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\begin{document}
\normalsize
\section{Lecture 13: Uniformity and sheaf properties}

\subsection{Introduction/global theory}

So far, we have avoided discussing the global theory of adic spaces, but this is an appropriate point to introduce the main idea
(which will be fleshed out at the start of the next lecture). Recall that for any Huber pair $(A,A^+)$ we have defined a presheaf of complete topological rings $\ccO_A$ on $X=\Spa(A,A^+)$: we first defined
\[
\ccO_A(X(T/s))=A\langle T/s\rangle
\]
for a finite subset $T\subset A$ and $s\in S$ such that $T\cdot A$ is open in $A$ (using a universal
mapping property in Proposition 12.3.3 to ensure that this is intrinsic to the pair $(A,A^+)$ and $X(T/s)$ merely as an open subset of
$X$ without reference to the choice of $s$ and $T$), and then for an arbitrary open set $U \subset X$ we define 
\[
\ccO_A(U)=\varprojlim_{W\subset U}\ccO_A(W),
\]
where $W$ varies through rational domains contained in $U$ and we equip this
with the inverse limit topology (subspace topology inside a direct product). 
We also define a presheaf $\ccO_A^+$ in the obvious parallel way.
In Proposition 12.3.7 we verified that the stalk $\ccO_{A,x}$ at a point $x\in X$ (throwing away topological information)
 is a local ring on which there is a canonical valuation $v_x$.

The particular way that we have defined $\ccO_A(U)$ for general open $U$ and the fact that the rational domains (by definition) form a basis for the topology on $X$ together imply that $\ccO_A$ is \emph{adapted} to the basis of rational domains, in Wedhorn's terminology \cite[\S8.9]{Wed}. In fact, the categories of sheaves of topological rings (or even of topological spaces)
on a topological space and sheaves relative to a chosen base of the topological space 
are equivalent \cite[III$_0$ 3.2.2]{ega}.
%See also \cite[Tag 009R]{Stacks}, though the result is only stated there for so-called ``sheaves of algebraic structures,'' which does not cover the case of topological rings. Stacks Project exercise: generalize as much as possible of the theory in the chapter ``Sheaves on Spaces'' to a setting which includes sheaves of topological rings and add it to the Stacks Project.
In particular, our recipe is the unique possible way to extend the definition of a Huber pair associated to a rational domain to all open sets compatibly with restriction maps. Thus for any presheaf $\ccF$ that is adapted to a basis $\cB$ on a topological space, $\ccF$ is a sheaf if and only if it is a sheaf when restricted to $\cB$; for the purposes of checking the sheaf axioms, we can focus exclusively on
rational domains and their {\em finite} covers
by rational domains (as finite covers are cofinal among all open covers, due to quasi-compactness
of rational domains). This is excellent news; otherwise, any explicit calculation would be hopeless.

In order to define global adic spaces, the general strategy is obvious: we want to say that something is an adic space if it is locally of the form 
``$\Spa(A,A^+)$'', where $(A,A^+)$ is a Huber pair. There are two issues that must be addressed to
make sense of this strategy.  First, we need to figure out in what underlying category we are working (for example, in the case of schemes this is the category of locally ringed spaces, in the case of formal schemes this is the category of locally topologially ringed spaces, and in the case of rigid-analytic spaces over a non-archimedean field $k$ this is the category of locally ringed ``G-topologized'' spaces with
$k$-algebra structure sheaf). Second, perhaps with these examples in mind, we note that in order to get any sort of reasonable theory off the ground we had better demand that the structure presheaves are, in fact, sheaves. 
The main work then is to prove that
the structure presheaf on $\Spa(A,A^+)$ is a sheaf for certain useful classes of Huber pairs $(A,A^+)$
(so the only pairs $(A,A^+)$ for which $\Spa(A,A^+)$ will be an adic space are those for which $\ccO_A$ is a sheaf).

Here's a solution to the first problem:

\begin{defin}
Let $\mathscr{V}$ be the category of triples $(X,\ccO_X,(v_x)_{x\in X})$, where $X$ is a topological space, $\ccO_X$ is a sheaf of topological rings such that each stalk $\ccO_{X,x}$ is a local ring, and $v_x$ is a valuation on the residue field $\kappa(x)$ of
$\ccO_{X,x}$. Morphisms $f:X\to Y$ in $\mathscr{V}$ are maps $(f,f^\flat)$ of locally topologically ringed spaces
(in particular, $f^{\flat}:\ccO_Y(U) \rightarrow \\ccO_X(f^{-1}(U))$ is continuous for all open $U \subset Y$) such that the induced
maps of residue fields $\kappa(f(x)) \rightarrow \kappa(x)$ are compatible with the valuations (equivalently,
induces a local inclusion between the valuation rings). 
\end{defin}

\begin{defin}
Given an object $(X,\ccO_X,(v_x)_{x\in X})$ in $\mathscr{V}$, 
we define the sheaf of integral structures $\ccO_X^+$ as follows. For any open $U$, set
\[
\ccO_X^+(U)=\{f\in\ccO_X(U):v_x(f)\le1\text{ for all }x\in U\}.
\]
\end{defin}

\begin{rem}
By Lemma 12.3.9, this definition agrees with the definition of the presheaf $\ccO_A^+$ on $\Spa(A,A^+)$.
\end{rem}

\begin{defin}
An \emph{affinoid adic space} is an object of $\mathscr{V}$ isomorphic to $\Spa(A,A^+)$ (with its associated structure sheaf and local valuations). An \emph{adic space} is an object $X$ of $\mathscr{V}$ such that there exists an open covering $\{U_i\}$ of $X$ 
such that each $U_i$ viewed in the evident manner as an object in $\mathscr{V}$ is an affinoid adic space.
\end{defin}

In particular, by the definition of $\mathscr{V}$, $\ccO_A$ is required to be a sheaf in order for $\Spa(A,A^+)$ to be an adic space. 
When this is the case, we call the Huber pair $(A,A^+)$ \emph{sheafy}. 

Our main goal in this lecture is to show that a large class of Huber pairs is sheafy so that the above definitions are not empty.

