_{i_k}$ is representable
by algebraic spaces and \'etale hence $X_{i_kj_k}$ has property $P$ by
Lemma \ref{lemma-property-goes-up-affine}.
Then $X_{i_1 j_1} \amalg \ldots \amalg X_{i_n j_n}$ is an
affine formal algebraic space with property $P$ (small detail
omitted on finite disjoint unions of affine formal algebraic spaces).
Hence we conclude by applying one of
Lemmas \ref{lemma-iff-countably-indexed},
\ref{lemma-iff-adic-star}, and
\ref{lemma-iff-noetherian}.
\end{proof}

\noindent
The previous lemma clears the way for the following definition.

\begin{definition}
\label{definition-types-formal-algebraic-spaces}
Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.
We say $X$ is {\it locally countably indexed},
{\it locally adic*}, or {\it locally Noetherian}
if the equivalent conditions of Lemma \ref{lemma-type-local}
hold for the corresponding property.
\end{definition}






\section{Morphisms and continuous ring maps}
\label{section-morphisms-rings}

\noindent
In this section we denote $\textit{WAdm}$ the category of
weakly admissible topological rings and continuous ring homomorphisms.
We define full subcategories
$$
\textit{WAdm} \supset \textit{WAdm}^{count} \supset
\textit{WAdm}^{adic*} \supset \textit{WAdm}^{Noeth}
$$
whose objects are
\begin{enumerate}
\item $\textit{WAdm}^{count}$: those weakly admissible topological rings
$A$ which have a countable fundamental system of neighbourhoods of $0$,
\item $\textit{WAdm}^{adic*}$: the adic topological rings which have
a finitely generated ideal of definition, and
\item $\textit{WAdm}^{Noeth}$: the adic topological rings which
are Noetherian.
\end{enumerate}
Clearly, the formal spectra of these types of rings are the basic
building blocks of locally countably indexed, locally adic*, and
locally Noetherian formal algebraic spaces.

\medskip\noindent
We briefly review the relationship between morphisms of
countably indexed, affine formal algebraic spaces and
morphisms of $\textit{WAdm}^{count}$.
Let $S$ be a scheme. Let $X$ and $Y$ be countably indexed,
affine formal algebraic spaces. Write $X = \text{Spf}(A)$
and $Y = \text{Spf}(B)$ topological $S$-algebras
$A$ and $B$ in $\textit{WAdm}^{count}$, see
Lemma \ref{lemma-countably-indexed}.
By Lemma \ref{lemma-morphism-between-formal-spectra}
there is a 1-to-1 correspondence between morphisms
$f : X \to Y$ and continuous maps
$$
\varphi : B \longrightarrow A
$$
of topological $S$-algebras. The relationship is given by
$f \mapsto f^\sharp$ and $\varphi \mapsto \text{Spf}(\varphi)$.

\medskip\noindent
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of
locally countably indexed formal algebraic spaces. Consider a
commutative diagram
$$
\xymatrix{
U \ar[d] \ar[r] & V \ar[d] \\
X \ar[r] & Y
}
$$
with $U$ and $V$ affine formal algebraic spaces and $U \to X$ and $V \to Y$
representable by algebraic spaces and \'etale. By
Definition \ref{definition-types-formal-algebraic-spaces} (and hence via
Lemma \ref{lemma-type-local}) we see that $U$ and $V$ are countably indexed
affine formal algebraic spaces. By the discussion in the previous
paragraph we see that $U \to V$ is isomorphic to $\text{Spf}(\varphi)$
for some continuous map
$$
\varphi : B \longrightarrow A
$$
of topological $S$-algebras in $\textit{WAdm}^{count}$.

