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# The geometry of diamonds

Laurent Fargues

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Laurent Fargues. The geometry of diamonds. 2023. hal-04235125

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Distributed under a Creative Commons CC0 1.0 - Universal - International License
TOKYO LECTURES ON THE GEOMETRY
OF DIAMONDS
Laurent Fargues

Institut de Mathématiques de Jussieu / CNRS.

E-mail : laurent.fargues@imj-prg.fr
TOKYO LECTURES ON THE GEOMETRY OF DIAMONDS

Laurent Fargues
<figure><img src="image_3.png" /></figure>
<figure><img src="image_4.png" /></figure>

Ce texte est dédié à Roland Huber à l'occasion de son départ à la retraite, en remerciement pour avoir envoyé son Habilitationsschrift à l'auteur en l'année 2000 et avoir répondu à ses questions.
<figure><img src="image_5.png" /></figure>
CONTENTS

Introduction.................................................................................. 1

1. Spectral spaces........................................................................ 3

1.1. Background: rigid geometry, 4 points of view.................................. 3

1.2. Spectral spaces: two equivalent definitions.................................. 5

1.3. The constructible topology.......................................................... 7

1.4. Projective limits of spectral spaces.................................................. 11

1.5. The equivalence between spectral spaces and pro-finite (T0) spaces.................. 13

1.6. Connected components.................................................................. 14

1.7. Quotients of spectral spaces.......................................................... 15

1.8. Constructible sheaves.................................................................. 17

1.9. Totally disconnected spectral spaces............................................. 18

1.10. The Berkovich spectrum.......................................................... 24

1.11. En résumé............................................................................. 27

2. Adic spaces.................................................................................. 29

2.1. Huber rings.................................................................................. 29

2.2. Affinoid rings.................................................................................. 36

2.3. The adic spectrum........................................................................ 37

2.4. Affinoid fields.................................................................................. 41

2.5. The presheaf of holomorphic functions............................................. 41

2.6. Analytic points.................................................................................. 43

2.7. Properties of the local rings and the residue fields.......................... 46

2.8. Overconvergent open subsets: the Berkovich spectrum.................. 47

2.9. The sheaf property........................................................................ 49

2.10. Adic spaces.................................................................................. 58

2.11. Analytic adic spaces........................................................................ 58

2.12. Canonical compactifications.......................................................... 59

2.13. The Zariski-Riemann-Raynaud point of view.................................. 60

2.14. Other structure sheaves on the adic spectrum: Henselian rigid spaces.................. 66

2.15. Étale morphisms of Noetherian analytic adic spaces.................. 70

2.16. Vector bundles on analytic adic spaces........................................ 73

3. Perfectoid spaces........................................................................ 75

3.1. Perfectoid rings.................................................................................. 75

3.2. Tilting.............................................................................................. 77

3.3. Integral perfectoid rings................................................................. 79

3.4. Untilting.......................................................................................... 80

3.5. Affinoid perfectoid rings................................................................. 84
<table>
  <tbody>
    <tr>
      <td>3.6. The cotangent complex point of view.</td>
      <td>85</td>
    </tr>
    <tr>
      <td>3.7. Identification of the adic spectra as topological spaces.</td>
      <td>86</td>
    </tr>
    <tr>
      <td>3.8. Sheafiness and acyclicity.</td>
      <td>87</td>
    </tr>
    <tr>
      <td>3.9. Perfectoid spaces.</td>
      <td>91</td>
    </tr>
    <tr>
      <td>3.10. Étale morphisms of perfectoid spaces.</td>
      <td>94</td>
    </tr>
    <tr>
      <td>3.11. Purity.</td>
      <td>94</td>
    </tr>
    <tr>
      <td>3.12. A criterion for a morphism to be an isomorphism.</td>
      <td>95</td>
    </tr>
    <tr>
      <td>3.13. André lemma and Zariski closed subsets.</td>
      <td>96</td>
    </tr>
    <tr>
      <td><b>4. The pro-étale topology.</b></td>
      <td>97</td>
    </tr>
    <tr>
      <td>4.1. Pro-étale perfectoid covers before Scholze.</td>
      <td>97</td>
    </tr>
    <tr>
      <td>4.2. Definition and elementary properties.</td>
      <td>100</td>
    </tr>
    <tr>
      <td>4.3. Affinoid pro-étale morphisms.</td>
      <td>101</td>
    </tr>
    <tr>
      <td>4.4. Totally disconnected perfectoid spaces.</td>
      <td>101</td>
    </tr>
    <tr>
      <td>4.5. Strictly totally disconnected perfectoid spaces.</td>
      <td>103</td>
    </tr>
    <tr>
      <td>4.6. Quasi-pro-étale morphisms.</td>
      <td>103</td>
    </tr>
    <tr>
      <td>4.7. The structure pro-étale sheaf.</td>
      <td>106</td>
    </tr>
    <tr>
      <td><b>5. The v-topology.</b></td>
      <td>107</td>
    </tr>
    <tr>
      <td>5.1. Definition and basic properties.</td>
      <td>107</td>
    </tr>
    <tr>
      <td>5.2. The v-sheaf O.</td>
      <td>107</td>
    </tr>
    <tr>
      <td>5.3. Subcanonicity.</td>
      <td>108</td>
    </tr>
    <tr>
      <td>5.4. A descent result.</td>
      <td>108</td>
    </tr>
    <tr>
      <td><b>6. Diamonds.</b></td>
      <td>109</td>
    </tr>
    <tr>
      <td>6.1. Definition and first properties.</td>
      <td>109</td>
    </tr>
    <tr>
      <td>6.2. Spa(Q<sub>p</sub>)°.</td>
      <td>109</td>
    </tr>
    <tr>
      <td>6.3. The diamond associated to a Q<sub>p</sub>-adic space.</td>
      <td>111</td>
    </tr>
    <tr>
      <td>6.4. The main result.</td>
      <td>111</td>
    </tr>
    <tr>
      <td>6.5. Diamonds as v-sheaves.</td>
      <td>112</td>
    </tr>
    <tr>
      <td>6.6. Subdiamonds.</td>
      <td>112</td>
    </tr>
    <tr>
      <td><b>7. Artin's criterion for spatial diamonds.</b></td>
      <td>113</td>
    </tr>
    <tr>
      <td><b>A. The cotangent complex and deformation theory.</b></td>
      <td>115</td>
    </tr>
    <tr>
      <td>A.1. Background on homotopical algebra.</td>
      <td>115</td>
    </tr>
    <tr>
      <td>A.2. Adjoint functors, comonads and the Bar construction.</td>
      <td>121</td>
    </tr>
    <tr>
      <td>A.3. The cotangent complex.</td>
      <td>125</td>
    </tr>
    <tr>
      <td>A.4. Basic properties of the cotangent complex.</td>
      <td>126</td>
    </tr>
    <tr>
      <td>A.5. Infinitesimal extensions of algebras.</td>
      <td>127</td>
    </tr>
    <tr>
      <td>A.6. Deformation theory.</td>
      <td>131</td>
    </tr>
    <tr>
      <td><b>Bibliography.</b></td>
      <td>133</td>
    </tr>
  </tbody>
</table>
# INTRODUCTION

Since its birth in the 60’s following the work of Tate, the field of rigid analytic geometry has undergone various mutations. In this text we present some of the latest one. This is not a course in rigid geometry, the purpose is not to be extensive but rather to focus on the key points of the theory that may be well known to the experts but not from everybody. Another purpose of these notes is to “clean up” the theory and organize it in a coherent way with the optimal hypothesis. There is a part pris: adic spaces, other points of view on rigid geometry being some kind of projection of the adic point of view. To motivate the reader we decided to fix a target: the so-called Artin criterion for spatial diamonds, one of the key tools of the work [28] among many others.

Those notes are derived from a course given by the author at the University of Tokyo in Fall 2022. The author would like to thank Takeshi Saito and Naoki Imai for giving him the opportunity to give those lectures.
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# CHAPTER 1

## SPECTRAL SPACES

### 1.1. Background: rigid geometry, 4 points of view

Historically there are 4 points of view on rigid analytic geometry that appeared since the 60's:

1. **Tate ([53]):** the underlying topological space is not a topological space but a site: the site of admissible open subsets and of admissible open coverings. Some good references for this point of view are [10], [30] and [9].

2. **Raynaud ([47]):** this is the point of view of formal Zariski-Riemann spaces, that is to say of admissible formal schemes up to admissible formal blow-ups. The series of articles [11], [12], [14] and [13] and the book [3] are good references.

3. **Berkovich ([6]):** The underlying topological space is (very often) locally compact (and very often) locally contractible. By definition this is an overconvergent theory compared to the preceding one: overconvergence, that is a property of coefficients i.e. sheaves, is forced at the space level. The first two main references are [6] and [7]. This can be completed by [24].

4. **Huber ([38]):** The underlying topological space is a locally spectral space in the sense of Hochster. The two articles [37] and [38] are the starting points. This can be completed by [39].

The link between those theories is summarized in the following diagram

<figure><img src="image_7.png" /></figure>
Here for $X$ a quasi-compact quasiseparated rigid analytic space à la Tate over a non-Archimedean field $K$:

* Raynaud chooses an integral model $\mathcal{X}$ (a $\pi_K$-adic formal scheme over $\mathcal{O}_K$ topologically of finite type) such that $X = \mathcal{X}_\eta$ (generic fiber) and the admissible topos of $X$ is the projective limit of the topos of all admissible formal blow up $\tilde{\mathcal{X}}$ of $\mathcal{X}$.

* The admissible topos $\tilde{\mathcal{X}}$ is spatial and the associated sober space is Huber's spectral space. Points are given by morphisms of topos $\mathrm{Ens} \to \tilde{\mathcal{X}}$ i.e. the points of the admissible topos.

* Huber's spectral space is the projective limit of the spectral spaces associated to all admissible formal blow up.

* The Berkovich space is obtained by taking the quotient of Huber's spectral space given by identifying two point $x$ and $y$ is $x$ is a specialization of $y$ or $y$ is a specialization of $x$ i.e. we identify any point $x$ with its maximal generalization given by a rank 1 valuation (that is to say with values in $\mathbb{R}$).