\subsection{A note on the category $\mathscr{V}$}

For Huber pairs $(A,A^+)$ and $(B,B^+)$, consider a continuous map $f:A \rightarrow B$ such that $f(A^+) \subset B^+$.
We get an induced continuous map $\varphi:Y = \Spa(B,B^+) \rightarrow \Spa(A,A^+)=X$ and for any rational domain
$W = X(T/s) \subset X$ and rational domain $U$ of $Y$ contained in $\varphi^{-1}(W)$ the universal property of 
the pair $(\ccO_A(W), \ccO_A(W)^+) = (A \langle T/s \rangle, A\langle T/s\rangle^+)$ over $(A,A^+)$ applied
to the composite map $$(A,A^+) \rightarrow (B,B^+) \rightarrow (\ccO_B(U), \ccO_B(U)^+)$$
provides a {\em unique} continuous map $$(\ccO_A(W), \ccO_A(W)^+) \rightarrow (\ccO_B(U),\ccO_B(U)^+)$$
over $f$.  Due to this uniqueness, as we vary such $U$ for fixed $W$ these maps are compatible under restriction, so we
obtain a {\em continuous} map
$$(\ccO_A(W), \ccO_A(W)^+) \rightarrow (\ccO_B(f^{-1}(W)), \ccO_B^+(f^{-1}(W))) = (f_{\ast}(\ccO_B)(W), f_{\ast}(\ccO_B^+)(W))$$
which is moreover compatible with restriction in $W$ (again via uniqueness).   Note
that it really is necessary to vary through rational domains $U$ inside $f^{-1}(W)$ because
the subset 
$f^{-1}(W)$ can fail to be rational in $Y$:  although certainly $f^{-1}(W)$
coincides with the subset $Y(f(T)/f(s))$ of points $y \in Y$ such that $y(f(t)) \le y(f(s)) \ne 0$ for all $t \in T$,
$f(T)\cdot B$ might fail to be open in $B$ (and so such inequalities need not define a rational domain).
For example, the map $(\bZ_p, \bZ_p) \rightarrow (\bZ_p[\![x]\!], \bZ_p[\![x]\!])$ (using
the $p$-adic and $(p,x)$-adic topologies respectively) with $T = \{p\}$ and $s = 1$
and $Y(f(T)/f(s))$ is not even quasi-compact! (This will be treated in more detail in Lecture 15
when we discuss adic spaces associated to rigid-analytic spaces.) 

Thus, we have built 
a map $Y \rightarrow X$ as adic spaces {\em if} $A$ and $B$ are sheafy.  (It is easy to check the compatibility
with valuations.)  This is readily seen to be compatible with composition, so it defines a contravariant functor 
from the category of sheafy Huber pairs (using continuous maps) into the category of affinoid adic spaces. 

The following proposition provides full faithfulness under a completeness hypothesis that is necessary
since by design $\ccO_A(\Spa(A,A^+)) = \widehat{A}$
(and recall as well the insensitivity of the topological space $\Spa(A,A^+)$ to completion of the Huber pair).

\begin{prop}[Proposition 2.1 in \cite{H2}]
Let $(A,A^+)$ and $(B,B^+)$ be sheafy Huber pairs with $B$ complete. The natural map
\[
\Hom((A,A^+),(B,B^+))\to\Hom(\Spa(B,B^+),\Spa(A,A^+))
\]
is a bijection. Here morphisms of Huber pairs are continuous homomorphisms $f:A\to B$ such that $f(A^+)\subseteq f(B^+)$, and morphisms on the right side are morphisms of adic spaces $($i.e., morphisms in the category $\mathscr{V}$$)$.
\end{prop}

In other words, the category of complete sheafy Huber pairs is anti-equivalent to the category of affinoid adic spaces. 

\begin{eg}
To see the importance of preserving the integral structure in the definition of a morphism of Huber pairs, consider the following example. Let $k$ be an algebraically closed non-archimedean field, $\bD_k=\Spa(k[t],k^0[t])$ the adic unit disc, and 
$\bD_k'=\Spa(k[t],A^+)$ where
\[
A^+=k^0+t\fm[t].
\]
This example was discussed in 11.3.13; the conclusion was that $\bD_k'$ consists of the union of $\bD_k$ with one additional type 5 point $v_{0,1^+}$ whose generization is the Gauss point. The identity map $k[t]\to k[t]$ induces a morphism of Huber pairs $(k[t],A^+)\to(k[t],k^0[t])$, which corresponds to the inclusion $\bD_k\hookrightarrow\bD_k'$. However, the identity does not induce a morphism of Huber pairs $(k[t],k^0[t])\to(k[t],A^+)$, which is good, as clearly $v_{0,1^+}$ has nowhere to go in the purported corresponding map $\bD_k'\to\bD_k$.
\end{eg}


\subsection{Huber's criteria}

The first sheafiness criteria were proven by Huber. In addition to proving the sheaf property, he also proved that higher cohomology of the structure sheaf vanishes on every rational subdomain, which is essential if one wants to calculate anything with sheaf cohomology.

\begin{thm}[Theorem 2.2 in \cite{H2}]\label{hubercoho}
Let $(A,A^+)$ be a Huber pair. Then $\ccO_A$ is a sheaf, and its higher cohomology vanishes on rational subdomains, if either of the following conditions holds:
\begin{enumerate}
\item[(i)] $\widehat{A}$ has a noetherian ring of definition.
\item[(ii)] $A$ is a \emph{strongly noetherian} Tate ring $($i.e., all rings $A\langle t_1,\ldots,t_n\rangle$ are noetherian$)$.
\end{enumerate}
\end{thm}

Recall that a Huber ring is Tate if it contains a topologically nilpotent unit. It is apparently not known whether there exist complete Tate rings that are noetherian but not strongly noetherian. An example of a (noncomplete) Tate ring that is noetherian but not strongly noetherian is given by the non-sheafy ring $A$ in Example \ref{nonsheafyexample} below (if it were strongly noetherian, it would be sheafy by \ref{hubercoho}!).

