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\begin{document}
\normalsize
\section{Lecture 15: Points and lft maps} 



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\subsection{A noetherian property}

Let $A$ be an affinoid algebraic over a non-archimedean field $k$ and $X = {\rm{Spa}}(A,A^0)$.  For any $x \in X$, the stalk
$\cO_{X,x}$ is the limit of the directed system $\{A_i\}$ of affinoid algebraic corresponding
to rational domains ${\rm{Spa}}(A_i, A_i^0) \subset X$ around $x$.  In particular, if $x$ is a classical point (i.e., corresponds
to a maximal ideal of $A$) then this limit likewise compute the stalk at $x$ on ${\rm{Sp}}(A)$, so by classical
rigid geometry \cite[7.3.2/7]{BGR} we know that $\cO_{X,x}$ is noetherian for such $x$ (with completion that naturally coincides
with the completion of $A$ at the maximal ideal associated to $x$).   But for more general $x$ we don't have the link to
rigid geometry and hence it is less clear if the stalk is noetherian.  Oddly, this result does not seem to be addressed in the literature,
so now we give Temkin's elegant proof of:

\begin{prop} For any $x \in X = {\rm{Spa}}(A,A^0)$, the local ring $\cO_{X,x}$ is Noetherian.
\end{prop}

\begin{proof}
The local ring at a point $x$ in an affinoid adic space $\Spa(A,A^+)$ for a classical affinoid algebra $(A,A^+)$ is the direct limit of a directed system $(A_i)$ for rational domains $\Spa(A_i,A^+)$ that contain $x$, and the transition maps $A_i \to A_j$ are flat by classical rigid-analytic geometry
\cite[7.3.2/6]{BGR}. 
Let $x_i \in \Spec(A_i)$ be the prime coming from the maximal ideal of the local limit $\cO_{X,x}$, so the directed system of algebraic localizations
$$B_i = (A_i)_{x_i}$$
has the same (local) limit.  Let $\fm_i$ denote the maximal ideal of $B_i$.

The advantage of considering the directed system ${B_i}$ is that since these are local noetherian
rings with flat local transition maps, by \cite[0$_{\rm{III}}$, 10.3.1.3]{EGA} the limit is noetherian if for sufficiently large $i$ we have 
$$\fm_i B_j = \fm_j$$
for all $j \ge i$. (This is the mechanism by which the noetherian property of henselizations of local noetherian rings is proved in \cite{EGA}, for example.)

Observe that $\dim(B_i) \leq \dim(A)$ for all $i$ since for a maximal ideal $\mathfrak{n}$ of $A_i$ containing $x_i$ and the point $\xi \in
{\rm{Sp}}(A_i) \subset {\rm{Sp}}(A) = X$ corresponding to $\mathfrak{n}$ we have
$\dim (B_i) \le \dim (A_i)_{\mathfrak{n}} = \dim \cO_{X,\xi} \leq \dim(A)$ (using that the local noetherian rings
$(A_i)_{\mathfrak{n}}$ and $\cO_{X,\xi}$ have the same completion
and hence the same dimension).  The dimension formula for flat local maps of local noetherian rings gives that 
$$\dim(B_j) = \dim(B_i) + \dim(B_j/\fm_i B_j) \geq \dim(B_i)$$
for all $j \geq i$.  Hence, the dimensions of the $B_i$'s are monotone in $i$ and bounded above, so 
for some large enough $i_0$ we have $\dim(B_i) = \dim(B_{i_0})$ for all $i \geq i_0$.  

Returning to the dimension formula,
the local noetherian fiber algebras $B_i/\fm_{i_0} B_j$ are 0-dimensional for all $i \geq i_0$.  
Thus, these fiber algebras have nilpotent maximal ideal, so to conclude that $\fm_i = \fm_{i_0} B_i$ for all $i \geq i_0$ it suffices 
to show that all fiber algebras of $\Spec(A_i) \to \Spec(A_{i_0})$ (e.g., the closed fiber, whose algebra admits $B_i/\fm_{i_0} B_i$ as an algebraic
localization) are reduced.  

More generally, for any map $A \to A'$ of classical affinoid algebras corresponding to an affinoid subdomain 
$\Sp(A') \hookrightarrow \Sp(A)$, and any $y'$  in $\Spec(A')$ and its image $y$ in $\Spec(A)$ we claim that for the local map
$$
(*)  \ \ \ A_x \longrightarrow A'_{y'}$$
all fibers algebras are geometrically regular, i.e., regular and remain so after any finite extension of the base field of the fiber algebra.  In other words, we claim that $(*)$ is a regular map of Noetherian rings.
Pick a maximal ideal $\fm'$ of $A'$ containing $y'$, so it contracts to a maximal ideal $\fm$ of $A$ containing $y$.  The map $(*)$ is a localization (on source and target) of the map
$$A_{\fm}\longrightarrow A'_{\fm'},$$
so it suffices to show that the latter flat local map is regular.  We know that the induced map on completions is an isomorphism by the classical theory, and to check the regularity property for a map of Noetherian rings it suffices to do after composing with a faithfully flat extension of the target
(since a noetherian ring is regular if it admits a faithfully flat regular extension ring).  Hence, it suffices to show the map from $A_{\fm}$ to its completion 
is a regular map. But this latter regularity is a consequence (and a key part in the proof) of the excellence of affinoid algebras,
an important result due to Kiehl.
\end{proof}