Huber's adic spaces are the most general objects we can consider, the theory is developed over any base (i.e. without a fixed base). For example:

* The analytic adic spectrum of $\mathbb{Z}_p[[x_1, \dots, x_n]]$ is a quasi-compact adic space that contains as an open subset the usual rigid analytic open ball

$$ \mathbb{B}_{\mathbb{Q}_p}^n $$

and whose complementary $\partial \mathbb{B}_{\mathbb{Q}_p}^n$ is a union of $n$ ($n-1$)-dimensional closed balls over $\mathbb{F}_p((T_1)), \dots, \mathbb{F}_p((T_n))$.

* Another example is the case of tubular neighborhoods. Let $X$ be any Noetherian scheme and $Y \subset X$ be Zariski closed. The formal completion

$$ \mathcal{X} = \tilde{X}_Y $$

is a formal scheme that is a weak approximation to what is a tubular neighborhood of $Y$ in $X$. One can associate to this formal scheme a much more subtle notion of tubular neighborhood: the adic space $\mathcal{X}^{ad}$ that contains the special fiber $Y \hookrightarrow \mathcal{X}^{ad}$ seen as a “discrete” adic space associated to a scheme (something not very different from a scheme) and the generic fiber, the punctured tubular neighborhood $\mathcal{X}^{ad} \setminus Y$ that is an analytic (the opposite of discrete) adic space,

$$ \underbrace{Y}_{\text{discrete}} \longrightarrow \mathcal{X}^{ad} \longleftarrow \underbrace{\mathcal{X}^{ad} \setminus Y}_{\text{analytic}} $$

If $K$ is a non-archimedean field, there is a functor $X \mapsto X^{ad}$ that is an equivalence

$$ \{\text{quasi-separated } K\text{-rigid spaces}\} \xrightarrow{\sim} \{\text{quasi-separated } K\text{-adic spaces locally of finite type}\}. $$

From this point of view the theory does not seem more powerful than Tate's theory when we work in a topologically of finite type framework over a non-archimedean field. Nevertheless, if $f: X \to Y$ is a separated quasi-compact morphism of quasi-separated rigid spaces Huber constructs a canonical "compactification"

$$ \begin{array}{ccc} X^{ad} & \xleftarrow{\substack{j \\ \text{open}}} & X_{Y^{ad}}^{ad} \\ \downarrow^{f^{ad}} & & \downarrow^{f^{ad,c}} \\ Y^{ad} & & \end{array} $$

that does not exist in Tate's world. The points of the boundary of this compactification $X_{Y^{ad}}^{ad} \setminus X^{ad}$ are higher rank valuations that do not show up in Tate or Berkovich theory. For example, the compactification of the closed ball $\mathbb{B}_K^1 \to \mathrm{Spa}(K)$ is obtained by adding a rank 2 valuation in the boundary that is a specialization of the Gauss norm. This construction is
essential in [39] to define $Rf_1$ as $R(f^{ad,c})_1 \circ j_1$ in étale cohomology and is unavailable in Berkovich's theory ([7]) where the compactly supported étale cohomology of an affinoid space is not defined.

Sometimes we will want to take an integral model of an adic space or consider it's Berkovich's spectrum as a locally compact topological space but at the end we will focus on adic spaces.

## 1.2. Spectral spaces: two equivalent definitions

To any complex analytic space one can associate a locally compact topological space after forgetting the complex analytic structure i.e. the structure sheaf of holomorphic functions. For smooth complex analytic spaces the associated topological space is a topological manifold.

For adic spaces, the associated topological space, after forgetting the structure sheaf of holomorphic functions, is a locally spectral space ([35], [51, Section 08YF]). Let us recall the following definition.

**Definition 1.2.1 (Hochster [35]).** — A topological space $X$ is spectral if

1. It is quasi-compact quasiseparated,

2. It is sober,

3. It has a base of quasi-compact open subsets.

Recall that in this definition:

* $X$ quasiseparated means that the intersection of two quasi-compact open subsets is quasi-compact,

* sober means that any irreducible closed subset has a unique generic point, that is to say
  — any irreducible closed subset has a generic point,
  — $X$ is $(T0)$.

The existence of a generic point for $Z$ closed irreducible means that
$$ \begin{array}{c} U \subset X \text{ open} \\ U \cap Z \neq \emptyset \end{array} \qquad U \cap Z \neq \emptyset. $$

* The fact that $X$ is $(T0)$ is equivalent to saying that the specialization relation $\leq$ is an order:
  $x \leq y$ and $y \leq x$ implies $x = y$.

**Remark 1.2.2.** — To any (Grothendieck) topos $\mathcal{X}$ one can associate a sober space $X$ whose points are exactly the set of points of $\mathcal{X}$, the morphisms of topoi

$$ \mathrm{Ens} \to \mathcal{X}, $$

and whose open subsets are induced by the subobjects of the final object of $\mathcal{X}$ i.e. if $U$ is a subobject of the final object of $\mathcal{X}$ we look at the points factorizing through $\mathcal{X}/U$ via $\mathcal{X}/U \to \mathcal{X}$. There is a natural morphism of topoi

$$ \tilde{X} \to \mathcal{X}. $$

When this is an equivalence we say that $\mathcal{X}$ is spatial. All of this only depends on the locale of subobjects of the final object of $\mathcal{X}$ associated to $\mathcal{X}$ (see [41]). One of the starting points of Huber's work is the discovery that the admissible topos of a qcqs rigid space à la Tate is in fact spatial ([37, Proposition 4.5]).

**Remark 1.2.3.** — Any Noetherian $(T0)$ topological space, for example the topological space associated to a Noetherian scheme, is spectral. Nevertheless, this is not the type of spaces that we will ultimately focus on in this text.
**Remark 1.2.4.** — The quasi-separatedness hypothesis is very important. In fact it is “difficult” to fall naturally on quasicompact non-quasiseparated schemes, the first basic example being two copies of $\text{Spec}(k[x_i]_{i \in \mathbb{N}})$ glued along $\text{Spec}(k[x_i]_{i \in \mathbb{N}}) \setminus V(x_i)_{i \in \mathbb{N}}$ (it is “easier” to fall on quasicompact non-quasiseparated algebraic spaces like $\mathbb{G}_{a,k}/\mathbb{Z}$ where $k$ is a characteristic zero field). But for adic spaces this can happen naturally. This is for example the case if you glue two closed balls $\mathbb{B}_K^1$ along $\mathbb{B}_K^1 \setminus \{0\}$, $K$ a non-archimedean field.

Profinite topological spaces are a particular case of spectral spaces: those are exactly the Hausdorff spectral spaces. They have a purely topological definition: those are the totally disconnected compact topological cases. The definition as a projective limit of finite sets is a combinatorial definition.

The same goes on more generally for spectral spaces.

<table><thead><tr><th>Topological definition</th><th>Combinatorial definition</th></tr></thead><tbody><tr><td>Totally disconnected compact topological spaces</td><td>Pro(finite sets)</td></tr><tr><td>Spectral topological spaces + qc maps</td><td>Pro(finite ordered sets)</td></tr></tbody></table>

Here we use the equivalence between finite ordered sets and finite (T0) topological spaces, the order relation corresponding to the specialization relation on the topological space. The finite (T0) spaces are exactly the finite spectral spaces. For example, the ordered set $\{s, \eta\}$ with $s \le \eta$ and $s \ne \eta$ corresponds to the spectral space that is the spectrum of a rank 1 valuation ring. More generally, the ordered set with $n$ elements $\{1 \le 2 \le \dots \le n\}$ corresponds to the spectrum of a rank $n$ valuation ring.

When we say that the definitions are equivalent this means that the projective limit functor induces an equivalence of categories between the right hand side of the preceding table and the left hand one: we are speaking of the pro-category of finite sets, resp. finite ordered sets. On the left hand side we have to take quasi-compact morphisms of spaces as morphisms to obtain an equivalence of categories.

**Theorem 1.2.5 ([35, Proposition 10]).** — There is an equivalence of categories

$\lim: \text{Pro(finite ordered sets)} \xrightarrow{\sim} \{\text{spectral topological spaces + qc continuous maps}\}$.

We will give a full proof of this theorem in Section 1.5 and write down an explicit inverse of the functor $\lim$, see Theorem 1.2.5.

**Example 1.2.6.** — Set $E_n = \{1, \dots, n\}^{\text{disc}} \cup \eta$ where $\{1, \dots, n\}^{\text{disc}}$ is the discrete ordered set (if $x \ne y$ then $x \not\le y$ and $x \not\ge y$) with $n$ elements and $\eta \ge 1, \eta \ge 2, \dots, \eta \ge n$. For $n \le m$ we define a morphism $E_m \to E_n$ by sending $\eta$ to $\eta$, $i \in \{1, \dots, m\}$ to $i$ if $i \le n$ and $i \mapsto \eta$ if $i > n$. This forms a projective system $E_1 \leftarrow E_2 \leftarrow \dots \leftarrow E_n \leftarrow \dots$.

Order the prime numbers $p_1, p_2, \dots$. Then $\text{Spec}(\mathbb{Z})$ corresponds to $\lim_{\leftarrow n \ge 1} E_n$ by sending $p_i$ to $i \in E_n$ if $i \le n$, $p_i \mapsto \eta \in E_n$ if $i > n$ and $\eta \mapsto \eta$.
<figure><img src="image_8.png" /></figure>

## 1.3. The constructible topology

**1.3.1. Definition.** — Recall the following definition.

**Definition 1.3.1.** — A constructible set in a spectral space $X$ is an element of the Boolean algebra generated by the quasi-compact open subsets.

The quasi-compact assumption is fundamental in the definition of a constructible set. One may be used to work with spectral spaces associated to Noetherian schemes where all open subsets are quasi-compact but when working with adic spaces the most basic objects like the closed ball contains many natural non-quasicompact open subsets like open balls. Moreover, when working with adic spaces, one has to forget at some point the intuition of the Zariski constructible subsets of Noetherian schemes. Typically if $Z$ is a constructible subset of the closed ball $\mathbb{B}_K^1$ as an adic space over the non-archimedean field $K$, if $0 \in Z$ then $\mathbb{B}_K^1(0, \varepsilon) \subset Z$ for some $\varepsilon > 0$, see Example 1.3.12.

According to the following lemma, the constructible subsets are exactly the finite disjoint unions of subsets of the form $U \cap F$ where $U$ is quasi-compact open and $F$ closed with $X \setminus F$ quasi-compact.

**Lemma 1.3.2.** — For a set $E$ and $F \subset P(E)$ containing $E$ and $\emptyset$, the Boolean algebra generated by $F$, that is to say the smallest subset of $P(E)$ containing $F$ that is stable under complementary and finite intersections, is the set of disjoint unions of sets of the form $S \setminus T$ with $S$ a finite intersection of elements of $F$ and $T$ a finite union of elements of $F$.