\begin{lemma}
\label{lemma-completion-in-sub}
Let $A \in \Ob(\textit{WAdm})$. Let $A \to A'$ be a ring
map (no topology). Let $(A')^\wedge = \lim_{I \subset A\text{ w.i.d}} A'/IA'$
be the object of $\textit{WAdm}$ constructed in
Example \ref{example-representable-morphism-from-completion}.
\begin{enumerate}
\item If $A$ is in $\textit{WAdm}^{count}$, so is $(A')^\wedge$.
\item If $A$ is in $\textit{WAdm}^{adic*}$, so is $(A')^\wedge$.
\item If $A$ is in $\textit{WAdm}^{Noeth}$ and $A'$ is Noetherian, then
$(A')^\wedge$ is in $\textit{WAdm}^{Noeth}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) is clear from the construction.
Assume $A$ has a finitely generated ideal of definition $I \subset A$.
Then $I^n(A')^\wedge =  \Ker((A')^\wedge \to A'/I^nA')$ by Algebra, Lemmas
\ref{algebra-lemma-hathat} and
\ref{algebra-lemma-hathat-finitely-generated}.
Thus $I(A')^\wedge$ is a finitely generated ideal of definition
and we see that (2) holds.
Finally, assume that $A$ is Noetherian and adic.
By (2) we know that $(A')^\wedge$ is adic.
By Algebra, Lemma \ref{algebra-lemma-completion-Noetherian-Noetherian}
we see that $(A')^\wedge$ is Noetherian. Hence (3) holds.
\end{proof}

\noindent
Let $P$ be a property of morphisms of $\textit{WAdm}^{count}$.
Consider commutative diagrams
\begin{equation}
\label{equation-localize}
\vcenter{
\xymatrix{
A \ar[r] & (A')^\wedge \\
B \ar[r] \ar[u]^\varphi & (B')^\wedge \ar[u]_{\varphi'}
}
}
\end{equation}
satisfying the following conditions
\begin{enumerate}
\item $A$ and $B$ are objects of $\textit{WAdm}^{count}$,
\item $A \to A'$ and $B \to B'$ are \'etale ring maps,
\item $(A')^\wedge = \lim A'/IA'$, resp.\  $(B')^\wedge = \lim B'/JB'$
where $I \subset A$, resp.\ $J \subset B$
runs through the weakly admissible ideals of definition of $A$, resp.\ $B$,
\item $\varphi : A \to B$ and $\varphi' : (A')^\wedge \to (B')^\wedge$
are continuous.
\end{enumerate}
By Lemma \ref{lemma-completion-in-sub} the topological rings
$(A')^\wedge$ and $(B')^\wedge$ are objects of $\textit{WAdm}^{count}$.
We say $P$ is a {\it local property} if the following axioms hold:
\begin{enumerate}
\item
\label{item-axiom-1}
for any diagram (\ref{equation-localize}) we have
$P(\varphi) \Rightarrow P(\varphi')$,
\item
\label{item-axiom-2}
for any diagram (\ref{equation-localize}) with $A \to A'$
faithfully flat we have
$P(\varphi') \Rightarrow P(\varphi)$,
\item
\label{item-axiom-3}
if $P(A \to B_i)$ for $i = 1, \ldots, n$, then
$P(A \to \prod_{i = 1, \ldots, n} B_i)$.
\end{enumerate}
Axiom (\ref{item-axiom-3})
makes sense as $\textit{WAdm}^{count}$ has finite products.

\begin{lemma}
\label{lemma-property-defines-property-morphisms}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of
locally countably indexed formal algebraic spaces over $S$.
Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$.
The following are equivalent
\begin{enumerate}
\item for every commutative diagram
$$
\xymatrix{
U \ar[d] \ar[r] & V \ar[d] \\
X \ar[r] & Y
}
$$
with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$
representable by algebraic spaces and \'etale, the morphism $U \to V$
corresponds to a morphism of $\textit{WAdm}^{count}$ with property $P$,
\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 each $X_{ji} \to Y_j$  corresponds
to a morphism of $\textit{WAdm}^{count}$ with property $P$, and
\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 and \'etale, and $X_i \to Y_i$ corresponds
to a morphism of $\textit{WAdm}^{count}$ with property $P$.
\end{enumerate}
\end{lemma}