To appreciate the significance of these two conditions, recall that one of Huber's motivations in the definition of the category of adic spaces was to construct a category that fully faithfully contained the categories of locally noetherian formal schemes and Tate's rigid-analytic spaces. We have not defined the functors that realize these embeddings, but when we do (in later lectures) we will see that (i) applies to all adic spaces that come from locally noetherian formal schemes, and (ii) applies to all adic spaces that come from rigid-analytic spaces. In particular, given the obvious functor from locally noetherian schemes to locally noetherian formal schemes that assigns the discrete topology to each quasi-compact open set,\footnote{and the so-called ``pseudo-discrete'' topology to each open set in general; for a discussion of this functor see \cite[Tag 0AHY]{Stacks}.} we see that the category of adic spaces fully faithfully contains the category of locally noetherian schemes. 

\begin{eg}
The adic space $\Spa(\bZ,\bZ)$ associated to the noetherian scheme $\Spec(\bZ)$ (where $A^+ = \bZ$ is given the discrete topology) is the final object in the category of adic spaces. There is exactly one valuation in $\Spa(\bZ,\bZ)$ above the closed point  $(p) \in \Spec(\bZ)$ 
(corresponding to the trivial valuation on $\bF_p$) and fiber over the generic point identified with the Rieman-Zariski space
$\Spv(\bQ,\bZ) = \Spec(\bZ)$ under which the generic point corresponds to the trivial valuation
and $(p)$ corresponds to the $p$-adic valuation ring $\bZ_{(p)}$ in $\bQ$.
\end{eg}

It should not be a surprise that Theorem \ref{hubercoho}(i) is proven using techniques in the cohomology of formal schemes borrowed from \cite[III]{ega}, whereas (ii) closely follows the proof of Tate's original acyclicity result for rigid-analytic spaces.

The best example to keep in mind when considering the usefulness of (ii) is the adic space associated to the Tate ring $\bC_p\langle t_1,\ldots, t_n\rangle$. Although this ring is noetherian (and very nice in several other respects as well), its topology is \emph{not} generated by a noetherian ring of definition because $\bC_p$ is not discretely valued. Tate rings over discretely valued fields such as $\bQ_p$ are sheafy by both (i) and (ii).

\begin{eg}\label{mixedchar}
Although we will ultimately care more about criterion (ii) than (i), here is a useful toy example of the kinds of mixed-characteristic adic spaces that are now being considered to study (integral) $p$-adic Hodge theory
via perfectoid spaces (see \cite{S2}). Let
\[
X=\Spa(\bZ_p[\![T]\!],\bZ_p[\![T]\!]),
\]
where $\bZ_p[\![T]\!]$ is given the $(p,T)$-adic topology. Everything in sight is noetherian, so (i) of Theorem \ref{hubercoho} proves that $X$ is an adic space. An non-analytic point corresponds to a valuation killing the ideal of definition
$(p,T)$, so there is exactly one such point $s$, namely the trivial valuation on the residue field $\mathbf{F}_p$.
If we consider the continuous map of topological spaces 
$X\to\Spec(\bZ_p[\![T]\!])$ sending a valuation to its support, this point $s$ is the unique point lying above the closed point
$(p,T)\in\Spec(\bZ_p[\![T]\!])$. In particular, $s$ is closed in $X$.

Now remove this closed point $s$, defining the open adic subspace
\[
Y=X-\{s\}.
\]
This consists entirely of analytic points. It possesses exactly one characteristic-$p$ point, corresponding to a copy of 
$\Spa(\bF_p(\!(T)\!),\bF_p[\![T]\!])$; all other points are characteristic zero. We can draw a useful picture of $Y$ by considering 
the inclusions $\Spa(\bF_p[\![T]\!],\bF_p[\![T]\!])\hookrightarrow X$ and $\Spa(\bZ_p,\bZ_p)\hookrightarrow X$, and imagining 
these two spaces as the horizontal and vertical axes of a quadrant in the plane. The horizontal axis is the locus ``$T=0$'' while 
the vertical axis is the locus ``$p=0$'' (i.e., the characteristic-$p$ part, or the special fiber over $\Spa(\bZ_p, \bZ_p)$). 
These axes in $X$ consist of 
exactly two points, and they intersect at the non-analytic point $s$, which we draw at the origin. All other points lie 
``somewhere between'' the horizontal and vertical axes in the first quadrant of the plane. 

Since $p$ and $T$ are topologically nilpotent, the following 
 rational domains of $X$ that lie inside $Y$ (consisting of ``rational sectors'' of this quadrant) cover subsets of interest: if we set
\[
Y_n^+=\{v\in Y:v(T^n)\le v(p)\ne0\}\quad \text{and}\quad Y_n^-=\{v\in Y:v(p^n)\le v(T)\ne0\},
\]
where $n$ varies over nonnegative integers, then $\bigcup_{n=1}^\infty Y_n^+$ is exactly the complement of the locus $p=0$ and $\bigcup_{n=1}^\infty Y_n^-$ is exactly the complement of the locus $T=0$. One can easily calculate that the complete Huber pairs
 associated to $Y_n^+$ and $Y_n^-$ are $(A^+[1/p],A^+)$ and $(B^+[1/T],B^+)$, respectively, where $A^+$ is the $p$-adic 
 completion of $\bZ_p[\![T]\!][T^n/p]$ and $B^+$ is the $p$-adic completion of $\bZ_p[\![T]\!][p^n/T]$. 
 All of these Huber rings are Tate, though the latters ones do {\em not} contain any field.

The moral of these calculations is that the complement of the locus $p=0$ in $Y$, which we would like to define as the generic fiber of $Y$,\footnote{To make this picture clearer, one can consider $Y$ as fibered over the adic space $\Spa(\bZ_p,\bZ_p)$ (our vertical axis), which consists of two points, one in characteristic $p$ and one in characteristic zero. The fiber over the characteristic $p$ point consists of the one point described above, while the fiber over the characteristic zero point is its complement, which we would like to define as the generic fiber of $Y$. One could define similar notions for the fibers of $Y$ over our horizontal axis $\Spa(\bF_p[\![T]\!],\bF_p[\![T]\!])$.} is covered by a \emph{countable union} of very nice rational domains, but is not itself a rational domain. In fact, it is not even quasi-compact, so it cannot possibly be an affinoid adic space. This is very different from the corresponding picture for schemes, where the generic fiber of an affine scheme over a mixed-characteristic ring is certainly affine! In particular, it would be a bad mistake to attempt to define the generic fiber 
of $X$ over $\Spa(\bZ_p, \bZ_p)$ to be 
as $\Spa$ of $\bZ_p[[x]][1/p]$ with some topology and some ``ring of integers''; as we have in fact already seen in Example 5.5.5.
\end{eg}