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\subsection{Points as morphisms}

Given a morphism of adic spaces $f:X\to Y$, and a point $y\in Y$, it is natural to want to view
the fiber $f^{-1}(y)$ as the underlying topological space of an ``adic space fiber''.  However, matters
are not quite as for schemes because in the context of adic spaces a field (equipped with extra structure) generally does
not give rise to a one-point space!  So the correct expectation is
that $f^{-1}(y)$ should coincide with the fiber over the unique closed point in a suitably-defined
``adic space fiber'' over $\Spa(\kappa(y),\kappa(y)^+)$ where $\kappa(y)$ is the residue field of $\cO_{Y,y}$ and 
$\kappa(y)^+$ is its valuation ring (the image of $\cO_{Y,y}^+$ in $\kappa(y)$); of course, we could also replace $\kappa(y)$ and $\kappa(y)^+$ with
their completions for this purpose.  Strictly speaking, to make sense of $\Spa(\kappa(y), \kappa(y)^+)$ we have to know
that $(\kappa(y), \kappa(y)^+)$ is a Huber pair, and we saw in 14.6 that this holds when $y$ is an analytic point.

Since we have not discussed fiber products of adic spaces (a topic which has some subtleties, since in contrast with schemes
it will not always exist), the preceding discussion merely serves as geometric motivaton for the algebraic considerations that follow.
In effect, just as a point of an affine scheme can be interpreted in terms of maps from the given ring to a field, we seek a similarly intepretation
for points of affinoid adic spaces.

Let $(A,A^+)$ be a Huber pair, $X:=\Spa(A,A^+)$ and $\Spa(A',{A'}^+)$ a rational domain of $X$ containing a point $x\in X$. Let $\fp_x$ be the support of $x$ in $A$ and set $\kappa(x):=\Frac(A/\fp_x)$ as a valued field using $v_x$. The following diagram is commutative:

\begin{displaymath}
\xymatrix{\kappa(x)\ar@{=}[d]\ar@{^(->}[r] & \kappa'(x)\ar@{=}[d]\\
		\Frac(A/\fp_x)\ar[d]_{v_x} & \Frac(A'/\fp_x')\ar[dl]^{v_x'}\\
		\Gamma_x & }
\end{displaymath}

Recall that $\kappa(x)$ and $\kappa'(x)$ have the same completion, as discussed in Lecture 14. In any global situation $\kappa(x)$ will mean 
$\cO_x/\fm_x$ or $(\cO_x/\fm_x)^{\wedge}$ (the context should make the intent clear). The
completion convention is, given the last observation, the safest to work with as it is computed using any affinoid open subspace around $x$. 

Let us now assume $x$ is an analytic point of $X$, so every generization of $x$ is analytic (as analyticity is an open condition)
and $(\kappa(x), \kappa(x)^+)$ is a Huber pair.  By Proposition 8.3.10, all generizations of $x$ in $X$ are {\em vertical} generizations.

Analyticity of $x$ ensures that $\kappa(x)$ with its $v_x$-topology is topologically a rank-1 valued field, where the rank-1 valuation ring 
is the subring $\kappa(x)^0$ of power-bounded elements (generally bigger than $\kappa(x)^+$).
Recall that there is a unique rank-1 vertical generization $\eta_x \in \Spv(\kappa(x))$ of $x$ that lies in the fiber through $x$
for $X\to\Spec(A)$, and that in general (by \S4.2 in Lecture 4 and \cite[10.1]{Mat}) the set
$$\{\text{generizations of}\ x \text{ in}\ \Spv(\kappa(x))\}$$
is in bijection with the set of valuation subrings $R$ of $\kappa(x)$ containing $\kappa(x)^+$ 
as well as with the set $\Spec(\kappa(x)^+)$ via the correspondence assigning to any prime ideal $\mathfrak{p}$ of $\kappa(x)^+$
the localization $R = \kappa(x)^+_{\mathfrak{p}}$. 

The set of all generizations of $x$ in $X$ (not just in $\Spv(A)$!) is therefore $$\Spa(\kappa(x),\kappa(x)^+)=\{\kappa(x)^+\subset R\subset\kappa(x)^0\}$$
under which $x$ is the unique closed point and $\eta_x$ is the unique generic point (corresponding to $\kappa(x)^0$); this does {\em not}
contain the generic point of $\Spv(\kappa(x))$ (i.e., the trivial valuation) since the latter is not continuous on $A$ and so does not lie in $X$ inside
$\Spv(A)$.
We are now going to focus on $\Spa(\kappa(x),\kappa(x)^+)$; as the topological space of all generizations of $x$ in $X$
(with the subspace topology from $X$) it is an analogue of $\Spec(\cO_{X,x})$ in the theory of schemes. 