\begin{proof}
It is clear that (1) implies (2) and that (2) implies (3).
Assume $\{X_i \to X\}$ and $X_i \to Y_i \to Y$ as in (3)
and let a diagram as in (1) be given.
Since $Y_i \times_Y V$ is a formal algebraic space
(Lemma \ref{lemma-fibre-products-general}) we may pick
coverings $\{Y_{ij} \to Y_i \times_Y V\}$ as in
Definition \ref{definition-formal-algebraic-space}.
For each $(i, j)$ we may similarly choose coverings
we can pick coverings $\{X_{ijk} \to Y_{ij} \times_{Y_i} X_i \times_X U\}$
as in Definition \ref{definition-formal-algebraic-space}.
Since $U$ is quasi-compact we can choose
$(i_1, j_1, k_1), \ldots, (i_n, j_n, k_n)$ such that
$$
X_{i_1 j_1 k_1} \amalg \ldots \amalg X_{i_n j_n k_n} \longrightarrow U
$$
is surjective. For $s = 1, \ldots, n$ consider the commutative diagram
$$
\xymatrix{
& & & X_{i_s j_s k_s} \ar[ld] \ar[d] \ar[rd] \\
X \ar[d] & X_{i_s} \ar[l] \ar[d] &
X_{i_s} \times_X U \ar[l] \ar[d] & Y_{i_s j_s} \ar[ld] \ar[rd] &
X_{i_s} \times_X U \ar[d] \ar[r] &
U \ar[d] \ar[r] & X \ar[d] \\
Y & Y_{i_s} \ar[l] &
Y_{i_s} \times_Y V \ar[l] & &
Y_{i_s} \times_Y V \ar[r] &
V \ar[r] & Y
}
$$
Let us say that $P$ holds for a morphism of countably indexed
affine formal algebraic spaces if it holds for the corresponding
morphism of $\textit{WAdm}^{count}$. Observe that the maps
$X_{i_s j_s k_s} \to X_{i_s}$, $Y_{i_s j_s} \to Y_i$ are given by completions
of \'etale ring maps, see Lemma \ref{lemma-etale}.
Hence we see that $P(X_{i_s} \to Y_{i_s})$ implies
$P(X_{i_s j_s k_s} \to Y_{i_s j_s})$ by axiom (\ref{item-axiom-1}).
By axiom (\ref{item-axiom-2}) (and the fact that identities are
faithfully flat ring maps) we conclude that $P(X_{i_s j_s k_s} \to V)$ holds.
By axiom (\ref{item-axiom-3}) we find that
$P(\coprod_{s = 1, \ldots, n} X_{i_s j_s k_s} \to V)$ holds.
Since the morphism $\coprod X_{i_s j_s k_s} \to U$ is surjective
by construction, the corresponding morphism of $\textit{WAdm}^{count}$
is the completion of a faithfully flat \'etale ring map, see
Lemma \ref{lemma-etale-surjective}.
One more application of axiom (\ref{item-axiom-2}) implies that
$P(U \to V)$ is true as desired.
\end{proof}

\begin{remark}[Variant for adic-star]
\label{remark-variant-adic-star}
Let $P$ be a property of morphisms of $\textit{WAdm}^{adic*}$.
We say $P$ is a {\it local property} if axioms
(\ref{item-axiom-1}), (\ref{item-axiom-2}), (\ref{item-axiom-3}),
hold for diagrams (\ref{equation-localize}) with $\varphi$ a
morphism of $\textit{WAdm}^{adic*}$. In exactly the same way
we obtain a variant of Lemma \ref{lemma-property-defines-property-morphisms}
for morphisms between locally adic* formal algebraic spaces over $S$.
\end{remark}

\begin{remark}[Variant for Noetherian]
\label{remark-variant-Noetherian}
Let $P$ be a property of morphisms of $\textit{WAdm}^{Noeth}$.
We say $P$ is a {\it local property} if axioms
(\ref{item-axiom-1}), (\ref{item-axiom-2}), (\ref{item-axiom-3}),
hold for diagrams (\ref{equation-localize}) with $\varphi$ a
morphism of $\textit{WAdm}^{Noeth}$. In exactly the same way
we obtain a variant of Lemma \ref{lemma-property-defines-property-morphisms}
for morphisms between locally Noetherian formal algebraic spaces over $S$.
\end{remark}

\begin{lemma}
\label{lemma-representable-property-rings}
Let $B \to A$ be an arrow of $\textit{WAdm}^{count}$.
The following are equivalent
\begin{enumerate}
\item[(a)] $B \to A$ is taut (Definition \ref{definition-taut}),
\item[(b)] for $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots$
a fundamental system of weak ideals of definitions there exist
a commutative diagram
$$
\xymatrix{
A \ar[r] & \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 \\
B \ar[r] \ar[u] & \ldots \ar[r] & B/J_3 \ar[r] \ar[u] &
B/J_2 \ar[r] \ar[u] & B/J_1 \ar[u]
}
$$
such that $A_{n + 1}/J_nA_{n + 1} = A_n$ and $A = \lim A_n$
as topological ring.
\end{enumerate}
Moreover, these equivalent conditions define a local property,
i.e., they satisfy axioms (\ref{item-axiom-1}), (\ref{item-axiom-2}),
(\ref{item-axiom-3}).
\end{lemma}

\begin{proof}
The equivalence of (a) and (b) is immediate. Below we will give an
algebraic proof of the axioms, but it turns out we've already proven
them. Namely, using Lemma \ref{lemma-representable-affine}
(a) and (b) translate to saying the corresponding morphism of
affine formal schemes is representable, and this condition is
``\'etale local on the source and target'' by
Lemma \ref{lemma-representable-by-algebraic-spaces-local}.