As a final and somewhat parenthetical remark, the space $X$ is a nice illustration of the following lemma, which states that a point is analytic if and only if it is ``locally Tate'':

\begin{lemma}
Let $(A,A^+)$ be a Huber pair and $x\in\Spa(A,A^+)$. Then $x$ is analytic if and only if there exists a rational domain $U$ containing $x$ such that $\ccO_A(U)$ is Tate.
\end{lemma}

\begin{proof}
First assume that $x$ is analytic, and let $\{f_1,\ldots,f_n\}$ generate an ideal of definition $I$ of a ring of definition of $A$. Without loss of generality, we may rearrange the $f_i$'s so that $v_x(f_i) \le v_x(f_1)$ for all $i$.
 Because $x$ is analytic, so the support of $v_x$ is not open, necessarily $v_x(I)$ is nonzero. In particular, $v_x(f_1)\ne0$. Hence, 
\[
U=X\left(\frac{\{f_1,\ldots,f_n\}}{f_1}\right)
\]
is a rational domain on which $f_1$ is a unit, and by design $x\in U$.  Since we can write $f_i= (f_i/f_1)\cdot f_1$ in $\ccO_A(U)$
with $f_i/f_1 \in \ccO^+_A(U)$, the $I$-adic topology on 
\[
\ccO_A^+(U)=A\left\langle\frac{\{f_1,\ldots,f_n\}}{f_1}\right\rangle^+,
\]
is the same as the $f_1$-adic-topology. Therefore $f_1$ is a topologically nilpotent unit in $\ccO_A(U)$, so $\ccO_A(U)$ is Tate.

In the other direction, since all points in $\Spa(B,B^+)$ are analytic when $B$ is Tate,
it suffices show that if $x$ is analytic ``from the point of view of $U$'' it is also analytic ``from the point of view of $\Spa(A,A^+)$.'' This is easy enough by tracking around ideals of definition: any ideal of definition $J$ of a ring of definition of $\ccO_A(U)$ cannot be killed by $v_x$, but any ideal of definition $I$ of a ring of definition of $A$ generates 
such a $J$, so $I$ cannot be killed by $v_x$ either. Therefore $x$ is analytic.
\end{proof}


\subsection{The criterion of Buzzard and Verberkmoes}

Another sheafiness criterion has been established for a particular class of Tate rings; namely, those that \cite{BuVe} calls \emph{stably uniform}. Most of our favorite examples of Tate rings contain a field; for example, affinoid algebras in rigid geometry or (as we will see) perfectoid algebras over perfectoid fields. Therefore \cite{BuVe} work under that hypothesis. However, it is unnecessary for the proofs to assume the presence of a field, as we will indicate, and there are examples where it is helpful to work in greater generality:

\begin{eg}
Perhaps the minimal example of a Tate ring that does not contain a field is $\bZ[X]_{(X)}$ with the $X$-adic topology. A considerably less silly example is the mixed-characteristic Example \ref{mixedchar}. An example ``from nature'' is the adic space avatar of the Fargues-Fontaine curve. 
\end{eg}

For the rest of this section, let $(A,A^+)$ be a Huber pair with $A$ Tate.

\begin{defin}
The pair $(A,A^+)$ is \emph{uniform} if the open subring $A^0$ of power-bounded elements is bounded.
\end{defin}

\begin{eg}\label{notuniform}
Examples of non-uniform Tate rings include non-reduced rings like $\bQ_p[\epsilon]/(\epsilon^2)$ with the $p$-adic topology. For such a ring we have
\[
A^0=\bZ_p\oplus\bQ_p\cdot\epsilon,
\]
which is clearly not bounded as it contains a $\bQ_p$-line.
\end{eg}

\begin{defin}
The pair $(A,A^+)$ is \emph{stably uniform} if for all rational $U\subseteq\Spa(A,A^+)$, the ring $\ccO_A(U)$ is uniform.
\end{defin}

Recall that the rings $\ccO_A(U)$ are automatically complete, but this is irrelevant as uniformity is preserved under completion. 
Also, the stably uniform property is intrinsic to the topological ring $A$ without reference to $A^+$: it is the assertion that
$A(T/s)$ (or equivalently its completion $A\langle T/s \rangle$) is uniform for any $s \in A$ and finite non-empty subset
$T \subset A$ such that $T \cdot A$ is open. 
We will see an example of a stably uniform pair that is not uniform in the next section.

\begin{thm}[Theorem 7 in \cite{BuVe}]\label{BVthm}
Let $(A,A^+)$ be a stably uniform Huber pair with $A$ Tate. Then $\ccO_A$ is a sheaf 
 and its higher sheaf cohomology on rational domains vanishes.
\end{thm}

The proof will be sketched later, after we work with some concrete examples. The above statement differs from that in \cite{BuVe} by  working in the generality of stably uniform Tate rings and including the statement about vanishing cohomology, which follows using the method of proof of Theorem 2.4.23 in \cite{KL} (which itself essentially follows Tate's original proof of the acyclicity theorem in classical rigid geometry).

The key usefulness of this result is that there are no finiteness assumptions whatsoever (aside from those in the definition of a Huber pair), so it can be applied to really huge rings like those that arise in the theory of perfectoid spaces. Useful rings that cannot be handled by the above theorem include most rings coming from formal schemes, which will not in general be Tate, and nonreduced rings coming from classical rigid geometry such as Example \ref{notuniform}. Both of these types of rings are sheafy by Huber's result.
(We will see immediately from the defintion that perfectoid algebras are always reduced.)


\subsection{Examples and counterexamples}

A natural reaction to Theorem \ref{BVthm} is to ask whether the stably uniform hypothesis on Tate rings is necessary; that is, can we come up with an example of a Tate ring that is not sheafy? The construction is not obvious, but one example is the following:

\begin{eg}[{\cite[\S\S4.1-2]{BuVe}}]\label{nonsheafyexample}
Let
\[
A=\bQ_p[T,1/T,Z]/(Z^2).
\]
This is a nice ring, finitely generated over a discretely-valued non-archimedean field, but we are going to define a rather nasty topology on it. Let $A_0$ be the $\bZ_p$-submodule of $A$ generated by all elements of the form
\begin{enumerate}
\item[(i)] $p^{|m|}T^m$, and
\item[(ii)] $p^{-|m|}T^mZ$
\end{enumerate} for all $m\in\bZ$. One checks easily that the elements in (i) and (ii) are linearly independent over $\bZ_p$
and that $A_0$ is actually a ring with $A_0[1/p]=A$. Put the topology on $A$ that has as a local basis at zero the subgroups $p^nA_0$ for $n\in\bZ$, so that in particular $A_0$ is an open subring of $A$ with ideal of definition $(p)$ and $(A,A_0)$ is a Huber pair. 