Since for analytic $x \in X$ as above the valuation subring $\kappa(x)^+$ is an open subring of the rank-1 valuation ring
$\kappa(x)^0$, we are motivated to make:

\begin{defin} \label{affinoid fld} An {\em affinoid field} $(k,k^+)$ is a pair consisting in a complete nonarchimedean field $k$ and an 
open valuation subring $k^+$ of $k^0$ (so ${\rm{Frac}}(k^+) = k$ due to openness of $k^+$ in $k$). 
\end{defin}

\begin{rem} Any affinoid field $(k,k^+)$ is a Huber pair.  Indeed, since $k^0$ is an intermediate valuation ring between
$k^+$ and $k = {\rm{Frac}}(k^+)$, necessarily $k^0$ arises from a valuation on the residue field of $k^+$ by
\cite[10.1]{Mat} and so the maximal ideal $k^{00}$ of $k^0$ is contained in that of $k^+$.  Thus, if $\varpi \in k^0$ is a pseudo-uniformizer
and $x \in k^{\times}$ then $\varpi^n/x \in k^{00} \subset k^+$ for all large $n$
(largeness depending on $x$), so $v(\varpi)^n \le v(x)$ where $v$ denotes the valuation for $k^+$.  It follows that
$v(\varpi^n k^+) \le v(x)$ for all large $n$, so the open ideals $\varpi^n k^+$ in $k^+$ are a base for the $v$-topology on $k$,
thereby verifying the Huber property.
\end{rem}

\begin{prop} \label{analytic bijection} Let $(A,A^+)$ be any Huber pair, and consider the set
$$\Sigma_{A}:=\{(A,A^+)\xrightarrow{\varphi}(k,k^+)\mid\varphi(A)\subset k\ \text{\rm{generates a dense subfield}}\}$$
where $(k,k^+)$ is some affinoid field, or equivalently the set of continuous valuations $v$ on $A$ for which $v(A^+) \le 1$ and 
the associated completion of ${\rm{Frac}}(A/\mathfrak{p}_v)$ has the topology of a rank-$1$ valued field.

There is a natural bijection
$$\Sigma_A\simeq|\Spa(A,A^+)_{\rm an}|$$
assigning to any $\varphi \in \Sigma_A$ the valuation $v_{\varphi}$ that is the image under $\Spa(\varphi)$ of the unique closed point in
$\Spa(k, k^+)$.
\end{prop}

\begin{proof} This is immediate from the preceding discussion since the valuation on $k$ associated to $k^+$ is the unique
closed point in $\Spa(k, k^+)$.
\end{proof}

As a corollary we can immediately globalize by working with affinoid open subspaces:

\begin{cor} For an adic space, the set $|X_{\rm{an}}|$ of analytic points is in natural bijection with the set of morphisms
$$\{f:\Spa(k,k^+)\to X\mid \kappa(x)\subset k\ \text{\rm{is dense}}\}$$
from the adic spectra of affinoid fields to $X$ by assigning to any such $f$ the image of the unique closed
point of $\Spa(k,k^+)$.
\end{cor}

This corollary is most useful when {\em all} points of $X$ are analytic. Since any map of adic spaces
carries analytic points to analytic points, if $X$ is equipped with a morphism to an adic space $S$ whose points are all
analytic then the same holds for $X$.  For instance, this happens when $S = \Spa(K, K^+)$ for an affinoid field
$(K, K^+)$, the most important example being when $K^+ = K^0$.

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\subsection{Locally of finite type maps}

In this section we discuss what a locally finite type morphism of adic spaces should be.  
One wrinkle is that classical rigid geometry over a complete nonarchimedean field $k$
does not provide much guidance, since (in hindsight) {\em all} morphisms in classical
rigid geometry are locally of finite type when upgraded to morphisms of adic spaces (since the adic spaces themselves
will be locally of finite type over $\Spa(k, k^0)$).

Let $(A,A^+)$ and $(B,B^+)$ be {\em complete} Huber pairs, with $X:=\Spa(B,B^+)$ and $S:=\Spa(A,A^+)$, and let $f:X\to S$ be a morphism. 
One context in which we certainly want to say that $f$ is ``finite type'' is if $f$ can be factored as a ``closed immersion'' into some relative unit
polydisc over $S$.  On the level of Huber pairs, this amounts to having a commutative diagram
\begin{displaymath}
\xymatrix{(A\langle y_1,\dots, y_n\rangle, A\langle y_1,\dots, y_n\rangle^+) \ar[r] & (B, B^+) \\
	(A,A^+)\ar[ur] \ar[u] & }
\end{displaymath}
where $A\langle y_1,\dots, y_n\rangle \rightarrow B$ is a topological quotient map (i.e., $B$ is topologically the quotient of
$A\langle y_1, \dots, y_n \rangle$ by a closed ideal) 
and the open image $B'$ of $A\langle y_1, \dots, y_n \rangle^+$ in $B$ is a ring of definition whose integral closure is $B^+$. 
(Here it is no different to work with the open subring $A^+\langle y_1, \dots, y_n \rangle$ whose integral closure in
$A \langle y_1, \dots, y_n \rangle$ in $A\langle y_1, \dots, y_n \rangle^+$ by definition.)