*Proof.* — The Boolean algebra generated by $F$ is the union of the Boolean algebras generated by the finite subsets of $F$. We can thus suppose that $F$ is finite. The Boolean algebra generated by $F$ is seen as a sub-$\mathbb{F}_2$-algebra of $\mathbb{F}_2^E$. More precisely, this is $A = \mathbb{F}_2[\mathbf{1}_S]_{S \in F}$ where $\mathbf{1}_S$ is the characteristic function of $S$. Now, since for any $a \in A$, $a^2 = a$, and $A$ is of finite type over $\mathbb{F}_2$, $A$ is an Artinian reduced ring with residue fields isomorphic to $\mathbb{F}_2$ at maximal ideals. We thus have, if $\hat{A} = \text{Hom}(A, \mathbb{F}_2)$,

$$A \xrightarrow{\sim} \mathbb{F}_2^{\hat{A}} \\ a \mapsto (\chi(a))_{\chi \in \hat{A}}$$

(this is a particular case of the so-called Stone representation theorem). Now, any set in the Boolean algebra generated by $F$ corresponding to a function $f: \hat{A} \to \mathbb{F}_2$ is written as

$$\coprod_{\substack{\chi \in \hat{A} \\ f(\chi)=1}} \bigcap_{\substack{S \in F \\ \chi(\mathbf{1}_S)=1}} S \setminus \bigcup_{\substack{S \in F \\ \chi(\mathbf{1}_S)=0}} S.$$

In fact, this union is clearly disjoint and the element

$$a = \sum_{\substack{\chi \in \hat{A} \\ f(\chi)=1}} \prod_{\substack{S \in F \\ \chi(\mathbf{1}_S)=1}} \mathbf{1}_S \cdot \prod_{\substack{S \in F \\ \chi(\mathbf{1}_S)=0}} (1+\mathbf{1}_S) \in A$$
is such that for all $\chi \in \hat{A}$, $\chi(a) = f(\chi)$. $\square$

<figure><img src="image_9.png" /></figure>

Let us note that in fact the locally closed constructible subsets of $X$ are exactly the subsets of the form $U \setminus V$ where $U$ and $V$ are quasi-compact open subsets of $X$, see Lemma 1.3.7.

**Definition 1.3.3.** — Let $X$ be a spectral space. The topological space

$X_{\text{cons}}$

is the set $X$ equipped with the topology that has as a base the set of constructible sets. Its closed subsets are the *pro-constructible* subsets and its open subsets are the *ind-constructible* subsets.

For example:

1. Any closed subsets is pro-constructible.

2. For any $x \in X$, the localization

$$ \begin{aligned} X_x &= \{y \in X \mid y \ge x\} \\ &= \bigcap_{\substack{U \ni x \\ \text{qc open}}} U \end{aligned} $$

is pro-constructible inside $X$.

3. For any $x \in X$, $\{x\}$ is pro-constructible in $X$ since

$$ \{x\} = X_x \cap \overline{\{x\}}. $$

4. If $X$ is a finite (T0) space then $X_{\text{cons}} = X_{\text{disc}}$.

The evident bijective map

$$ X_{\text{cons}} \to X $$

is continuous quasi-compact. Let us moreover remark that the correspondence $X \mapsto X_{\text{cons}}$ is functorial with respect to quasi-compact maps of spectral spaces. We will see in Proposition 1.3.8 that it has a definition as an adjoint functor.

**Remark 1.3.4.** — The geometry of constructible and pro-constructible subsets for quasi-compact quasi-separated schemes goes back to Grothendieck, see [34, Section 1.9]. For example, the compacity of the constructible topology for qc qs schemes is [34, Proposition 1.9.15]. Hochster is the one who realized that this has nothing to do with schemes and brought out the notion of spectral spaces.

**1.3.2. Compacity of the constructible topology.** — The first result that is used all the time is the following.

**Theorem 1.3.5 ([35, Theorem 1]).** — The topological space $X_{\text{cons}}$ is compact totally disconnected.

*Proof.* — If $x, y \in X$ are distinct, up to permuting $x$ and $y$, there exists a quasi-compact open subset $U$ containing $x$ and not $y$. Then $U$ and $X \setminus U$ are disjoint constructible subsets with $x \in U$ and $y \in X \setminus U$. We deduce that $X_{\text{cons}}$ is Hausdorff.

Since a base of the topology of $X_{\text{cons}}$ is made of open and closed subsets, $X_{\text{cons}}$ is moreover totally discontinuous.

It thus remains to prove that $X_{\text{cons}}$ is quasi-compact. For this we use the following version of Alexander's subbase quasi-compacity criterion. Recall that for any set $S$ a subset $\mathcal{F} \subset \mathcal{P}(S)$ is said to have the finite intersection property if for any $A \subset \mathcal{F}$ finite non-empty one has $\bigcap_{E \in A} E \neq \emptyset$.
Let $T$ be a topological space that has a subbase $\mathcal{B}$ satisfying: for any $\mathcal{F} \subset \mathcal{B}$ maximal among the subsets of $\mathcal{B}$ having the finite intersection property, one has $\bigcap_{U \in \mathcal{F}} U \neq \emptyset$. Then $T$ is quasicompact.

We apply this to $X_{\text{cons}}$ with $\mathcal{B}$ the set of open quasi-compact subsets of $X$ and their complements. Let $\mathcal{F}$ be such a maximal subset. Let us note

$$F = \bigcap_{\substack{Z \in \mathcal{F} \\ \text{closed}}} Z.$$

The quasi-compacity of $X$ implies that the closed set $F$ is non-empty. Let us prove it is irreducible. Suppose

$$F = F_1 \cup F_2$$

with $F_1$ and $F_2$ closed strictly contained in $F$. The open subset $X \setminus F_1$ satisfies $(X \setminus F_1) \cap F_1 = \emptyset$ and $(X \setminus F_1) \cap F_2 \neq \emptyset$. The same type of properties holds for $X \setminus F_2$. From this we deduce that, up to shrinking $X \setminus F_1$ and $X \setminus F_2$, there exists $U_1$ and $U_2$ quasi-compact open subsets satisfying

$$U_1 \cap F_1 = \emptyset, U_1 \cap F_2 \neq \emptyset, U_2 \cap F_2 = \emptyset, U_2 \cap F_1 \neq \emptyset.$$

Since $F_2 \subsetneq X \setminus U_1$ and $F_1 \subsetneq X \setminus U_2$, one has

$$(1) \qquad X \setminus U_1 \notin \mathcal{F}, X \setminus U_2 \notin \mathcal{F}.$$

Moreover if $Z_1, \dots, Z_n \in \mathcal{F}$ and $Z'_1, \dots, Z'_{n'} \in \mathcal{F}$ satisfy

$$Z_1 \cap \dots \cap Z_n \cap X \setminus U_1 = \emptyset \text{ and } Z'_1 \cap \dots \cap Z'_{n'} \cap X \setminus U_2 \neq \emptyset$$

then if $W = Z_1 \cap \dots \cap Z_n \cap Z'_1 \cap \dots Z'_{n'}$ one has

$$W \cap X \setminus U_1 = \emptyset \text{ and } W \cap X \setminus U_2 = \emptyset$$

and thus since $F \subset X \setminus U_1 \cup X \setminus U_2$, $W \cap F = \emptyset$. Now, one has

$$W \cap F = \bigcap_{\substack{Z \in \mathcal{F} \\ \text{closed}}} (W \cap Z)$$

and thus since $W$ is quasicompact as a closed subset of a quasicompact open subset of $X$, there exists $Z''_1, \dots, Z''_{n''} \in \mathcal{F}$ satisfying

$$Z_1 \cap \dots \cap Z_n \cap Z'_1 \cap \dots Z'_{n'} \cap Z''_1 \cap \dots \cap Z''_{n''} = \emptyset.$$

This is impossible since $\mathcal{F}$ has the finite intersection property. From this we deduce that $\mathcal{F} \cup \{X \setminus U_1\}$ or $\mathcal{F} \cup \{X \setminus U_2\}$ has the finite intersection property. The equation (1) then contradicts the maximality of $\mathcal{F}$. We thus deduce that $F$ is irreducible.

Let now $\xi \in F$ be its generic point. For $U \in \mathcal{F}$ open, $U \cap F \neq \emptyset$ since $\{Z \cap U | Z \in \mathcal{F} \text{ closed}\}$ has the finite intersection property and $U$ is quasicompact. One then has

$$\xi \in \bigcap_{\substack{U \in \mathcal{F} \\ \text{open}}} U \cap F = \bigcap_{Z \in \mathcal{F}} Z$$

and this is thus non-empty. $\square$

**Example 1.3.6.** — The spectral space $\text{Spec}(\mathbb{Z})_{\text{cons}}$ is identified with the Alexandroff compactification $\mathbb{N} \cup \{\infty\}$ of the discrete space $\mathbb{N}$. The morphism $\mathbb{N} \cup \{\infty\} \to \text{Spec}(\mathbb{Z})$ sends $n \in \mathbb{N}$ to the $(n+1)$th prime number and $\infty$ to the generic point $\eta$.
For $X$ spectral, the open/closed subsets of $X_{cons}$ are the constructible subsets of $X$. Thus, as a profinite set one has simply

$$X_{cons} = \lim_{\substack{X=\bigcup_{i \in I} Z_i \\ \text{finite constructible partition}}} I$$

where the indexing category is the one of finite sets $I$ together with a surjective constructible function $X \to I$ and the morphisms are commuting diagrams

<figure><img src="image_10.png" /></figure>

Let us give a simple very first application of the preceding result.

**Lemma 1.3.7.** — The locally closed constructible subsets of $X$ are the $U \cap F$ where $U$ is a quasi-compact open subset and $F$ is closed with quasi-compact complementary.

*Proof.* — Let $Z$ be a locally closed constructible subset of $X$. Each point $z$ of $Z$ has a quasi-compact open neighborhood $U_z$ such that $U_z \cap Z$ is closed in $U_z$. Since $Z$ is compact in $X_{cons}$, it is quasi-compact and we can thus find a quasi-compact open subset $U$ of $X$ containing $Z$ such that $Z$ is closed in $U$. Since $Z$ is constructible in $X$, it is constructible in $U$ and thus $V = U \setminus Z$ is quasi-compact since it is quasi-compact in $U_{cons}$. We then have $Z = U \setminus V$ and the result is proved. □

Finally, let us note the following categorical characterization of the constructible topology whose proof is elementary.