Let
\[
X=\Spa(A,A_0).
\]
The space $X$ has a cover by the rational domains
\[
U=\{v\in X:v(T)\le1\}\quad\text{and}\quad V=\{v\in X:v(T)\ge1\}.
\]

We claim that the element $Z\in A$ considered as a global section on $X$ is nonzero but that its restrictions to both $U$ and $V$ are zero. If this is the case, $\ccO_A$ is clearly not a sheaf, as it fails the ``uniqueness of gluing'' (or ``locality'') axiom.

The proof of the claim is straightforward; we know that 
\begin{align*}
\ccO_A(X)&=\varprojlim_nA/p^nA_0,\\
\ccO_A(U)&=\varprojlim_nA/(p^nA_0[T]), \\ 
\ccO_A(V)&=\varprojlim_nA/(p^nA_0[1/T]).
\end{align*}
We verify that $Z\ne0$ in $\ccO_A(X)$ because it is not contained in $p^nA_0$ for any nonnegative $n$. But we claim that $Z\in p^nA_0[T]$ and $Z\in p^nA_0[1/T]$ for all nonnegative $n$, which implies that $Z=0$ in both $\ccO_A(U)$ and $\ccO_A(V)$! 
More specificallly, is enough to write $Z$ as $p^n$ multiplied by an element in $A_0$ multiplied by non-negative integral powers of both 
$T$ and $1/T$:
\[
Z=p^n\cdot(p^{-n}T^{-n}Z)\cdot T^n\quad\text{and}\quad Z=p^n\cdot(p^{-n}T^nZ)\cdot T^{-n}.
\]
Note that $p^{-n}T^{-n}Z$ is a monomial of the form (ii) by taking $m=-n$, and $p^{-n}T^nZ$ is a monomial of the form (ii) by taking $m=n$.

In this example, there is nothing special about the field $\bQ_p$; the same calculation works for any non-archimedean field with topology given by a rank-one valuation (and $p$ replaced by any pseudo-uniformizer). 
A similar construction also works over $\bZ$; see Rost's example at the end of \cite[\S1]{H2}.

If one actually calculates the rings of sections $\ccO_A(U)$ and $\ccO_A(V)$, one finds that
\[
\ccO_A(U)=\widehat{\bQ_p[T,1/T]}
\]
with a topology generated by the subring $\bZ_p[T,p/T]$, and
\[
\ccO_A(V)=\widehat{\bQ_p[T,1/T]}
\] 
with a topology generated by the subring $\bZ_p[pT,1/T]$. That is, $U$ is the adic space associated to the annulus $\{|p|\le|T|\le1\}$ and $V$ is the adic space associated to the annulus $\{1\le|T|\le|1/p|\}$. They do not glue properly, however; in particular, $\ccO_A(X)$ contains the nilpotent element $Z$.

One can ``perfectify'' this example to get a topological space that is locally perfectoid (in fact, having a cover by rational domains that are just perfectoid annuli) but not globally perfectoid (because perfectoid algebras will be shown to be sheafy): we let
\[
A'=(\varinjlim_{T\mapsto T^p}\bQ_p[T,1/T])[Z]/(Z^2),
\]
(which is often abbreviated as $\bQ_p[T^{1/p^\infty},T^{-1/p^\infty},Z]/(Z^2)$) and let $A_0'$ be the $\bZ_p$-submodule of $A'$ generated by elements of the form $p^{|n|}T^n$ and $p^{-|n|}T^nZ$, for all $n\in \bZ[1/p]$. The example then goes through as before.
\end{eg}

We have seen a failure of locality in the presheaf $\ccO_A$. It is also possible to exhibit a Tate ring $A$ such that $\ccO_A$ fails the gluing axiom:

\begin{eg}[{\cite[\S4.4]{BuVe}}]
Let
\[
A=\bQ_p[T,1/T,Z_1,Z_2,\ldots],
\]
and take $A_0$ to be the free $\bZ_p$-submodule of $A$ generated by elements of the form $p^dT^a\prod_iZ_i^{e_i}$, for $a,d\in\bZ$ and each $e_i\in\bZ_{\ge0}$, subject to the following conditions:
\begin{itemize}
\item[(i)] If all $e_i=0$, then $d=|a|$.
\item[(ii)] If $\sum_i e_i=1$, then $d=|a|-2\min\{\sum_i ie_i,|a|\}$.
\item[(iii)] Otherwise, $d=|a|-2\sum_iie_i$.
\end{itemize}
It is a considerably more tedious process this time to check that $A_0$ is in fact a subring, though obviously
 $A_0[1/p]=A$. Once this has been done, we can as above define a topology on $A$ generated by the basis at zero consisting of the subgroups $p^nA_0$ for $n\in\bZ$. We let the rational subdomains $U$ and $V$ of $\Spa(A,A_0)$ be defined exactly as in the previous example.

The key idea here is that the element $\sum_iZ_i$ converges on $U$ and on $V$, but does not converge on all of $X$, so there are two local sections that agree on intersections but that do not glue. To verify the former claim, we note that for all $n\ge0$,
\[
Z_n=p^n\cdot(p^{-n}T^{-n}Z_n)\cdot T^n\in p^nA_0[T],
\]
so $Z_n\to0$ in $A_0[T]$ as $n\to\infty$ and therefore the (non-archimedean) sum converges on $U$. Note that the element $p^{-n}T^{-n}Z_n$ satisfies condition (ii) above. Similarly,
\[
Z_n=p^n\cdot(p^{-n}T^nZ_n)\cdot T^{-n}\in p^nA_0[1/T],
\]
so $Z_n\to0$ in $A_0[1/T]$ as $n\to\infty$ and therefore the sum converges on $V$. The element $p^{-n}T^nZ_n$ likewise satisfies condition (ii).  It is clear that these sections over $U$ and $V$ coincide over $U \cap V$.

To verify that $\sum_iZ_i$ does not ``make sense'' in $\ccO_A(X)$ is more complicated, but the idea is straightforward. We define functions $\rho_i:A\to\bQ_p$ that send each element of $A$ to its coefficient for the degree-1 monomial $Z_i$. It is easy to check that each $\rho_i$ is continuous, so it extends to a function on the completion $\ccO_A(X)$. Now I claim that for any $r\in\ccO_A(X)$, we must have
\[
\lim_{i\to\infty}\rho_i(r)=0.
\]
This follows because $r$ is the limit of a Cauchy sequence of elements of $A$, and each element of $A$ has no $Z_N$ coefficient at all for large enough $N$, so a simple approximation argument suffices. The hypothetical global section ``$\sum_iZ_i$''
(by which we really mean any hypothetical gluing of that sum viewed over $U$ and $V$)
cannot satisfy this condition, and therefore cannot exist.
\end{eg}

It seems unlikely that there exists a \emph{noetherian} counterexample to the gluing axiom.