We should allow ``varying polyradii'' in these relative polydiscs, much as in classical rigid geometry we consider
polydiscs given by conditions $\{|y_i| \le r_i\}$ for $r_i \in \surd{|k^{\times}|}$ (not requiring $r_i \in |k^{\times}|$). 
So for nonempty finite subsets $T_1,\dots, T_n\subset A$ such that $T_iA\subset A$ is open we should also allow ourselves to use 
the topological $A$-algebra
$$A\langle X\rangle_T:=\{\sum a_{\nu} X^{\nu} \in A[\![X]\!] 
\mid \text{\rm{for every }} U, a_{\nu} \in T^{\nu} U \text{\rm{ for all but finitely many }} \nu\}$$
where $U$ varies through additive open subgroups of $A$, having as 
a base of opens $$U\langle X\rangle_T:=\{\sum a_{\nu}X^{\nu}\in A\langle X\rangle_T\mid a_{\nu}\in T^{\nu}U 
\text{\rm{ for all }} \nu\}$$
and ring of definition $A^+\langle X \rangle_T$.
We recall that $A\langle X\rangle_T$ is Huber, complete when $A$ is complete,
and is initial among Huber rings over $A$ equipped with an ordered $n$-tuple
of elements $x_1,\dots, x_n$ such that $t_ix_i$ is power-bounded for all $t_i\in T_i$ and for all $1 \le i \le n$.

Finally, given a Huber pair $(A,A^+)$, we define $A\langle X\rangle_T^+$ to be the integral closure of the 
ring of definition $A^+\langle X\rangle_T\subset A\langle X\rangle_T$.

\begin{defin} \label{tft} Given a map $(A,A^+)\to (B,B^+)$ of complete Huber pairs, we say it is 
{\em  of finite type} if there exists $T_1, \dots, T_n$ as above and a map of Huber pairs
$$(A\langle X\rangle_T, A\langle X\rangle_T^+) \to (B, B^+)$$
such that the map on Huber rings is an open surjection and the map $A\langle X \rangle_T^+ \rightarrow B^+$ is integral;
equivalently, $B = A\langle X \rangle_T/J$ for some closed ideal $J$ and $B^+$ is the integral closure in $B$
of the open image of $A^+\langle X\rangle_T$.
\end{defin}

\begin{eg} \label{eg tft} For a complete Huber pair $(A, A^+)$ and an element $s \in A$ and a non-empty finite subset $T = \{t_1,\dots, t_n\}
\subset A$ such that
$T \cdot A \subset A$ is open, consider the rational subdomain
$$A\langle T/s\rangle=A\langle X_1, \dots, X_n\rangle_T/\overline{(sX_i-t_i)}$$
with $A\langle T/s\rangle^+$ as usual. Then $(A\langle T/s\rangle, A\langle T/s\rangle^+)$ is of  finite type over $(A,A^+)$. 
\end{eg}

It is important to note that when the base ring is Tate (e.g., a non-archimedean field), the role of the polyradii can be suppressed:

\begin{prop}\label{tateft} If $A$ is Tate then for $(B,B^+)$  of finite type over $(A,A^+)$ we can always choose $T_i=\{1\}$ for all $i$;
i.e., for any $T_1, \dots, T_n$ there exists a continuous open 
$A$-algebra surjection $A\langle Y \rangle \twoheadrightarrow A\langle X\rangle_T$
such that $A^+\langle Y \rangle \rightarrow A^+\langle X\rangle_T$ is integral.
\end{prop}

\begin{proof}  The idea is to identify $\Spa(A\langle X\rangle_T, A\langle X\rangle_T^+)$ with a rational domain in some 
$\Spa(A\langle Y\rangle, A\langle Y\rangle^+)$, as then we can apply Example \ref{eg tft} to conclude.

First we shall make a ``change of coordinates'' to bound the $T_i$'s relative to a ring of definition, as follows.
Let $A_0 \subset A$ be a ring of definition, and 
pick a topologically nilpotent unit $\varpi\in A$.  Let $m \ge 0$ be an integer large enough so that
$\varpi^mt_i\in A_0\subset A$ for all $t_i \in T_i$ ($1 \le i \le n$). 
By replacing $X_i$ with $\varpi^{-m}X_i$ and replacing $T_i$ with $\varpi^mT_i$ for all $i$, 
we may thereby assume $t_i\in A_0$ for all $t_i\in T_i$ and for all $i$.

Recall that $A\langle X\rangle_T$ is universal among Huber rings over $A$ equipped
with an ordered $n$-tuple of elements $x_1, \dots, x_n$ such that
$t_ix_i$ is power-bounded for all $t_i\in T_i$ and all $i$. We have rigged that all elements of each $T_i$
are power-bounded, and the only open ideal in $A$ is the unit ideal (by the Tate hypothesis!), so $T_i A = A$ for all $i$. 
Letting $T_i X_i$ denote the finite subset of elements $t_i X_i \in A \langle X \rangle$,
we get a finite subset 
$$\Sigma:=T_1\cup\dots\cup T_n\cup T_1X_1\cup\dots\cup T_nX_n \subset A\langle X\rangle^0$$
that generates an open ideal of $A\langle X\rangle$ (even the unit ideal).  Therefore it makes sense to form the completed rational localization
$$(A\langle X\rangle)\langle \Sigma/1 \rangle$$
that is a Huber ring and uniquely isomorphic to $A\langle X\rangle_T$
as a topological $A\langle X \rangle$-algebra via their universal properties as
topological $A$-algebras. It is straightforward to chase the ``integral structures'' to check that this achieves
the original idea with a rational domain in a relative unit polydisc over $\Spa(A, A^+)$.
\end{proof}