**Proposition 1.3.8.** — The functor

$$(-)_{cons} : \{\text{spectral spaces + qc maps}\} \to \{\text{profinite spaces}\}$$

is a right adjoint to the inclusion of the category of profinite spaces in the category of spectral spaces equipped with the quasi-compact maps. The natural map $X_{cons} \to X$ is one of the adjunction maps, the other being the identification between $X_{cons}$ and $X$ for $X$ profinite.

**1.3.3. First applications of the compacity of the constructible topology.** — Theorem 1.3.5 has plenty of applications that we use frequently. They are the basic tools we use when we work with spectral spaces. Let's cite some of them.

**Corollary 1.3.9.** — 1. Let $f: X \to Y$ be a qc (continuous) map of spectral spaces. For any $Z \subset X$ pro-constructible, $f(Z)$ is pro-constructible. In particular, the image of $f$, $f(X)$, is pro-constructible.

2. For $Z$ pro-constructible in $X$,

$$\bar{Z} = \bigcup_{z \in Z} \overline{\{z\}}$$

that is to say the specializations of elements of $Z$. In particular

$Z$ closed $\iff$ $Z$ stable under specialization.

In particular for $f$ as in point (1), its image is closed iff it is stable under specialization.

3. If $Z$ is pro-constructible inside $X$ then $Z$ (equipped with the induced topology) is spectral.

4. Any non-empty spectral space has a closed point.

5. Any non-empty spectral space has a maximal point.
*Proof.* — Point (1) is evident. Let us note by the way that it gives a canonical factorization

$$X \xrightarrow{f} f(X) \hookrightarrow Y$$

in the category of spectral spaces with $qc$ maps as morphisms.

Let us verify point (2). If $x \in X$ is not a specialization of an element of $Z$ then $X_x \cap Z = \emptyset$. This means

$$\bigcap_{\substack{U \ni x \\ \mathrm{qc\ open}}} (U \cap Z) = \emptyset.$$

But for such a $U$, $U \cap Z$ is closed in the compact topological space $Z_{cons}$. We deduce that a finite sub-intersection is empty and thus there exists such a $U$ as before with $U \cap Z = \emptyset$. Thus $x \notin \bar{Z}$.

Point (3) is easily deduced from the compacity of $Z$ in $X_{cons}$.

For point (4) we can use Zorn lemma. More precisely, if $(x_i)_{i \in I}$ is a chain of elements of $X$ in the sense that for $i, j \in I$, $x_i \le x_j$ or $x_j \le x_i$, then

$$\bigcap_{i \in I} \overline{\{x_i\}} \neq \emptyset.$$

In fact, this is an intersection of closed subsets of the compact topological space $X_{cons}$ and thus if this intersection where empty a finite sub-intersection would be empty. This is impossible because of the chain condition. Such a chain is thus bounded below by an element of $X$.

For point (5) we gain use Zorn lemma. Let $(x_i)_{i \in I}$ be a chain of elements of $X$. If

$$\bigcap_{i \in I} X_{x_i} = \emptyset$$

then a finite sub-intersection is empty since for all $i$, $X_{x_i}$ is a closed subset of $X_{cons}$. This is impossible because of the chain condition and thus $(x_i)_{i \in I}$ is bounded above. $\square$

**Remark 1.3.10.** — Points (1) and (2) give rise to the valuation criterion of properness. Point (4) is the analog of the fact that any non-zero ring has a maximal ideal. Point (5) is the analog of the fact that any non-zero ring has a minimal prime ideal.

**Example 1.3.11.** — For any $x \in X$ a spectral space, the localization of $X$ at $x$, $X_x$, is pro-constructible and thus spectral with a unique closed point $x$.

**Example 1.3.12.** — 1. Let $X$ be a spectral space, $Z \subset X$ constructible and $x \in Z$. If $X_x \subset Z$ then there exists a neighborhood $U$ of $x$ contained in $Z$. In fact, $\bigcap_{U \ni x} U \cap (X \setminus Z) = \emptyset$ and using the compactness of $(X \setminus Z)_{cons}$ one concludes.

2. For example, any maximal point lying in $Z$ constructible inside $X$ spectral has a neighborhood contained in $Z$.

3. For example, let $X$ be a quasi-separated adic space of finite type over $Spa(K)$ with $K$ a non-archimedean field, i.e. the adic space associated to a quasi-compact quasi-separated Tate rigid space over $K$. Let $Z \subset |X|$ be constructible. For any classical point $x \in X$, $x \in Z$ implies a neighborhood of $x$ is contained in $Z$. Typically, if $Z \subset |\mathbb{B}_K^d|$ is constructible, then for any $x \in \mathbb{B}^d(K)$, $x \in Z \Rightarrow \exists \varepsilon > 0$, $\mathbb{B}^d(x, \varepsilon) \subset Z$.

## 1.4. Projective limits of spectral spaces

When dealing with spectral spaces projective limits of spectral spaces show up very often, already in Theorem 1.2.5. But there is more to it when working with perfectoid spaces and diamonds. In fact, one of the basic notions we will use is the so-called pro-étale topology on perfectoid spaces. This involves projective limits of perfectoid spaces as the name suggests.
**1.4.1. Spectrality of the projective limit.** — We will use the following elementary result all the time.

**Proposition 1.4.1.** — If $(X_i)_{i \in I}$ is a cofiltered projective system of spectral spaces with quasi-compact transition maps then the topological space $\varinjlim_{i \in I} X_i$ is spectral.

*Proof.* — From Tychonoff theorem we deduce that

$$ \prod_{i \in I} X_i $$

is quasi-compact quasi-separated and has a base of quasi-compact open subsets. It is moreover clearly (T0). Let now $Z \subset \prod_{i \in I} X_i$ be an irreducible closed subset. We have to prove that

$$ \bigcap_{\substack{U \text{ qc open} \\ U \cap Z \neq \emptyset}} U \cap Z $$

is non-empty. But such a set $U$, $U \cap Z$ is closed in the compact topological space $(\prod_{i \in I} X_i)_{\text{cons}}$. Since $Z$ is irreducible, the family $\{U \cap Z \mid U \text{ qc open}, U \cap Z \neq \emptyset\}$ has the finite intersection property. We conclude that $Z$ has a generic point. Thus, we have proven that $\prod_{i \in I} X_i$ is spectral.

For $i, j \in I$ with $i \geq j$ let us note $p_{ij}: X_i \to X_j$ the transition map. One has

$$ \varprojlim_i X_i = \bigcap_{i \geq j} (p_{ij} \times \text{Id})^{-1} \Delta_{X_j} $$

where $\Delta_{X_j} \subset X_j \times X_j$ is the diagonal. Since $(p_{ij} \times \text{Id})$ is quasi-compact, Lemma 1.4.2 implies that $\varinjlim_i X_i$ is pro-constructible in $\prod_{i \in I} X_i$. Applying point (3) of Corollary 1.3.9 we deduce that $\varinjlim_i X_i$ is spectral. $\square$

Since a spectral space $X$ is quasi-separated, $\Delta: X \to X \times X$ is a quasi-compact map of spectral spaces and thus its image is pro-constructible according to point (1) of Corollary 1.3.9. Nevertheless, we can verify this directly as we do in the following lemma.

**Lemma 1.4.2.** — For $X$ a spectral space, the diagonal $\Delta_X \subset X \times X$ is pro-constructible.

*Proof.* — One has since $X$ is (T0),

$$ X \times X \setminus \Delta_X = \bigcup_{U \text{ open qc}} [(U \times X \setminus U) \cup (X \setminus U \times U)] $$

and this is thus ind-constructible. $\square$

**1.4.2. Some properties of the projective limit.** — Let us start by giving a simple immediate application of the adjunction property of Proposition 1.3.8.

**Proposition 1.4.3.** — For $(X_i)_{i \in I}$ a cofiltered projective system of spectral spaces with quasi-compact transition maps,

$$ \varinjlim_{i \in I} X_{i, \text{cons}} \xrightarrow{\sim} (\varinjlim_{i \in I} X_i)_{\text{cons}}. $$

The following result is elementary and will be used all the time.

**Proposition 1.4.4.** — Let $X = \varinjlim_{i \in I} X_i$ be a qc cofiltered limit of spectral spaces. One has

$$ \varinjlim_{i \in I} \{\text{qc open subsets of } X_i\} \xrightarrow{\sim} \{\text{qc open subsets of } X\} $$

$$ \varinjlim_{i \in I} \{\text{constructible subsets of } X_i\} \xrightarrow{\sim} \{\text{constructible subsets of } X\}. $$
*Proof.* — The surjectivity of the map is immediate from the quasi-compacity of X, resp. the compacity of X<sub>cons</sub> and the fact that the open/closed subsets of the constructible topology are the construcible subsets of the original topology + Proposition 1.4.3.

Let us now prove the injectivity. Let us note $p_i : X \to X_i$ and for $j \ge i$, $p_{ji} : X_j \to X_i$. Let $U, V \subset X$ be constructible subsets such that

$$p_i^{-1}(U) = p_j^{-1}(V).$$

Let us set for $j \ge i$, $Z_j = p_{ji}(X_j)$. The sequence $(Z_j)_{j \ge i}$ is a decreasing sequence of pro-constructible subsets of $X_i$. The formula $p_i^{-1}(U) = p_j^{-1}(V)$ is equivalent to

$$\bigcap_{j \ge i} Z_j \cap U = \bigcap_{j \ge i} Z_j \cap V.$$

We thus have

$$\cap_{j \ge i} Z_j \cap (U \setminus V) = \emptyset, \quad \cap_{j \ge i} Z_j \cap (V \setminus U) = \emptyset.$$

Using the compacity of the constructible topology we deduce that some finite sub-intersections of those two intersections are empty and thus, since $(I, \le)$ is cofiltered, there exists $j \ge i$ such that

$$(U \setminus V) \cap Z_j = \emptyset, \quad (V \setminus U) \cap Z_j = \emptyset.$$

From this we deduce that

$$p_{ji}^{-1}(U) = p_{ji}^{-1}(V).$$

□

**Example 1.4.5.** — Let $\text{Spec}(A) = \lim_{\leftarrow i \in I} \text{Spec}(A_i)$ be a cofiltered limit of affine schemes. Any quasi-compact open subset $U$ of $\text{Spec}(A)$ is a finite union of principal open subsets, $U = \cup_{\alpha \in S} D(f_\alpha)$ where the set of indices $S$ is finite. Now for each $\alpha \in S$ we can chose $i_\alpha$ such that $f_\alpha$ is the image of some $g_\alpha \in A_{i_\alpha}$. We can then chose, since $S$ is finite and $I$ cofiltered, some $j \in I$ such that for all $\alpha \in S$, $j \ge i_\alpha$. Let $h_\alpha \in A_j$ be the image of $g_\alpha$ in $A_j$. One then has: $U$ is the reciprocal image of the quasi-compact open subset $\cup_{\alpha \in S} D(h_\alpha) \subset \text{Spec}(A_j)$.