\medskip\noindent
Let a diagram (\ref{equation-localize}) be given.
By Example \ref{example-representable-morphism-from-completion}
the maps $A \to (A')^\wedge$ and $B \to (B')^\wedge$
satisfy (a) and (b).

\medskip\noindent
Assume (a) and (b) hold for $\varphi$. Let $J \subset B$ be a weak ideal
of definition. Then the closure of $JA$, resp.\ $J(B')^\wedge$
is a weak ideal of definition $I \subset A$, resp.\ $J' \subset (B')^\wedge$.
Then the closure of $I(A')^\wedge$ is a weak ideal of definition
$I' \subset (A')^\wedge$. A topological argument shows that $I'$ is also
the closure of $J(A')^\wedge$ and of $J'(A')^\wedge$.
Finally, as $J$ runs over a fundamental system of weak ideals of definition
of $B$ so do the ideals $I$ and $I'$ in $A$ and $(A')^\wedge$.
It follows that (a) holds for $\varphi'$. This proves (\ref{item-axiom-1}).

\medskip\noindent
Assume $A \to A'$ is faithfully flat and that (a) and (b) hold for $\varphi'$.
Let $J \subset B$ be a weak ideal of definition. Using (a) and (b)
for the maps $B \to (B')^\wedge \to (A')^\wedge$ we find that the
closure $I'$ of $J(A')^\wedge$ is a weak ideal of definition.
In particular, $I'$ is open and hence the inverse image of $I'$
in $A$ is open. Now we have (explanation below)
\begin{align*}
A \cap I'
& =
A \cap \bigcap (J(A')^\wedge + \Ker((A')^\wedge \to A'/I_0A')) \\
& =
A \cap \bigcap \Ker((A')^\wedge \to A'/JA' + I_0 A') \\
& = \bigcap (JA + I_0)
\end{align*}
which is the closure of $JA$ by Lemma \ref{lemma-closed}.
The intersections are over weak ideals of definition $I_0 \subset A$.
The first equality because a fundamental system of neighbourhoods of
$0$ in $(A')^\wedge$ are the kernels of the maps $(A')^\wedge \to A'/I_0A'$.
The second equality is trivial. The third equality because $A \to A'$
is faithfully flat, see
Algebra, Lemma \ref{algebra-lemma-faithfully-flat-universally-injective}.
Thus the closure of $JA$ is open. By Lemma \ref{lemma-topologically-nilpotent}
the closure of $JA$
is a weak ideal of definition of $A$. Finally, given a weak
ideal of definition $I \subset A$ we can find $J$ such that
$J(A')^\wedge$ is contained in the closure of $I(A')^\wedge$
by property (a) for $B \to (B')^\wedge$ and $\varphi'$.
Thus we see that (a) holds for $\varphi$. This proves (\ref{item-axiom-2}).

\medskip\noindent
We omit the proof of (\ref{item-axiom-3}).
\end{proof}

\begin{lemma}
\label{lemma-representable-local-property}
Let $P=$"taut" viewed as a property of morphisms of $\textit{WAdm}^{count}$.
Then under the assumptions of
Lemma \ref{lemma-property-defines-property-morphisms} the equivalent conditions
(1), (2), and (3) are also equivalent to the condition
\begin{enumerate}
\item[(4)] $f$ is representable by algebraic spaces.
\end{enumerate}
\end{lemma}

\begin{proof}
Property $P$ is a local property by
Lemma \ref{lemma-representable-property-rings}.
By Lemma \ref{lemma-representable-affine} condition $P$ on morphisms
of $\textit{WAdm}^{count}$ corresponds to ``representable
by algebraic spaces'' for the corresponding morphisms of
countably indexed affine formal algebraic spaces.
Thus the lemma follows from
Lemma \ref{lemma-representable-by-algebraic-spaces-local}.
\end{proof}

\begin{remark}[Tautness for adic-star]
\label{remark-taut-adic-star}
For a morphism $\varphi : A \to B$ of $\textit{WAdm}^{adic*}$
the condition ``$\varphi$ is taut'' is equivalent to
\begin{itemize}
\item[$(*)$] the topology on $B$ is $I$-adic for some ideal
of definition $I$ of $A$.
\end{itemize}
This follows from Lemma \ref{lemma-taut-is-adic}.
\end{remark}