As an application of the preceding proposition, we get a first result towards characterizing adic spaces associated
to affinoid algebras over a non-archimedean field $k$ 
among general adic spaces over $\Spa(k, k^0)$:

\begin{cor} \label{tft over aff field} Let $k$ be a nonarchimedean field, $A$  a $k$-affinoid algebra, and $(A,A^+)$ is a Huber pair, 
so $A^+\subset A^0$ is an open subring. The map $(k,k^0)\to (A,A^+)$ is  of finite type if and only if $A^+=A^0$.
\end{cor}

\begin{proof} Since there exists an open surjection 
$$k\langle X\rangle \twoheadrightarrow k\langle X\rangle/J\simeq A$$
and we have seen that the associated map $k\langle X\rangle^0\to A^0$ is integral, 
the ``if'' direction is verified.  Conversely, assume $(k,k^0)\to (A,A^+)$ is  of finite type, so by
Proposition \ref{tateft} over the base $(k, k^0)$ there exists an open map
$$k\langle X\rangle \twoheadrightarrow A$$
which induces an integral map
$$k\langle X\rangle^0 \to A^+.$$
But composing it with the inclusion $A^+\subset A^0$ is known to give an integral map
(as always for integral maps, such as surjections, between $k$-affinoid algebras), so $A^+ \rightarrow A^0$
is integral.  Since $A^+$ integrally closed in $A$, we get $A^+=A^0$ as desired.
\end{proof}



\begin{defin} A morphism of adic spaces $f : X \to Y$ 
is {\em locally of  finite type} if, for any $x\in X$, there exists an affinoid open subspaces $U$ around $x$ and 
$V$ around $f(x)$ such that $f(U)\subseteq V$ and the induced homomorphism of 
complete Huber pairs $(\cO_Y(V), \cO^+_Y(V))\to (\cO_X(U),\cO^+_X(U))$ 
is  of finite type as in Definition \ref{tft}.
\end{defin}

By Example \ref{eg tft}, the open immersion of a rational domain into an affinoid adic space is
locally of  finite type.  Thus, in general {\em every} open immersion of adic spaces
is locally of  finite type since the rational domains give a base of open subspaces of an
affinoid adic space.  

It is clear that in order for the notion ``locally of  finite type'' to be useful, in the affinoid setting it has to coincide
with ``of finite type''.  The analogous fact for schemes is quite elementary, but for adic spaces
the proof is much harder (involving tricky approximation arguments) and is an important theorem of Huber: 

\begin{thm} Assume that $(A,A^+)$ and $(B,B^+)$ are complete Huber pairs such that each of $A$ and $B$ is strongly Noetherian, or each has Noetherian ring of definition. Then a map $(A,A^+)\to (B,B^+)$ which is locally of finite type is in fact of finite type.
\end{thm}

The only published proof appears to be \cite[Thm.\,3.8.15]{H3}, and unfortunately \cite{H3} is difficult to obtain 
(e.g., the only copy in the United States is in the University of Michigan math library).  
In a later version of these notes I will include a proof (in English).