Finally, one would like a counterexample proving that ``stably uniform'' cannot be replaced by ``uniform'' in the statement of Theorem \ref{BVthm}, or failing that at least an example of a Tate ring that is uniform but not stably uniform, to show that the condition is not empty. The latter construction is easier and is described below; for an example of the former see \cite[\S4.6]{BuVe}.

\begin{eg}[{\cite[\S4.5]{BuVe}}]
To construct a uniform Tate ring that is not stably uniform, first let
\[
\tilde A=\bQ_p[T,1/T,Z],
\]
and let $A_0$ be the free $\bZ_p$-submodule generated by elements of $A$ of the form $(pT)^a(pZ)^b$, where $b\ge0$ and $a\ge-b^2$. Again, one verifies that $A_0$ is actually a ring, and one sets $A=A_0[1/p]$ (this time, $A\ne\tilde A$; we'll never get $1/T$ by inverting $p$, for instance). Topologize $A$ using the subgroups $p^nA_0$ as before.

We first claim that $A$ is uniform. In fact, we claim that $A_0=A^0$. By \cite[Lemma 11]{BuVe}, any compatible grading on $A$ and $A_0$ induces a grading on $A^0$, so to show this equality it suffices to check on each graded part separately. This is easy: if $\lambda\cdot(pT)^a(pZ)^b\in A^0$ with $\lambda\in\bQ_p$, then the powers $\lambda^n(pT)^{an}(pZ)^{bn}$ are bounded. But such powers escape $p^{-N}A_0$ for every $N$ unless $\lambda\in\bZ_p$, which in turn implies that $\lambda\cdot (pT)^a(pZ)^b$ was in $A_0$ after all. We have proven that $A$ is uniform.

Next, with $U$ defined as usual to be $\{v\in\Spa(A,A_0):v(T)\le1\}$, the ring $\ccO_A(U)$ is not uniform. This is true because now the element $Z/p^n$ is power-bounded for all $n$, so we get a $\bQ_p$-line in $\ccO_A(U)^0$. Indeed, we calculate
\[
\left(\frac{Z}{p^n}\right)^{n+1}=(pT)^{-n^2-2n-1}(pZ)^{n+1}T^{n^2+2n+1}\in\widehat{A_0[T]}=\ccO_A(U)^+.
\]
As the $(n+1)$th power is inside the ``closed unit ball'' defining the topology on $\ccO_A(U)$, 
it follows that $Z/p^n$ is power-bounded.
\end{eg}

\subsection{Sketch of the proof}

There are two steps in the proof of Theorem \ref{BVthm}, following Tate. First, one verifies the sheaf property for a very specific type of cover. Specifically, we wish to verify that for a uniform and Tate Huber pair $(A,A^+)$, the sheaf property holds with respect to covers of the form
\[
U=\{v\in X:v(t)\ge1\}\quad\text{and}\quad V=\{v\in X:v(t)\le1\},
\]
where $t$ is any element of $A$ and $X=\Spa(A,A^+)$. Second, one verifies that we can reduce the general case to this specific case, this time under a stably uniform hypothesis.

The heart of the proof is the first step, so we will consider it in more detail. We will deal first with the corresponding sequence of uncompleted exact rings, which for the cover $\{U,V\}$ reads
\begin{align}\label{uncomplete}
0\to A\xrightarrow{\epsilon} A(t/1)\oplus A(1/t)\xrightarrow{\delta}A(t/1, 1/t) = 
A(\{t^2,1\}/t)\to0.
\end{align}
Here $\epsilon$ is the localization map to both factors, $\delta$ is simply subtraction of the second factor from the first,
and the ring notation is as defined in Lectures 6 and 7; for example, 
\[
A\left(\frac{t}{1}\right)=A\left(\frac{\{1,t\}}{1}\right)=A[X]_{\{1,t\}}/(1-X),
\]
where $A[X]_{\{1,t\}}$ is the polynomial ring $A[X]$ equipped with the topology where a basis of open sets around zero is given by the sets
\[
U[\{1,t\}\cdot X]:=\left\{\sum_{\text{finite}}a_iX^i:a_i\in t^j\cdot U \mbox{ for some } j \le i\right\}
\]
where $U$ runs over all open additive subgroups $U\subseteq A$. This construction makes sense by Proposition 6.3.5 because $\{1,t\}$ generates the unit ideal, which is certainly open. By the discussion in \S7.6, a ring of definition for $A[X]_{\{1,t\}}$ is given by 
$A_0[\{1,t\}\cdot X]$ equipped with the $J\cdot A_0[\{1,t\}\cdot X]$-adic topology where $J$ is an ideal of definition for 
a ring of definition $A_0$ of $A$. Taking the quotient, we find that a ring of definition for $A(t/1)$, denoted by $A_0[t/1]$, is given by $A_0[t]$ with a certain topology on it.

With all relevant notation in place, it is clear that $\epsilon$ and $\delta$ are continuous and 
we may proceed to simplify things dramatically by using the Tate hypothesis. Let $u$ be a topologically nilpotent unit in $A$
and fix a ring of definition $A_0$. Replacing $u$ with $u^n$ for sufficiently large $n$
so that $u$ lies in the open subring $A_0$. By Proposition 7.5.3, $A_0$ has the $u$-adic topology. 
Applying the same reasoning to the  rational localizations (and noting that $u$ is still a topologically nilpotent unit in $A(t/1)$), we find that the topology on the ring of definition $A_0[t/1]\subset A(t/1)$ is also $u$-adic, and $A_0[t/1][1/u]$ is equal to all of $A(t/1)$.

The same reasoning can be applied to the rings $A(1/t)$ and $A(t,1/t)$, proving that their rings of definition $A_0[t/1]$ and $A_0[t/1,1/t]$ both have the $u$-adic topology and that inverting $u$ yields the ambient ring in each case. 
It is a trivial matter to verify that (\ref{uncomplete}) is an exact sequence of rings: the underlying sequence of rings is just
\[
0\to A\to A\oplus A[1/t]\to A[1/t]\to0
\]
in which composition of the second and third maps is clearly zero
whereas if an element of $(r,s)\in A\oplus A[1/t]$ maps to zero then we must have $r=s$ 
in $A[1/t]$ (and in particular $s$ is  the image of $r \in A$), so $(r,s)$ comes from $A$.