\section{Morphisms of finite type}
\label{section-finite-type}

\noindent
Due to how things are setup in the Stacks project, the following
is really the correct thing to do and stronger notions should have
a different name.

\begin{definition}
\label{definition-finite-type}
Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of formal algebraic
spaces over $S$.
\begin{enumerate}
\item We say $f$ is {\it locally of finite type}
if $f$ is representable by algebraic spaces and is locally
of finite type in the sense of
Bootstrap, Definition \ref{bootstrap-definition-property-transformation}.
\item We say $f$ is of {\it finite type} if $f$ is locally of finite type and
quasi-compact (Definition \ref{definition-quasi-compact-morphism}).
\end{enumerate}
\end{definition}

\begin{lemma}
\label{lemma-characterize-finite-type}
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 $f$ is of finite type,
\item $f$ is representable by algebraic spaces and is of finite type in
the sense of
Bootstrap, Definition \ref{bootstrap-definition-property-transformation}.
\end{enumerate}
\end{lemma}

\begin{proof}
This follows from Bootstrap, Lemma
\ref{bootstrap-lemma-transformations-property-implication},
the implication ``quasi-compact $+$ locally of finite type
$\Rightarrow$ finite type'' for morphisms of algebraic spaces, and
Lemma \ref{lemma-quasi-compact-representable}.
\end{proof}

\begin{lemma}
\label{lemma-finite-type-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 locally of finite type,
\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 and \'etale, $U \to X$
is surjective, and $U \to V$ is locally of finite type,
\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 vertical arrows
representable by algebraic spaces and \'etale, the morphism
$U \to V$ is locally of finite type,
\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 locally of finite type 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 and \'etale, such that $X_i \to Y_i$ is
locally of finite type, and
\item add more here.
\end{enumerate}
\end{lemma}

\begin{proof}
In each of the 5 cases the morphism $f : X \to Y$ is representable
by algebraic spaces, see
Lemma \ref{lemma-representable-by-algebraic-spaces-local}.
We will use this below without further mention.

\medskip\noindent
It is clear that (1) implies (2) because we can take $U = X$ and $V = Y$.
Conversely, assume given a diagram as in (2). Let $T$ be a scheme and
let $T \to Y$ be a morphism. Then we can consider
$$
\xymatrix{
U \times_Y T \ar[d] \ar[r] & V \times_Y T \ar[d] \\
X \times_Y T \ar[r] & T
}
$$
The vertical arrows are \'etale and the top horizontal arrow
is locally of finite type as base changes of such morphisms.
Hence by Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-finite-type-local} we conclude that
$X \times_Y T \to T$ is locally of finite type. In other words
(1) holds.

\medskip\noindent
Assume (1) is true and consider a diagram as in (3).
Then $U \to Y$ is locally of finite type
(as the composition $U \to X \to Y$, see
Bootstrap, Lemma \ref{bootstrap-lemma-composition-transformation-property}).
Let $T$ be a scheme and let $T \to V$ be a morphism.
Then the projection $T \times_V U \to U$ factors as
$$
T \times_V U = (T \times_Y U) \times_{(V \times_Y V)} V
\to T \times_Y U \to U
$$
The second arrow is locally of finite type (see above) and
the first is the base change of the diagonal $V \to V \times_Y V$
which is locally of finite type by
Lemma \ref{lemma-diagonal-morphism-formal-algebraic-spaces}.