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
%\subsection{Appendix A: rigid varieties and adic spaces}
%
%To construct a functor $\mathfrak{R}_k\rightarrow\mathfrak{A}_k$ we find convenient to stick to the more abstract setting of ringed sites, and thus we may view a morphism of adic spaces $X\rightarrow Y$ as inducing a morphism of ringed sites $(|X|,\cO_X)\rightarrow(|Y|,\cO_Y)$.
%\paragraph{Construction of the functor $(-)^{\rm ad}:\mathfrak{R}_k\rightarrow\mathfrak{A}_k$} \ \\
%
%Let us notice that if $X=\Sp(k)$ and $Y=\Spa(k,k^{0})$, then we have an equivalence of ringed sites
%$$(|X|,\cO_X)=(|Y|,\cO_Y)$$
%We call this site $S$. We now discuss some of the details of Huber's proof of the following
%\begin{prop} \label{eta} Let $X\in\textup{ob}(\mathfrak{R}_k)$. There exists an adic space $X^{\rm ad}\in\textup{ob}(\mathfrak{A}_k)$ and a morphism
%\begin{displaymath}
%\xymatrix{(|X^{\rm ad}|,\cO_{X^{\rm ad}})\ar[r]^{\eta_X} & (|X|,\cO_X)}
%\end{displaymath}
%of ringed sites over $S$, such that $X^{\rm ad}$ is locally of finite type over $\Spa(k,k^{0})$, $\eta$ is locally coherent, and the following universal property holds true. If $Z$ is an adic space locally of finite type over $\Spa(k,k^{0})$ and 
%\begin{displaymath}
%\xymatrix{(|Z|,\cO_Z)\ar[r]^{\vartheta} & (|X|,\cO_X)}
%\end{displaymath}
%is a locally coherent morphism of ringed sites over $S$, then there exists a unique morphism
%\begin{displaymath}
%\xymatrix{Z\ar[r]^f & X^{\rm ad}}
%\end{displaymath}
%in the category $\mathfrak{A}_k$, making the following diagram commute:
%\begin{displaymath}
%\xymatrix{(|Z|,\cO_{Z})\ar[r]^{f^{\rm s}}\ar[rd]_{\vartheta} & (|X^{\rm ad}|,\cO_{X^{\rm ad}})\ar[d]^{\eta_X}\\
%							& (|X|,\cO_X)}
%\end{displaymath}
%\end{prop}
%Before discussing the proof, let us state and prove one more result which eventually assigns the functor $\mathfrak{R}_k\rightarrow\mathfrak{A}_k$.
%\begin{prop} Let $f:X\rightarrow Y$ be a morphism in the category $\mathfrak{R}_k$. Then the induced morphism of ringed sites over $S$
%\begin{displaymath}
%\xymatrix{(|X|,\cO_X)\ar[r]^{f^{\rm s}} & (|Y|,\cO_Y)}
%\end{displaymath}
%is locally coherent.
%\end{prop}
%\begin{proof}
%For every quasi-compact open subset $U\subset X$, the restriction $f\vert_U$ is quasi-compact as a morphism in the category of adic spaces over $\Spa(k,k^{0})$, meaning that the morphism $|X|\rightarrow|Y|$ on the level of $S$-sites is locally coherent. It obviously respect the ringed structures $\cO_X$ and $\cO_Y$ on $|X|$ and $|Y|$.
%\end{proof}
%Indeed, by Proposition \ref{eta}, we have the natural morphisms $\eta_X$ and $\eta_Y$. If we choose $Z=X^{\rm ad}$ and $\vartheta=f^{\rm s}\circ\eta_X$, by the universal property we have as in the following diagram
%\begin{displaymath}
%\xymatrix{(|X^{\rm ad}|,\cO_{X^{\rm ad}})\ar[d]_{\boxed{\exists ! h}}\ar[r]^{\eta_X} & (|X|,\cO_X)\ar[d]^{f^{\rm s}}\\
%		(|Y^{\rm ad}|,\cO_{Y^{\rm ad}})\ar[r]_{\eta_Y} & (|Y|,\cO_Y)}
%\end{displaymath}
%If we set $f^{\rm ad}:=h$, by universality of $h$ this indeed says $(-)^{\rm ad}$ is a functor $\mathfrak{R}_k\rightarrow\mathfrak{A}_k$.\\ \ \\
%Now we sketch the proof of Proposition \ref{eta}.
%\begin{proof}[Sketch of proof of Proposition \ref{eta}]
%We can assume $X$ is affinoid, ie. $X=\Sp(A)$ for some Tate tft $k$-algebra $A$. Let us pick $x\in A$. It corresponds to a maximal ideal $\fm_x$ of $A$. Let $k':=A/\fm$. $k'/k$ is a finite extension, and therefore we can uniquely extend the valuation on $k$ to $k'$. The composite
%\begin{displaymath}
%\xymatrix{A\ar[r]\ar@/_1.3pc/[rr] & A/\fm =k'\ar[r] & \bR^+_{>0}\cup\{0\}}
%\end{displaymath}
%assigns a valuation on $A$. Since $A$ is a Tate $k$-algebra, such a valuation will be $\leq 1$ on $A^{0}$, and therefore defines a point of $\Spa(A,A^{0})$. We set $$X^{\rm ad}:=\Spa(A,A^{0}).$$
%This assigns a natural one-to-one correspondence between rational subsets of $X$ and rational subsets of $X^{\rm ad}$.
%Via this correspondence, given $U$ a rational subset of $X$ and $V$ the corresponding rational subset of $X^{\rm ad}$, we have isomorphisms of $A$-algebras
%\begin{displaymath}
%\xymatrix{\cO_X(U)\ar[r]^{\varphi_U} & \cO_{X^{\rm ad}}(V)}
%\end{displaymath}