Unfortunately, the uncompleted groups are not what we are interested in, and in general completing a sequence of topological abelian
groups does not preserve exactness (the whole point is that we do not want to assume restrictive finiteness hypotheses!). Fortunately, Lemme 2 of III.2.12 in \cite{Bou} comes to the rescue. It states that if an exact sequence of topological abelian groups
 has \emph{strict} maps then completion preserves exactness.\footnote{There is an additional technical condition: we require that there exists a countable basis of the topology around zero. This is always satisfied for Huber rings, whose topology around zero is given by powers of a given ideal in a ring of definition}  We recall that a map of topological abelian groups
  $f:B\to C$ is {\em strict} if the quotient topology on $f(B)$ from $B$ coincides with the subspace topology from $C$. It is easy to see that $f$ is strict if and only if it is continuous and and the induced map $B\to f(B)$ is open.

In our case, the map $\delta:A(t/1)\oplus A(1/t)\to A(t/1,1/t)$ is always strict: by translating and scaling by powers of $u$, it suffices to check that the image of $A_0[t/1]\oplus A_0[1/t]$ is open in $A(t/1,1/t)$, which is the case because this image is precisely $A_0[t/1,1/t]$. However, it may not be the case that $\epsilon:A\to A(t/1)\oplus A(1/t)$ is strict; in particular, this is where the uniformity hypothesis comes in.

Let's see what we would need in order for $\epsilon$ to be strict. Again by translating and scaling, it would suffice to show that the image of $A_0$ is open, which is the case if and only if it contains a set of the form $u^n(A_0[t/1]\oplus A_0[1/t])$ for some $n$ (as these sets form a basis around zero). In other words, we seek some $n$ such that $u^n(\phi^{-1}(A(t/1))\cap\psi^{-1}(A(1/t)))$ lies in $A_0$, where $\phi:A\to A(t/1)$ and $\psi:A\to A(1/t)$ are the canonical maps.

This is proven in two stages. First, we will show that for any Tate ring, if a global section is locally integral, then it is globally power-bounded. This establishes that $\phi^{-1}(A(t/1))\cap\psi^{-1}(A(1/t))$ is contained in $A^0$. Next, we use uniformity, which implies that $u^nA^0$ is contained in $A_0$ for some $n$. Putting the two together, we would obtain strictness of 
(\ref{uncomplete}) and therefore exactness of the completed sequence
\begin{align}\label{completed}
0\to\ccO_A(X)\to\ccO_A(U)\oplus\ccO_A(V)\to\ccO_A(U\cap V)\to0.
\end{align}
Thus, we need the following lemma, applied to $t_1=1,t_2=t$:

\begin{lemma}[generalization of Lemma 3 in \cite{BuVe}]\label{locallyintegral}
Let $A$ be a Tate ring and let $t_1,\ldots,t_n\in A$ be elements that generate the unit ideal. Consider the rational localizations
\[
A_i=A(\{t_1,\dots,t_n\}/t_i)
\]
$($a topological ring structure on $A_{t_i}$$)$
together with the canonical maps $\phi_i:A\to A_i$ and the associated open subrings
\[
A_{i,0}=A_0\left[\frac{t_1}{t_i},\ldots,\frac{t_n}{t_i}\right].
\]
If an element $r\in A$ satisfies $\phi_i(r)\in A_{i,0}$ for all $i$ then $r\in A^0$.
\end{lemma}

\begin{proof}
The proof goes through exactly as in \cite{BuVe}, except instead of taking an $\eta$ in the base field (which we are not assuming to exist) such that $\eta t_i\in A_0$ for all $\eta$, we take $\eta$ to be a large enough power of a topologically nilpotent unit $u$, and likewise for $\theta$
in their argument.
\end{proof}

Having proven sheafiness for the cover $\{U,V\}$ (under the hypothesis of uniformity), we wish to bootstrap to the general case (under the hypothesis of \emph{stable} uniformity). The basic method is standard and goes back to Tate, so we give only a sketch. 

First one deals only with $\ccO_A$ as a presheaf of abelian groups (ignoring the topological aspect). The method to prove that it is actually a sheaf follows the classical case very closely: first one proves the sheaf property for so-called \emph{Laurent covers}, which are covers of the form $\{U_I\}$ where $I$ runs over all subsets of $\{1,2,\ldots,n\}$, $t_1,t_2,\ldots,t_n$ are elements of $A$, and
\[
U_I=\{v\in X:v(t_i)\le1\text{ for }i\in I,v(t_i)\ge1\text{ for }i\notin I\}.
\]
This is proved by induction on $n$; the base case $n=1$ has been of course proven above. Then, by using a refinement lemma exactly paralleling Lemmas 8.2.2/2-4 in \cite{bgr}, one bootstraps to so-called \emph{rational covers}, which are those considered in Lemma \ref{locallyintegral}, and then to arbitrary covers by rational domains. By the discussion in Lecture 12, this suffices to show that $\ccO_A$ is a sheaf of abelian groups.

To prove that $\ccO_A$ is actually a sheaf of topological rings, by the discussion at the end of Lecture 12 we need only check for that for a cover $\{U_i\}$ of a rational domain $U$ by rational domains, the induced map
\begin{align}\label{embedding}
\ccO_A(U)\to\prod_i\ccO_A(U_i)
\end{align}
is a topological embedding (i.e., strict). We can refine this to a finite rational cover by the same refinement lemma as before (using
 quasi-compactness of rational domains). Then the same argument as in the case of the cover $\{U,V\}$, using Lemma \ref{locallyintegral} to 
 relate $\prod_i\ccO_A(U_i)^+$ with power-bounded elements of $\ccO_A(U)$ and using strict uniformity (which we now need, as we are dealing with an arbitrary rational domain $U$) to control the power-bounded elements of $\ccO_A(U)$ by  $\ccO_A(U)^+$, 
 we conclude that \ref{embedding} is a topological embedding. This completes the proof of sheafiness.