## 1.5. The equivalence between spectral spaces and pro-finite (T0) spaces

We can now state and prove the full form of Theorem 1.2.5.

**Theorem 1.5.1.** — The projective limit functor induces an equivalence of categories

$$\lim_{\leftarrow} : \text{Pro}(\text{finite ordered sets}) \xrightarrow{\sim} \{\text{spectral topological spaces} + \text{qc continuous maps}\}.$$

An inverse sends a spectral space $X$ to the pro-finite (T0) space

$$\lim_{\leftarrow A \in \mathcal{I}_X} \text{Im}(X \to \{s, \eta\}^A)$$

where

* $\mathcal{I}_X$ is the ordered set of finite subsets of the set of quasi-compact maps $X \to \{s, \eta\}$ i.e. the ordered set of finite subsets of the set of quasicompact open subsets of $X$,

* the map $X \to \{s, \eta\}^A$ sends $x$ to $(f(x))_{f \in A}$.
*Proof.* — Proposition 1.4.1 implies that the functor $\lim_{\leftarrow}$ takes values in the category of spectral spaces. Let now $X$ be a spectral space. There is a diagram

<figure><img src="image_11.png" /></figure>

According to Proposition 1.4.3 $h$ is an homeomorphism. The set $\mathcal{I}_X$ is identified with a subset of the set of finite subsets of the set of continuous quasi-compact maps $X_{\text{cons}} \to \{0,1\}$ that separates the point of $X_{\text{cons}}$. According to Theorem 1.3.5 $X_{\text{cons}}$ is profinite and thus $h \circ g$ and thus $g$ are homeomorphisms (see Lemma 1.5.2). We deduce that the quasi-compact continuous map $f$ is bijective. Now, if $U$ is a quasi-compact open subset of $X$, and $\chi_U : X \to \{s, \eta\}$ is defined by $\chi_U^{-1}(\eta) = U$, then $f(U)$ is the reciprocal image of $\eta \in \text{Im}(X \to \{s, \eta\}^{\chi_U})$ in the factor $A = \{\chi_U\}$. It is thus open and we deduce that $f$ is a homeomorphism.

The full faithfulness of the functor $\lim_{\leftarrow}$ is easy and left to the reader. $\square$

**Lemma 1.5.2.** — Let $P$ be a profinite set and $\mathcal{I}$ a subset of the set of finite subsets of the set of continuous quasi-compact maps from $P$ to $\{0,1\}$. Suppose $A, B \in \mathcal{I}$ implies $A \cup B \in \mathcal{I}$ and for $x, y \in P$ there exists $A \in \mathcal{I}$ and $f \in A$ such that $f(x) \neq f(y)$. Then,

$$P \xrightarrow{\sim} \lim_{\leftarrow A \in \mathcal{I}} \text{Im}(P \to \{0,1\}^A)$$

is an homeomorphism.

*Proof.* — The proof is left to the reader. $\square$

**Remark 1.5.3.** — Let $X = \lim_{\leftarrow i} X_i$ with $X_i$ a finite (T0) space. As a particular case of Propositions 1.4.3 and 1.4.4 we have the following dictionary between the topological and combinatorial description of spectral spaces:

1. There is an identification

$$X_{\text{cons}} = \lim_{\leftarrow i} X_{i, \text{disc}}.$$

2. There is a bijection

$$\lim_{\leftarrow i \in \mathcal{I}} \{\text{open subsets of } X_i\} \xrightarrow{\sim} \{\text{qc open subsets of } X\}.$$

3. There is a bijection

$$\lim_{\leftarrow i \in \mathcal{I}} \{\text{subsets of } X_i\} \xrightarrow{\sim} \{\text{constructible subsets of } X\}.$$

## 1.6. Connected components

Let $X$ be a spectral space and write $X = \lim_{\leftarrow i \in I} X_i$ with the $X_i$ finite (T0) spaces and the limit is cofiltered.

The basic results about connected components of spectral spaces are the following (see for example [40, Lemma 2.4.1]).

**Proposition 1.6.1.** — 1. Every connected component of $X$ is the intersection of the open/closed subsets containing it.
2. The set $\pi_0(X)$ equipped with the quotient topology is profinite and $\pi_0(X) \xrightarrow{\sim} \lim_{\leftarrow i \in I} \pi_0(X_i)$ as a profinite set.

3. The surjection

$$X \to \pi_0(X)$$

identifies the functor $X \mapsto \pi_0(X)$ as a left adjoint to the inclusion

$$\{\text{Hausdorff spectral spaces}\} = \{\text{profinite spaces}\} \hookrightarrow \{\text{spectral spaces}\}.$$

*Proof.* — The result is easily deduced from Theorem 1.5.1 once one has verified that if $(X_i)_{i \in I}$ is a cofiltered projective system of connected finite (T0) spaces then $X = \lim_{\leftarrow i \in I} X_i$ is connected. But open/closed subsets of $X$ are quasi-compact and thus $\lim_{\leftarrow i \in I} \{\text{open/closed subsets of } X_i\} \xrightarrow{\sim} \{\text{open/closed subsets of } X\}$. From this we deduce that $X$ is connected. $\square$

**Example 1.6.2.** — If $R$ is a ring, writing $R = \lim_{i \in I} R_i$, a filtered colimit of Noetherian rings, one has

$$\text{Spec}(R) = \lim_{i \in I} \text{Spec}(R_i)$$

and thus

$$\pi_0(\text{Spec}(R)) = \lim_{i \in I} \underbrace{\pi_0(\text{Spec}(R_i))}_{\text{finite set}}$$

as a profinite set. The Boolean algebra of open/closed subset of the profinite set $\pi_0(\text{Spec}(R))$ is then identified with the one of idempotents of $R$.

## 1.7. Quotients of spectral spaces

Taking quotient of spectral spaces is a natural operation when dealing with diamonds that are pro-étale quotients of perfectoid spaces.

**1.7.1. Quotient maps.** — Next proposition is a key point in the theory of analytic adic spaces. In fact, all morphisms of analytic adic spaces are generalizing (contrary to morphisms of schemes that are generalizing when they are flat but not in general). This fact plays a key role in the theory of diamonds.

**Proposition 1.7.1.** — Let $f: X \to Y$ be a quasicompact surjective map of spectral spaces. If $f$ is generalizing then $f$ is quotient map i.e. if $R$ is the equivalence relation $R = X \times_Y X \subset X \times X$ then $X/R \to Y$ is an homeomorphism.

*Proof.* — Let $V$ be a subset of $Y$ such that $f^{-1}(V)$ is open. We have to prove that $V$ is open. Since $f$ is surjective,

$$V = Y \setminus f(X \setminus f^{-1}(V)).$$

Since $f$ is quasicompact and $X \setminus f^{-1}(V)$ closed and thus pro-constructible,

$$f(X \setminus f^{-1}(V))$$

is pro-constructible. Let now

$$y \in f(X \setminus f^{-1}(V))$$

and $z \in Y$ such that $y \ge z$. If $z \in V$ write $z = f(x)$. Since $f$ is generalizing we can find $x' \in X$ such that $x' \ge x$ and $f(x') = y$. Since $f^{-1}(V)$ is open we deduce that $x' \in f^{-1}(V)$ and thus $y \in V$ which is impossible. From this we deduce that $f(X \setminus f^{-1}(V))$ is stable under specializations and thus closed since pro-constructible. This proves the result. $\square$
**Example 1.7.2.** — Let $f: X \to Y$ be a flat morphism of quasicompact quasiseparated schemes. If $|f|: |X| \to |Y|$ is surjective then this is a quotient map.

**Example 1.7.3.** — If $f: X \to Y$ is a quasicompact generalizing map of spectral spaces then $f: X \to \text{Im}(f)$ is a quotient map since $\text{Im}(f)$ is pro-constructible in $Y$ and thus spectral.

**1.7.2. Quotient by an equivalence relation.** — We are now dealing with “the converse” to Proposition 1.7.1.

**Proposition 1.7.4 ([48, Lemma 2.9, Lemma 2.10]).** — Let $X$ be a spectral space and $R \subset X \times X$ be a pro-constructible equivalence relation such that both maps $R \twoheadrightarrow X$ are generalizing.

1. If the topological space $X/R$ has a basis of open subsets whose preimages in $X$ are quasi-compact then $X/R$ is spectral and the map $X \to X/R$ is quasicompact generalizing.

2. If both maps $R \twoheadrightarrow X$ are open then $X/R$ is spectral and $X \to X/R$ is quasicompact open.

*Proof.* — Let us verify point (1). It is clear that $X/R$ is quasicompact and that each point has a basis of neighborhoods made of quasicompact retrocompact open subsets. Let us verify it is (T0). Let $x_1, x_2 \in X$ having distinct images in $X/R$. Let us note $s, t: R \to X$ for the two maps defining the equivalence relation, and

$$Z_1 = s(t^{-1}(x_1)), Z_2 = s(t^{-1}(x_2))$$

the orbits of $x$ and $y$ under $R$. We thus have $Z_1 \cap Z_2 = \emptyset$. Since $R$ is pro-constructible inside $X \times X$, $s, t: R_{\text{cons}} \to X_{\text{cons}}$ are continuous maps of compact topological spaces. Since $\{x_1\}$ and $\{x_2\}$ are pro-constructible we deduce that $Z_1$ and $Z_2$ are pro-constructible subsets of $X$. For $i = 1, 2$, since $Z_i$ is pro-constructible, its closure $\overline{Z_i}$ is the set of its specializations. Since $t$ is generalizing we have

$$\overline{t^{-1}(Z_i)} = t^{-1}(\overline{Z_i}).$$

We deduce that

$$s(t^{-1}(\overline{Z_i})) = \overline{Z_i}$$

and thus $\overline{Z_i}$ is stable under $R$. Now, if

$$Z_1 \cap \overline{Z_2} = \emptyset$$

then $U = X \setminus \overline{Z_2}$ is an open subset stable under $R$ whose image in $X/R$ is an open subset containing $x_1$ and not $x_2$. The same goes on if $\overline{Z_1} \cap Z_2 = \emptyset$, the image in $X/R$ of $X \setminus \overline{Z_1}$ is an open subset containing $x_2$ but not $x_1$. We are thus reduced to proving that the situation

$$Z_1 \cap \overline{Z_2} = \emptyset \text{ and } \overline{Z_1} \cap Z_2 = \emptyset$$

is impossible. For this consider the pro-constructible subset

$$Z_1 \cup Z_2$$

of $X$. Being pro-constructible it is a spectral space and thus contains a maximal point (Corollary 1.3.9). Up to permuting $x_1$ and $x_2$ and replacing $x_1$ by this maximal point we can suppose that $x_1$ is maximal in $Z_1 \cup Z_2$. Since $Z_1$ is the orbit of one point and $\overline{Z_2}$ is stable under $R$, the condition $Z_1 \cap \overline{Z_2} \neq \emptyset$ implies that $Z_1 \subset \overline{Z_2}$. This implies that $x_1$ is a specialization of an element of $Z_2$ and thus $x_1 \in Z_2$ by maximality of $x_1$. This is impossible and we conclude that $X/R$ is (T0).