\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 locally of finite type by (3)
and we see that (4) is true. This concludes the proof.
\end{proof}

\begin{example}
\label{example-finite-type-from-finite-type-ring-map}
Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over $S$.
Let $A \to A'$ be a finite type ring map. Then
$$
(A')^\wedge = \lim_{I \subset A\ w.i.d.} A'/IA'
$$
is a weakly admissible ring and the corresponding morphism
$\text{Spf}((A')^\wedge) \to \text{Spf}(A)$ is representable,
see Example \ref{example-representable-morphism-from-completion}.
If $T \to \text{Spf}(A)$ is a morphism where $T$ is a quasi-compact
scheme, then this factors through $\Spec(A/I)$ for some weak
ideal of definition $I \subset A$ (Lemma \ref{lemma-factor-through-thickening}).
Then $T \times_{\text{Spf}(A)} \text{Spf}((A')^\wedge)$
is equal to $T \times_{\Spec(A/I)} \Spec(A'/IA')$ and
we see that $\text{Spf}((A')^\wedge) \to \text{Spf}(A)$ is
of finite type.
\end{example}




\section{Closed immersions}
\label{section-closed-immersions}

\noindent
Here is the definition.

\begin{definition}
\label{definition-closed-immersion}
Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of formal algebraic
spaces over $S$. We say $f$ is a {\it closed immersion}
if $f$ is representable by algebraic spaces and a closed immersion
in the sense of
Bootstrap, Definition \ref{bootstrap-definition-property-transformation}.
\end{definition}

\begin{lemma}
\label{lemma-closed-immersion-into-McQuillan}
Let $S$ be a scheme. Let $X$ be a McQuillan affine formal algebraic space
over $S$. Let $f : Y \to X$ be a closed immersion of formal algebraic spaces
over $S$. Then $Y$ is a McQuillan affine formal algebraic space and
$f$ corresponds to a continuous homomorphism $A \to B$ of weakly admissible
topological $S$-algebras which is taut, has closed kernel, and has
dense image.
\end{lemma}

\begin{proof}
Write $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological ring.
Let $I_\lambda$ be a fundamental system of weakly admissible ideals
of definition in $A$. Then $Y \times_X \Spec(A/I_\lambda)$ is
a closed subscheme of $\Spec(A/I_\lambda)$
and hence affine (Definition \ref{definition-closed-immersion}).
Say $Y \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda)$.
The ring map $A/I_\lambda \to B_\lambda$
is surjective. Hence the projections
$$
B = \lim B_\lambda \longrightarrow B_\lambda
$$
are surjective as the compositions $A \to B \to B_\lambda$ are surjective.
It follows that $Y$ is McQuillan by
Lemma \ref{lemma-mcquillan-affine-formal-algebraic-space}.
The ring map $A \to B$ is taut by Lemma \ref{lemma-representable-affine}.
The kernel is closed because $B$ is complete and $A \to B$ is
continuous. Finally, as $A \to B_\lambda$ is surjective for all $\lambda$
we see that the image of $A$ in $B$ is dense.
\end{proof}

\noindent
Even though we have the result above, in general we do not know how
closed immersions behave when the target is a
McQuillan affine formal algebraic space, see Restricted Power Series,
Remark \ref{restricted-remark-questions}.

\begin{example}
\label{example-closed-immersion-from-quotient}
Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over $S$.
Let $K \subset A$ be a closed ideal. Setting
$$
B = (A/K)^\wedge = \lim_{I \subset A\ w.i.d.} A/(I + K)
$$
the morphism $\text{Spf}(B) \to \text{Spf}(A)$ is representable, see
Example \ref{example-representable-morphism-from-completion}.
If $T \to \text{Spf}(A)$ is a morphism where $T$ is a quasi-compact
scheme, then this factors through $\Spec(A/I)$ for some weak
ideal of definition $I \subset A$ (Lemma \ref{lemma-factor-through-thickening}).
Then $T \times_{\text{Spf}(A)} \text{Spf}(B)$
is equal to $T \times_{\Spec(A/I)} \Spec(A/(K + I))$ and
we see that $\text{Spf}(B) \to \text{Spf}(A)$ is a closed immersion.
The kernel of $A \to B$ is $K$ as $K$ is closed, but
beware that in general the ring map $A \to B = (A/K)^\wedge$
need not be surjective.
\end{example}





\section{Proper morphisms}
\label{section-proper}

\noindent
Here is the definition we will use.

\begin{definition}
\label{definition-proper}
Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of formal algebraic
spaces over $S$. We say $f$ is {\it proper}
if $f$ is representable by algebraic spaces and is proper in the sense of
Bootstrap, Definition \ref{bootstrap-definition-property-transformation}.
\end{definition}

\noindent
It follows from the definitions that a proper morphism is of finite type.













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