%The rational subsets form a basis for the $G$-topology on $X$, as well as for the topology on $X^{\rm ad}$. Therefore the correspondence between rational subsets of $X$ and $X^{\rm ad}$ respectively, actually extends uniquely to a morphism of sites
%\begin{displaymath}
%\xymatrix{|X|\ar[r]^{\psi} & |X^{\rm ad}|}
%\end{displaymath}
%so that we can produce also sheaf isomorphisms
%\begin{displaymath}
%\xymatrix{\cO_X\ar[r]^{\varphi\ \ \ \ \ }_{\sim\ \ \ \ \ } & \psi_*(\cO_{X^{\rm ad}})}
%\end{displaymath}
%We define
%$$\eta_X:=(\psi,\varphi).$$
%This is indeed a morphism of ringed sites $(|X^{\rm ad}|,\cO_{X^{\rm ad}})\rightarrow (|X|,\cO_X)$ over $S$. Finally, one just checks the universal property, given a locally coherent morphism $\vartheta:(|Z|,\cO_Z)\rightarrow (|X|,\cO_X)$ in the category of ringed sites. Existence of $f^{\rm s}$ is given aready in \cite{SGA4}, on the level of ringed sites (here we use $\vartheta$ is locally coherent). Since, actually, $Z$ is an adic space, $f^{\rm s}$ actually descends to a morphism of ringed {\em spaces} $f:(Z,\cO_Z)\rightarrow(X^{\rm ad},\cO_{X^{\rm ad}})$, and we are left to show that $f$ is a morphism in $\mathfrak{A}_K$. The argument eventually makes use of the finiteness assumption on $Z$ as an adic space over $\Spa(k,k^{0})$ to use the fact that $k$-algebra homomorphisms between Tate rings of tft over $k$ are continuous.
%\end{proof}
%
%\paragraph{$(-)^{\rm ad}$ embeds $\mathfrak{R}_k$ as a full subcategory of $\mathfrak{A}_k$}\ \\
%
%We have indeed already proved this, if we consider Proposition \ref{eta} carefully. The key step in defining $X^{\rm ad}$ as an adic space is in the assignement
%$$\Sp(A)\mapsto\Spa(A,A^{0})$$
%which was suggested by simply considering the points of $\Sp(A)$, and observing that they give rise to a point in $\Spa(A,A^{0})$. If, again, we pick a point $x\in \Sp(A)$, and consider the corresponding maximal ideal $\fm_x\subset A$, we are naturally given a finite field extension $A/\fm_x$ of $k$. The crucial step is that there exists a unique valuation on $A/\fm_x$ extending the valuation on $k$! In other words, due to this, $X^{\rm ad}$ is uniquely determined by $X$ as an adic space! Functoriality of this assignement is given already on the level of ringed sites by \cite{EGA4}.
%
%\paragraph{An important consequence.}\ \\
%There is a key step we have not enough extensively discussed in the sketch of proof. Given a rigid-analytic variety $X$ over $k$, according to the Proposition $X^{\rm ad}$ is of finite type over $\Spa(k,k^{0})$. This is important, as it allows us to see that $X$ is an affinoid rigid-analytic variety over $k$ if and only if $X^{\rm ad}$ is an affinoid adic space over $\Spa(k,k^{0})$.\\
%In fact, let $X\in\textup{ob}(\mathfrak{R}_k)$ be such that $X^{\rm ad}$ is affinoid. Therefore, the morphism of affinoid rings
%$$(k,k^{0})\longrightarrow(\cO_{X^{\rm ad}}(X^{\rm ad}),\cO_{X^{\rm ad}}^+(X^{\rm ad}))$$
%is of tft. The adic space $X^{\rm ad}$ is isomorphic over $\Spa(k,k^{0})$ to
%$$\Spa(\cO_{X^{\rm ad}}(X^{\rm ad}),\cO_{X^{\rm ad}}(X^{\rm ad})^{\circ})=\Sp(\cO_{X^{\rm ad}}(X^{\rm ad}))^{\rm ad}.$$
%Call $A:=\cO_{X^{\rm ad}}(X^{\rm ad})$. Since $(-)^{\rm ad}$ is fully faithful, we can say $\Sp(A)$ is isomorphic to $X$ as a rigid-analytic variety over $k$ only if we have $A^+=A^{0}$. By this finiteness condition, this is indeed the case, by Corollary \cite{tft over aff field}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\subsection{Appendix B: formal schemes and adic spaces}
%
%
%We wish, now, to construct a functor from the category of formal schemes to the category of adic spaces. We prove the key results, and sketch some of the consequences of the main Theorem of this section, which the reader can find in \cite{H2}.
%
%\begin{defin} Let $\fX$ be a formal scheme that is adic in the sense that the topology on the coordinate ring of every affine open
%subspace is defined by a finitely generated ideal.  We say it is {\em sheafy} is for any affine 
%open subspace  $U=\Spf(A)\subset\fX$, the adic ring $A$ is such that $\Spa(A,A)$ is sheafy.
%\end{defin}
%
%Let $Y$ be an adic space. We can obviously consider $\cO_Y^+$ as a sheaf of topological rings by endowing $\cO_Y^+(U)$ with the subspace topology of $\cO_Y(U)$, for any $U\subset Y$ open. Then $(Y,\cO_Y^+)$ is still a locally and topologically ringed space. Every morphism of adic spaces $f:X\rightarrow Y$ induces a morphism $f^+:(X,\cO_X^+)\to(Y,\cO_Y^+)$ of locally and topologically ringed spaces. 