Let now $Z \subset X/R$ be an irreducible closed subset. To prove it has a generic point we have to prove that

$$\bigcap_{\substack{U \subset X/R \text{ open} \\ U \cap Z \neq \emptyset}} U \cap Z \neq \emptyset.$$
Let $\pi: X \to X/R$ be the projection. One has to verify that

$$ \bigcap_{\substack{U \subset X/R \text{ open} \\ U \cap Z \neq \emptyset}} \pi^{-1}(U) \cap \pi^{-1}(Z) \neq \emptyset. $$

This is immediately deduced from the fact that $\pi^{-1}(Z)_{\text{cons}}$ is compact and the fact that the preceding intersection has the finite intersection property since $Z$ is irreducible.

We have thus proven that $X/R$ is spectral. The fact that $X \to X/R$ is quasicompact generalizing is easy and left to the reader.

Point (2) is deduced from point (1) and the fact that for $U$ a quasicompact open subset of $X$, its orbit $s(t^{-1}(U))$ is quasicompact open. $\square$

**Example 1.7.5.** — Let $G$ be a profinite group acting continuously on a spectral space $X$ in the sense that the map

$$ G \times X \to X $$

is continuous. Let $U$ be a quasicompact open subset of $X$. By continuity of the action for any point $x$ of $U$ there exists an open subgroup $H$ of $G$ and a neighborhood $V$ of $x$ such that $H.V \subset U$. Using the quasicompacity of $U$ we deduce the existence of an open subgroup $H$ of $G$ such that $H.U = U$. From this we deduce that $G.U$ is a finite union of translates of $U$ and is thus quasicompact open. We deduce from this that the continuous map

$$ \begin{gathered} G \times X \to X \times X \\ (g, x) \mapsto (g.x, x) \end{gathered} $$

is quasicompact and thus its image $R$ is pro-constructible. Both projections from $R$ to $X$ are clearly open since the map $(g, x) \mapsto g.x$ is open. Applying Proposition 1.7.4 we deduce that

$$ X/G $$

is a spectral space.

## 1.8. Constructible sheaves

If $X = \varinjlim_{i \in I} X_i$ is a qc cofiltered limit of spectral spaces, then

$$ \varinjlim_{i \in I} \{\text{qc open subsets of } X_i\} \xrightarrow{\sim} \{\text{qc open subsets of } X\}. $$

Since quasi-compact open subsets form a base of the topology we deduce an equivalence of topos

$$ \widetilde{X} \xrightarrow{\sim} \varinjlim_{i \in I} \widetilde{X}_i. $$

We can then apply [2, Exp.VI Sec.8] to deduce that for any sheaf of abelian groups $\mathcal{F}$ on $X$ with associated $\mathcal{F}_i$ on $X_i$, $i \in I$, one has

$$ (2) \qquad \varinjlim_{i \in I} R\Gamma(X_i, \mathcal{F}_i) \xrightarrow{\sim} R\Gamma(X, \mathcal{F}). $$

**Example 1.8.1.** — Let $f: X \to Y$ be a qc map of spectral spaces, $\mathcal{F}$ a sheaf of abelian groups on $X$ and $y \in Y$. Note $X_y = X \times_Y Y_y$. One has

$$ X_y = \varinjlim_{\substack{U \ni y \\ \text{open qc}}} f^{-1}(U). $$

From this we deduce that

$$ (Rf_*\mathcal{F})_y = R\Gamma(X_y, \mathcal{F}|_{X_y}). $$
The following proposition is easy.

**Proposition 1.8.2.** — 1. When the $X_i$, $i \in I$, are finite, via pullback to $X$, there is an equivalence of categories

$$ \lim_{i \in I} \widetilde{X}_i \xrightarrow{\sim} \{\text{constructible sheaves on } X\} $$

where a constructible sheaf is a sheaf that is constant along a locally closed constructible stratification.

2. There is an equivalence of categories

$$ \lim_{\rightarrow} \mathrm{Ind}(\mathrm{constructible\ sheaves\ on\ } X) \xrightarrow{\sim} \widetilde{X} $$

that identifies the category of sheaves on $X$ as the ind-category of constructible sheaves.

3. The constructible sheaves of abelian groups with finite type stalks are the compact objects of the category of sheaves of abelian groups on $X$ and this last category is compactly generated.

4. The category of constructible sheaves of abelian groups on $X$ is the smallest sub-abelian category of sheaves of abelian groups on $X$ stable under extension an containing the sheaves $j\mathbb{Z}$ for $j$ the inclusion of a locally closed constructible subsets and $M$.

## 1.9. Totally disconnected spectral spaces

Totally disconnected perfectoid spaces are an essential notion in the theory of diamonds. We will often use the slogan “pro-étale locally, any perfectoid space is totally disconnected”. Here we first introduce the notion for spectral spaces.

**1.9.1. Generalities.** — Let $X$ be a spectral space. We say that an open covering

$$ X = \bigcup_{i \in I} U_i $$

of $X$ splits if

$$ \coprod_{i \in I} U_i \rightarrow X $$

has a section. This means that we can find a collection of open subsets $(V_i)_{i \in I}$ with $V_i \subset U_i$ and

$$ X = \coprod_{i \in I} V_i. $$

**Theorem 1.9.1 (F. [48, Lemma 7.2]).** — For $X$ a spectral space the following are equivalent:

1. Each connected component of $X$ has a unique closed point.

2. Each open covering of $X$ splits.

3. For any topological space $Y$ and any surjective continuous map $f: Y \rightarrow X$ that is a local isomorphism on $Y$, $f$ has a section.

4. For any sheaf of sets $\mathcal{F}$ on $X$, $\mathcal{F}(X) \neq \emptyset$.

5. For any sheaf of groups $\mathcal{G}$ on $X$, $H^1(X, \mathcal{G}) = \{\ast\}$.

6. For any sheaf of abelian group $\mathcal{F}$ on $X$ and any $i > 0$, $H^i(X, \mathcal{F}) = 0$ i.e. any sheaf of abelian groups is acyclic.
Proof. — Let us verify the equivalence between points (2) and (6). Suppose point (2) is verified and let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Point (2) implies immediately the vanishing of the Chech cohomology of $X$,

$$ \forall i > 0, \tilde{H}^i(X, \mathcal{F}) = 0. $$

Since $\tilde{H}^1(X, \mathcal{F}) \xrightarrow{\sim} H^1(X, \mathcal{F})$ we deduce that $H^1(X, \mathcal{F}) = 0$. This being true for all $\mathcal{F}$, using a standard “décalage cohomologique” method by induction on $i$ starting with $i = 1$, we deduce that

$$ \forall i > 0, H^i(X, \mathcal{F}) = 0. $$

In the other direction, suppose point (6) is verified. Let $X = \cup_{\alpha} U_{\alpha}$ be an open covering of $X$. Note $j_{\alpha}: U_{\alpha} \hookrightarrow X$ the inclusion. There is an epimorphism

$$ \bigoplus_{\alpha} j_{\alpha}! \mathbb{Z} \to \mathbb{Z}. $$

By hypothesis, the global section functor $\Gamma(X; -)$ is exact and we deduce that

$$ \bigoplus_{\alpha} \Gamma(X, j_{\alpha}! \mathbb{Z}) \to \Gamma(X, \mathbb{Z}) $$

is surjective. We deduce that one can write

$$ (3) \qquad 1 = \sum_{\alpha} f_{\alpha} $$

as locally constant functions from $X$ to $\mathbb{Z}$ where $\text{supp}(f_{\alpha}) \subset U_{\alpha}$. Since $X$ is quasicompact and $f_{\alpha}$ locally constant, $V_{\alpha} = \text{supp}(f_{\alpha}) \subset U_{\alpha}$ is open and closed. The Equation (3) then shows that $X = \cup_{\alpha} V_{\alpha}$. There is factorization $\coprod_{\alpha} V_{\alpha} \to \coprod_{\alpha} U_{\alpha} \to X$. Since $X$ is quasi-compact we can suppose that our coverings are finite. Since any finite covering of a topological space by open/closed subsets splits we deduce that $(U_{\alpha})_{\alpha}$ splits and we deduce that point (2) is verified. We have thus proven (1) $\Leftrightarrow$ (6).

Let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Note

$$ \pi: X \to \pi_0(X). $$

Since $\pi_0(X)$ is profinite any covering of $\pi_0(X)$ splits and thus, as we just saw, any sheaf on $\pi_0(X)$ is acyclic. We thus have for $i \ge 0$,

$$ (4) \qquad H^i(X, \mathcal{F}) = H^0(\pi_0(X), R^i \pi_* \mathcal{F}). $$

Moreover, for $c \in \pi_0(X)$ one has

$$ \pi^{-1}(c) = \lim_{\substack{\xrightarrow{\quad} \\ \mathcal{U} \ni c}} \pi^{-1}(\mathcal{U}). $$
open/closed

We deduce from this that

$$ (5) \qquad (R\pi_*\mathcal{F})_c = R\Gamma(\pi^{-1}(c), \mathcal{F}), $$

see Equation (2) in Section 1.8.