To conclude that the higher cohomology of $\ccO_A$ vanishes on rational domains, one can use a \v{C}ech-theoretic approach that also mirrors Tate's original acyclicity theorem: we have already shown that the sequence (\ref{completed}) is right-exact, thereby handling
the case of a \v{C}ech cover consisting of two open sets, and we can use the same refinement lemmas to reduce to this case. (Note that cohomology is computed in the category of abelian sheaves, so we are free to disregard the topologies of the rings in question.) For a detailed argument in a slightly restricted setting (but which is perfectly general and applies to the setting of sheafy Tate rings just as easily), see \cite[\S2.4]{KL}, especially Theorem 2.4.23.\qed

\subsection{A corollary of the proof, and one more example}

The following result and example are almost surely useless, as we will be dealing exclusively with sheafy rings from now on, but are at least moderately amusing.

\begin{cor}[Corollary 5 in \cite{BuVe}]
Let $X=\Spa(A,A^+)$, where $(A,A^+)$ is a Huber pair with $A$ Tate. If $f\in\ccO_A(X)$ and there exists a cover of $X$ by open sets such that $f|_{U_i}=0$ for all $i$, then $f$ is topologically nilpotent. 
\end{cor}

\begin{proof}
As usual, it suffices to check on covers by rational domains, and by one of the refinement lemmas it suffices to check on rational covers (i.e., covers of the form described in Lemma \ref{locallyintegral}). By that Lemma, an element that is locally zero is certainly globally power-bounded. Applying this to $u^{-1}f$, where $u$ is a topologically nilpotent unit, we see that $u^{-1}f$ is power-bounded and hence $f=u\cdot u^{-1}f$ is topologically nilpotent.
\end{proof}

\begin{eg}[{\cite[\S4.3]{BuVe}}]
We have seen an example of a locally zero element that is globally nilpotent (but nonzero) in Example \ref{nonsheafyexample} above. One can also give an example of a Tate ring with a locally zero element that is not globally nilpotent (although by the above corollary, it must be topologically nilpotent). One takes
\[
A=\bQ_p[T,1/T,Z]
\]
and $A_0$ to be the $\bZ_p$-subalgebra generated by the elements $pT$, $p/T$, $p^{-n}T^{a(n)}$, and $p^{-n}T^{-b(n)}$, where $a(n)$ and $b(n)$ are certain sequences that increase extremely rapidly. One topologizes $A$ by taking the $p$-adic topology on $A_0$ as usual. It is easy to check that $Z$ restricts to zero on the familiar rational subdomains $U$ and $V$, but one has to work harder
to check that for each $e$ there is a number $M(e)$ such that $Z^e\notin p^{M(e)}A_0$, thereby verifying 
that $Z$ is not nilpotent in the completion of $A$. Details can be found in \cite{BuVe}.
\end{eg}

\begin{thebibliography}{dieci}
%\bibitem[Bo]{oldbosch} S. Bosch, ``Orthonormalbasen in der nichtarchimedischen Funktionentheorie'', Manuscripta Math. {\bf 1} (1969), pp. 35--57.
\bibitem[BGR]{bgr} S.\,Bosch, U.\,G\"untzer,\,R.\,Remmert, {\em Non-archimedean analysis}, Springer-Verlag.
\bibitem[Bou]{Bou} N. Bourbaki, Alg\'ebre commutative, chap. I - VII, Springer (1989); chap. VIII- IX, Masson (1983); chap. X, Masson (1998).
\bibitem[BV]{BuVe} K. Buzzard, A. Verberkmoes, {\em Stably uniform affinoids are sheafy}, arxiv preprint (2014).
%\bibitem[Bosch]{Bosch} S. Bosch, {\em Lectures on Formal and Rigid Geometry}. Springer Lecture Notes in Mathematics, vol. 2105.
%\bibitem[Con]{Con} B. Conrad - ``Several approached to non-archimedean geometry'', Arizona Winter School lecture notes available at: \texttt{http://math.stanford.edu/$\sim$conrad/papers/aws.pdf}.
\bibitem[EGA]{ega} A. Grothendieck, EGA, Publ.\,Math.\,IHES 1961--68.
%\bibitem[dJ]{dJ} A.J.\,de Jong, ``Crystalline Dieudonn\'e theory via rigid and formal geometry'', IHES {\bf 82} (1995), 5--96.
%\bibitem[Mat]{Mat} H. Matsumura - {\em Commutative Ring Theory}. Vol. 8 of Cambridge Studies in Advanced Mathematics (1989)
%\bibitem[MM]{MM} S. MacLane, I. Moerdijk - {\em Sheaves in Geometry and Logic: a first introduction to topos theory}. Springer Science $\&$ Business Media, 1992.
%\bibitem[HuKne]{HuKne} R. Huber, M. Knebusch, {\em On valuation spectra}, Contemporary Mathematics vol. 155 (1994).
%\bibitem[AM]{AM} M. F. Atiyah, I. G. MacDonald, {\em Introduction to Commutative Algebra}.
%\bibitem[H1]{H1} R. Huber, {\em Continuous valuations}.
\bibitem[H2]{H2} R. Huber, {\em A generalization of formal schemes and rigid-analytic varieties}.
\bibitem[KL]{KL} K. Kedlaya, R. Liu, {\em Relative $p$-adic Hodge theory: Foundations}.
%\bibitem[Mi]{mihara} T.\,Mihara, ``On Tate acyclicity and uniformity of Berkovich spectra and adic spectra'', arxiv preprint (2014).
%\bibitem[S1]{S1} P. Scholze, {\em Perfectoid spaces}.
\bibitem[S2]{S2} P. Scholze, {\em Lectures on $p$-adic geometry, Fall 2014}, notes by J. Weinstein.
\bibitem[SP]{Stacks} A. J. de Jong, {\em Stacks Project}.
%\bibitem[vdP]{vdP}  M. van der Put - {\em Cohomology on affinoid spaces.} Compositio Math. 45 (1982), 165 - 198.
%\bibitem[vdPS]{vdPS}  M. van der Put, P. Schneider - {\em Points of rigid analytic varieties.} J. reine angew. Math. 434 (1993), 127 - 157.
%\bibitem[Zar2]{Zar2} O. Zariski - {\em The reduction of the singularities of an algebraic surface}. Ann. Of Math. (2) 40 (3): 639-689 (1939).
%\bibitem[Zar3]{Zar3} O. Zariski {\em Reduction of the singularities of algebraic three dimensional varieties}. Ann. Of Math. (2) 45 (3) (1944).
\bibitem[Wed]{Wed} T. Wedhorn, {\em Adic Spaces}, unpublished lecture notes.


%\bibitem[Po]{poonen} B.\,Poonen, {\em Maximally complete fields}, L'enseignement Math.\,{\bf 39} (1993), pp.\,87--106.



\end{thebibliography}






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