%
%\begin{prop} Let $\fX=\Spf(A)$ be a sheafy affine adic formal scheme. Then, for any adic space $Z$, the map
%$$\Hom((Z,\cO_Z^+),(\fX,\cO_{\fX}))\longrightarrow \Hom_{\rm cont}(A,\Gamma(Z,\cO_Z^+))$$
%sending a morphism $(f,f^{\sharp})$ of locally and topologically ringed spaces to a continuous homomorphism $f_{\fX}$ of adic rings, is a bijection.
%\end{prop}
%
%This gives some hints on how to define a functor from some category of adic formal schemes to the category of adic spaces.
%\begin{thm} Let $\fX$ be a sheafy adic formal scheme. There exists an adic space $\fX^{\rm ad}$ and a morphism of locally and topologically ringed spaces
%\begin{displaymath}
%\xymatrix{(\fX^{\rm ad},\cO_{\fX^{\rm ad}}^+)\ar[r]^{\alpha} & (\fX,\cO_{\fX})}
%\end{displaymath}
%satisfying the following universal property. For every adic space $Z$ and for every morphism $f:(Z,\cO_Z^+)\to (\fX,\cO_{\fX})$ of locally and topologically ringed spaces, there exists a unique morphism $f^{\rm ad}:Z\to\fX^{\rm ad}$ making the following diagram commute
%\begin{displaymath}
%\xymatrix{(Z,\cO_Z^+)\ar[rd]_{f^{\rm ad}}\ar[r]^f & (\fX,\cO_{\fX})\\
%				& (\fX^{\rm ad}, \cO_{\fX^{\rm ad}})\ar[u]_{\boxed{\alpha}}}
%\end{displaymath}
%\end{thm}
%\begin{proof}
%Morphisms of adic spaces glue together, so it suffices to 
%prove this when $\fX=\Spf(A)$. Define $$\fX^{\rm ad}:=\Spa(A,A).$$
%The identity $A\to A$ induces a morphism of locally and topologically ringed spaces $\alpha:(\fX^{\rm ad},\cO_{\fX^{\rm ad}})\to (\fX,\cO_{\fX})$. For every adic space $Z$, we have natural bijections
%\begin{flalign*}
%\Hom_{\mathbf{Adic}}(Z,\fX^{\rm ad})=&\ \Hom_{\mathbf{Affinoid}}((A,A),(\Gamma(Z,\cO_Z),\Gamma(Z,\cO_Z^+)))\\
%		=&\ \Hom_{\mathbf{TopRng}}(A,\Gamma(Z,\cO_Z^+))\\
%		=&\ \Hom_{\mathbf{LTRSp}}((Z,\cO_Z^+),(\fX^{\rm ad},\cO_{\fX^{\rm ad}}^+)).
%\end{flalign*}
%where, respectively, $\mathbf{Adic}$ is the category of adic spaces, $\mathbf{Affinoid}$ is the category of affinoid rings, $\mathbf{TopRng}$ is the category of topological rings and $\mathbf{LTRSp}$ is the category of locally and topologically ringed spaces. 
%It is easy to check that $\alpha$ and $\fX^{\rm ad}$ satisfy the required universal property
%since for any morphism of adic spaces $f:Z\to\fX^{\rm ad}$, we apply the bijections above 
%to the composition $\alpha\circ f^+$ to obtain the desired $f^{\rm ad}$.
%\end{proof}
%
%As a consequence, $\fX\mapsto \fX^{\rm ad}$ assigns a functor from the category of sheafy adic formal schemes to the category of adic spaces. 
%In fact, let $f:\fX\to\fY$ be a morphism of sheafy  adic formal schemes. The universal property applied to the morphism
%\begin{displaymath}
%\xymatrix{(\fX^{\rm ad},\cO_{\fX^{\rm ad}}^+)\ar@/_1.0pc/[rr]\ar[r]^{\alpha} & (\fX,\cO_{\fX})\ar[r]^f & (\fY,\cO_{\fY})}
%\end{displaymath}
%yelds a unique morphism $f^{\rm ad}:\fX^{\rm ad}\to\fY^{\rm ad}$ of adic spaces, making the following diagram commute in the category of locally and topologically ringed spaces
%\begin{displaymath}
%\xymatrix{(\fX^{\rm ad},\cO_{\fX^{\rm ad}}^+)\ar[d]_{\alpha_{\fX}}\ar[r]^{f^{\rm ad +}} & (\fY^{\rm ad},\cO_{\fY^{\rm ad}}^+)\ar[d]^{\alpha_{\fY}}\\
%		(\fX,\cO_{\fX})\ar[r]_f & (\fY,\cO_{\fY})}
%\end{displaymath}
%Uniqueness of $f^{\rm ad}$ implies its functorialty, so $\fX\mapsto\fX^{\rm ad}$ is a functor.
%
%\paragraph{A special example}\ \\
%
%Just to fix ideas on how different are adic spaces from formal schemes, let us consider the following very basic example. We let $A$ be an adic ring with finitely generated ideal of definition $I$. We define
%$$X_{\rm triv}:=\{x\in\Spa(A,A)\mid v_x\ \textup{is trivial}\}.$$
%Then the map sending $v_x$ to its support yields a homeomorphism
%$$X_{\rm triv}\simeq\Spf(A).$$
%The inclusion $X_{\rm triv}\subset X$ is spectral, because, explicitly, the intersection of a rational subset $U\left(\frac{f_1,...,f_n}{g}\right)$ with $X_{\rm triv}$ is just $D^+(g)$!
%To sum up, we have the following
%\begin{thm} The functor $\fX\mapsto \fX^{\rm ad}$ from the category of locally noetherian formal schemes to the category of adic spaces is fully faithful.
%\end{thm}
%
%





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