Suppose $X$ is connected. Then the equivalence between points (1) and (2) is immediate. In fact, if $X$ has two distinct closed point $x_1$ and $x_2$ then the covering $X = X \setminus \{x_1\} \cup X \setminus \{x_2\}$ can not split since $X$ is connected. Reciprocally, if $X$ has a unique closed point $x$ then $X = X_x$ since for any $y \in X$, $\overline{\{y\}}$ has a closed point (see Corollary 1.3.9). Thus, if $X = \cup_i U_i$ is an open covering, as soon as $x \in U_i$ for some index $i$, $X = U_i$ and thus the covering splits.

Let now $X$ be any spectral space. If point (2) is verified for $X$ then it is verified if we replace $X$ by any closed subspace of $X$. Since any connected component of $X$ is closed, from the case of a connected space established before, we deduce that point (1) is verified. Reciprocally, suppose point (1) is verified. Let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Using Equation (5), the case of a connected space treated before and (2) $\Rightarrow$ (6), we deduce that $R^i \pi_* \mathcal{F} = 0$ for $i > 0$. Equation
(4) then implies that $H^i(X, \mathcal{F}) = 0$ for $i > 0$. Using (6) ⇒ (2) we deduce that (1) is verified. We thus have proven (1) ⇔ (6).

We have established (1) ⇔ (2) ⇔ (6). The equivalence (3) ⇔ (4) is obtained using the “espace étalé” of a sheaf of sets. The implication (4) ⇒ (5) is deduced from the fact that $H^1(X, \mathcal{G})$ classifies $\mathcal{G}$-torsors and those are trivial iff they have a section. The implication (5) ⇒ (6) is obtained using “décalage cohomologique”. Finally, (2) ⇒ (3) is deduced from the fact that for such a map $Y \to X$ as in point (3), there exists an open covering $(U_i)_i$ of $X$ such that $\coprod_i U_i \to Y \to X$. □

**Definition 1.9.2.** — The spectral spaces satisfying the equivalent conditions of theorem 1.9.1 are called totally disconnected.

Those spaces will play a key role in the theory of diamonds where we will use the slogan “pro-étale locally any perfectoid spaces is totally disconnected” very often.

**Example 1.9.3.** — For $X$ spectral and $x \in X$, $X_x$ is connected and totally disconnected!

**1.9.2. The particular case of analytic spectral spaces.** — Let us begin with a definition. This is motivated by the nature of spectral spaces associated to analytic adic spaces. Here we enter a world that is very different from the one of Noetherian schemes. It happens a Noetherian topological space is analytic, for example this is the case for the topological space of a dimension 1 Noetherian scheme or the spectrum of a finite height valuation ring, but this is not the case in general.

**Definition 1.9.4.** — A spectral space $X$ is analytic if for any $x \in X$, the ordered set $X_x$ is a chain i.e. is a totally ordered set for the specialization relation.

The following property is particular to analytic spectral spaces since it is already false for general finite (T0) spaces.

**Proposition 1.9.5.** — Let $X$ be an analytic totally disconnected spectral space. Then any pro-constructible subset of $X$ is totally disconnected.

*Proof.* — Let $Z \subset X$ be pro-constructible. Let $C$ be a connected component of $Z$. Let $D$ be the connected component of $X$ containing $C$. Since $D$ has a unique closed points, $(D, \leq)$ is a chain. Thus, $(C, \leq)$ is a chain and thus has a unique closed point. □

Using this simple result we obtain the more surprising one that follows. We put it to illustrate the difference between Noetherian schemes and analytic adic spaces that will show up later.

**Proposition 1.9.6.** — Let $j : Z \hookrightarrow X$ be the inclusion of a pro-constructible set inside an analytic spectral space, for example the inclusion of a quasi-compact open subset. Then for any sheaf of abelian groups $\mathcal{F}$ on $Z$ and $i > 0$,

$$R^i j_* \mathcal{F} = 0.$$

*Proof.* — Let $x \in X$. According to Example 1.8.1 one has $(R^i j_* \mathcal{F})_x = H^i(X_x \cap Z, \mathcal{F})$. The spectral space $X_x$ is analytic totally disconnected and thus according to Proposition 1.9.5, $X_x \cap Z$ is totally disconnected. The result follows. □

Typically, if $j$ is the inclusion of a quasi-compact open subset of $X$ an analytic spectral space then $R^i j_* \mathcal{F} = 0$ for $i > 0$. For non-analytic spectral spaces this is evidently false. Typically, if $k$ is a field and $j : \mathbb{A}_k^2 \to \{(0,0)\} \hookrightarrow \mathbb{A}_k^2$ then $R^1 j_* \mathcal{O} \neq 0$.

**1.9.3. w-local spaces.** —
**1.9.3.1. Pro-étale maps.** — Start with $X$ a spectral space. We fix a base $\mathcal{B}$ of the topology of $X$ that is

• stable under finite intersections,

• made of quasi-compact open subsets,

• such that if $U \in \mathcal{B}$ any open/closed subset of $U$ is in $\mathcal{B}$.

Let us start with a definition that will take on its full meaning in the context of perfectoid spaces and their pro-étale topology.

**Definition 1.9.7.** — 1. A map of spectral spaces $Y \to X$ is $\mathcal{B}$-étale if it is isomorphic to a map $\coprod_{i \in I} U_i \to X$ with $I$ finite and for each $i, U_i \in \mathcal{B}$.

2. A morphism $Y \to X$ of spectral spaces is $\mathcal{B}$-pro-étale if it is isomorphic to a cofiltered limit of $\mathcal{B}$-étale spaces over $X$.

A morphism

<figure><img src="image_12.png" /></figure>

with $I$ and $J$ finite, $U_i, V_j \in \mathcal{B}$, is given by a decomposition $U_i = \coprod_{j \in J} U_{ij}$ for each $i \in I$ where $U_{ij} \subset V_j$. From this we deduce the following lemma.

**Lemma 1.9.8.** — 1. If $f : Y \to X$ is $\mathcal{B}$-étale then $f^{-1}\mathcal{B} = \{f^{-1}(U) | U \in \mathcal{B}\}$ is a base of the topology of $Y$ satisfying analogous assumptions to the one satisfied by $\mathcal{B}$.

2. If we have a diagram

<figure><img src="image_13.png" /></figure>

with $f$ and $f \circ g$ $\mathcal{B}$-étales then $g$ is $f^{-1}\mathcal{B}$-étale and is in particular quasi-compact.

3. The category of $\mathcal{B}$-étale spaces over $X$ is cofiltered:

• the equalizer of a pair of maps between $\mathcal{B}$-étale spaces over $X$ is $\mathcal{B}$-étale over $X$,

• For $Y \to X$ and $Y' \to X$ that are $\mathcal{B}$-étale, $Y \times_X Y' \to X$ is $\mathcal{B}$-étale.

**Example 1.9.9.** — For any $x \in X$, $X_x \hookrightarrow X$ is $\mathcal{B}$-pro-étale for any choice of $\mathcal{B}$.

**Proposition 1.9.10.** — The projective limit functor induces an equivalence of categories

$\lim_{\leftarrow} : \text{Pro}(\mathcal{B}\text{-étale spaces over } X) \xrightarrow{\sim} \{\mathcal{B}\text{-pro-étale spaces over } X + \text{qc maps}\}$.

*Proof.* — The essential surjectivity is the definition of $\mathcal{B}$-pro-étale spaces over $X$. For the full faithfulness it suffices to prove that

$$ \operatorname{Hom}_X(\lim_{\underset{i \in I}{\longrightarrow}} Y_i, Z) = \lim_{\underset{i \in I}{\longrightarrow}} \operatorname{Hom}_X(Y_i, Z) $$

but this is easily deduced from Proposition 1.4.4. $\square$

<figure><img src="image_14.png" /></figure>

**Remark 1.9.11.** — Suppose $X = |\text{Spec}(R)|$ for a ring $R$. Let us choose for $\mathcal{B}$ the base of principal open subsets $\{D(f)\}_{f \in R}$. Then, if $f : Y \to X$ is $\mathcal{B}$-pro-étale, $Y$ inherits automatically a canonical structure of an affine scheme as a cofiltered limit of affine schemes.
**1.9.3.2. w-localization.** — We now want to construct $f : X' \to X$ such that

1. $X'$ is spectral totally disconnected,

2. $f$ is qc surjective $\mathcal{B}$-pro-étale.

If we can do this we will say that “pro-étale locally, $X$ is totally disconnected”.

One functorial way to do this is to introduce the category of w-local spectral spaces ([8]). In fact, if we want to functorially associate to $X$ a totally disconnected spectral space $X'$ with a morphism $X' \to X$, the simplest way is to find a subcategory $\mathcal{C}$ of the category of totally disconnected spectral spaces such that the inclusion $\mathcal{C} \hookrightarrow \{\text{spectral spaces}\}$ has a right adjoint. But

1. if we take for $\mathcal{C}$ the category of profinite spaces this is too small and $X_{cons} \to X$ is not pro-étale in general,

2. if we take for $\mathcal{C}$ the full sub-category of totally disconnected spectral spaces there is no such adjoint.

The category of w-local spectral spaces is a nice category in between the two preceding one that works well (but this is a not a full sub-category).

**Definition 1.9.12 ([8]).** — We say that $X$ is w-local if it is totally disconnected and the set of closed points $X_c$ of $X$ is closed. The category of w-local spectral spaces has objects the w-local spectral spaces and morphisms the qc maps $f: X \to Y$ such that $f(X_c) \subset Y_c$.

**Example 1.9.13.** — For $X$ spectral and $x \in X$, $X_x$ is w-local with only one closed point.

For $X$ w-local the bijection $\pi_{|X_c}: X_c \to \pi_0(X)$ is in fact an homeomorphism and thus the surjection $\pi: X \to \pi_0(X)$ has a continuous section. It it not satisfied in general for a totally disconnected spectral space.

**Example 1.9.14.** — Consider the profinite set $\overline{\{\frac{1}{n} | n \ge 1\}}$. We add a point $\eta$ and set $X = \overline{\{\frac{1}{n} | n \ge 1\}} \cup \{\eta\}$ where

• a basis of neighborhoods of $\frac{1}{n}$ is given by the set with one element that is $\{\frac{1}{n}, \eta\}$,

• a basis of neighborhoods of 0 is $\overline{\{\frac{1}{n} | n \ge N\}} \cup \{\eta\}$ when $N \ge 1$ varies,

• a basis of neighborhoods of $\eta$ is $\overline{\{\frac{1}{n} | n \ge N\}} \cup \{\eta\}$ when $N \ge 1$ varies.

Then $X$ is spectral totally disconnected with $\pi_0(X) = \overline{\{\frac{1}{n} | n \ge 1\}}$ as a profinite set. But this is not w-local.

**Proposition 1.9.15.** — The natural functor from the category of w-local spectral spaces to the one of spectral spaces has a right adjoint $X \mapsto X^{wl}$. Moreover, $X^{wl} \to X$ is $\mathcal{B}$-pro-étale.

*Proof.* — The construction of the adjoint is easy. In fact, seeing the category of spectral spaces as the pro-category of finite (T0) spaces, it suffices to define $X^{wl}$ for $X$ finite (by right adjunction, the w-localization functor has to commute with projective limits). But then one verifies that

$$X^{wl} = \coprod_{x \in X} X_x$$

works for $X$ a finite (T0) space. The key point here is that if $Y$ is w-local and $f: Y \to X$ is qc with $X$ finite (T0) then $f|_{Y_c}: Y_c \to X$ is a qc map from a profinite space to a finite (T0) space, thus a locally constant map. This implies $f$ factorizes through $\coprod_{x \in X} X_x$.

The fact that $X^{wl} \to X$ is pro-étale is more complicated. We refer to [48, Lemma 7.13] for this. □
Finally let us note that via the descriptions of the functors $X \mapsto X_{\text{cons}}$ and $X \mapsto X^{\text{wl}}$ on Pro(finite ($T0$) spaces), we have

$$ \pi_0(X^{\text{wl}}) = X_{\text{cons}}. $$

We thus have two qc maps

<figure><img src="image_15.png" /></figure>

Next proposition is left to the reader.

**Proposition 1.9.16.** — The map $X^{\text{wl}} \to X_{\text{cons}} \times X$ identifies $X^{\text{wl}}$ with $\{(x, y) \in X_{\text{cons}} \times X \mid x \le y\}$ equipped with the induced product topology.

Thus, $X^{\text{wl}}$ is the set of generalizations

$$ \Delta^{\text{gen}} = \bigcap_{\substack{U \supset \Delta \\ \text{neighborhood}}} U $$

of the diagonal $\Delta = \{(x, x) \mid x \in X\} \subset X_{\text{cons}} \times X$.

**1.9.4. Another construction.** — The morphism $X^{\text{wl}} \to X$ is not as nice as we would want for applications to perfectoid spaces: this is not open in general. For applications to perfectoid spaces we need another construction that provides an open pro-étale morphism $X' \to X$ with $X'$ totally disconnected.

**Example 1.9.17.** — The map $\text{Spec}(\mathbb{Z}_p)^{\text{wl}} \to \text{Spec}(\mathbb{Z})$ is not open. In fact, let $\pi : \text{Spec}(\mathbb{Z})^{\text{wl}} \to \pi_0(\text{Spec}(\mathbb{Z}))_{\text{cons}} = \text{Alexandrov compactification of Spm}(\mathbb{Z})_{\text{disc}}$ where the point at infinity is $\eta$ (see Example 1.3.6). Then for a prime number $p$, $\{p\}$ is open in $\text{Spm}(\mathbb{Z})_{\text{disc}}$ and $\pi^{-1}(p)$ is the open subset $\{(p \le p), (p \le \eta)\}$. Its image in $\text{Spec}(\mathbb{Z})$ is $\{p, \eta\}$ that is not open.

**Remark 1.9.18.** — The reason why $X^{\text{wl}} \to X$ is not open in general is the following. In the limit $X^{\text{wl}} = \lim_{i \to \infty} X_i \to X$ with $X_i \to X$ $\mathscr{B}$-étale and thus open, the transition maps $X_i \to X_j$ for $i \ge j$ are not surjective and $X^{\text{wl}} \to X_i$ is not surjective in general although $X^{\text{wl}} \to X$ is surjective.

**Proposition 1.9.19.** — There exists an open $\mathscr{B}$-pro-étale map $X' \to X$ with $X'$ totally disconnected.

*Proof.* — We are going to define a sequence

$$ \dots \to X_n \xrightarrow{p_n} \dots \to X_1 \xrightarrow{p_1} X_0 = X $$

where $p_n$ is $\mathscr{B}_{n-1} := (p_1 \circ \dots \circ p_{n-1})^{-1}\mathscr{B}$-pro-étale. Suppose $X_n$ is defined. Let $E$ be the ordered set of finite subsets of the finite subsets of $\mathscr{B}_n$ that form a cover of $X_n$. Define

$$ X_{n+1} = \lim_{\substack{V \in E \\ V = \{V_1, \dots, V_n\}}} \prod_{U \in V_1} U \times_{X_n} \dots \times_{X_n} \prod_{U \in V_i} U. $$

Now set $X' = \lim_{n \ge 0} X_n$. Since all morphisms in the transition maps in our projective limits are open surjective, $X' \to X$ is open. Any finite cover of $X'$ by quasi-compact open subsets comes by pullback from such a cover of $X_n$ for some $n \ge 0$. Any such cover of $X_n$ is refined by a finite cover
consisting of elements of $\mathcal{B}_n$. Let $W \subset \mathcal{B}_n$ be such a finite subset that makes a cover of $X_n$. As soon as $W \in \mathcal{V} = \{V_1, \dots, V_i\}$, there is a section

<figure><img src="image_16.png" /></figure>

and there is thus a section

<figure><img src="image_17.png" /></figure>

Our cover $W$ thus splits after pullback to $X_{n+1}$. It thus splits after pullback to $X'$. Our original cover of $X'$ is thus refined by a cover that splits and it thus splits. □

**1.10. The Berkovich spectrum**

In this section we dig further into the structure of the spectral topological spaces we will be interested in at the end: the analytic one, see Definition 1.9.4. They happen to have a nice compact (Hausdorff) quotient that corresponds to the Berkovich spectrum $X^B = \mathcal{M}(A)$ when one considers the spectral space $X = \text{Spa}(A, A^+)$ with $A$ a Tate ring. But nevertheless, this quotient exists and is well behaved for more general analytic spectral spaces than the preceding adic spaces, for example for spatial diamonds.

**1.10.1. Compactness and adjunction.** — Let us give the very general definition of the Berkovich quotient of a spectral space.

**Definition 1.10.1.** — For $X$ spectral define $X^B$ to be the quotient of $X$ by the equivalence relation generated by the order relation that is the specialization. We equip it with the quotient topology.

The open subsets of $X^B$ are thus in bijection with the the open subsets of $X$ stable under specialization (the so-called partially proper, resp. overconvergent, resp. wide open, subsets in Huber's terminology, resp. classical rigid geometry terminology, resp. Coleman's terminology).

We now make the following assumption: $X$ is analytic, see Definition 1.9.4. One has in fact the following elementary result that simplifies the description of the quotient $X^B$.

**Lemma 1.10.2.** — If $X$ is analytic then any $x \in X$ has a unique maximal generalization, $x^{max}$.

*Proof.* — The uniqueness is immediate from the chain property. Now, if $\cap_{y \ge x} X_y = \emptyset$, since for all $y \ge x$, $X_y$ is closed in the compact topological space $(X_x)_{\text{cons}}$, a finite sub-intersection is empty. This is impossible using the chain property. One deduces that $\cap_{y \ge x} X_y = \{x^{max}\}$ for $x^{max}$ maximal. □
Now, the equivalence relation defining the quotient $X^B$ of $X$ is simply

$$x \sim y \iff x^{\max} = y^{\max}.$$

Thus, as a set, $X^B$ is identified with the set of maximal points of $X$.

**Lemma 1.10.3.** — Let $X$ be an analytic spectral space and $Z$ a pro-constructible subset. Let $Z^{gen}$ be the set of generalizations of the elements of $Z$. Then, $Z^{gen}$ is pro-constructible and the open subset

$$X \setminus \overline{Z^{gen}}$$

is stable under specialization.

*Proof.* — The set $Z^{gen}$ is the intersection of all open subsets containing $Z$. But now, since $Z$ is quasi-compact,

$$Z^{gen} = \bigcap_{\substack{U \supset Z \\ \text{qc open}}} U$$

and it is thus pro-constructible. The closed set $\overline{Z^{gen}}$ is thus the set of specializations of the elements of $Z^{gen}$ (Corollary 1.3.9). Now, if $x \in X \setminus \overline{Z^{gen}}$ and $x \ge y$. If $y \in \overline{Z^{gen}}$ there exists $w \in X$ and $z \in Z$ such that $w \ge y$ and $w \ge z$. Since $X_y$ is a chain we have either $x \ge w$ or $w \ge x$. If $x \ge w$ then $x \ge z$ and thus $x \in Z^{gen}$ which is impossible. If $w \ge x$ then $x \in \overline{Z^{gen}}$ which is impossible. We thus have $y \notin \overline{Z^{gen}}$. $\square$

**Proposition 1.10.4.** — If $X$ is analytic, $X^B$ is a compact (Hausdorff) space.

*Proof.* — We just have to prove that $X^B$ is Hausdorff. Let $x \ne y$ be two maximal points of $X$. We have, $\overline{\{x\}} \cap \overline{\{y\}} = \emptyset$. We can thus find $U$ and $V$ two quasi-compact open subsets satisfying

$$\overline{\{x\}} \subset U, U \cap \overline{\{y\}} = \emptyset, \overline{\{y\}} \subset V, V \cap \overline{\{x\}} = \emptyset.$$

Let us now consider

$$U' = X \setminus (\overline{(X \setminus U)^{gen}} \cup \overline{V}).$$

According to Lemma 1.10.3 applied to $Z = X \setminus U$ and $Z = V$, this is an open subset of $X$ stable under specializations and contained in $U$. One has $x \notin \overline{V}$ since $\overline{V}$ is the set of specializations of the elements of $V$, $x \notin V$ and $x$ is maximal. The relation $x \in (\overline{X \setminus U})^{gen}$ implies $x$ is a specialization of an element of $(X \setminus U)^{gen}$ which implies $x \in (X \setminus U)^{gen}$ by maximality of $x$. But this last relation implies there exists $y \in X \setminus U$ such that $x \ge y$ which is impossible since $\overline{\{x\}} \subset U$. We thus have proven that $x \in U'$. In the same way, if

$$V' = X \setminus (\overline{(X \setminus V)^{gen}} \cup \overline{U}),$$

this is an open neighborhood of $y$ stable under specializations. We now have $U' \cap V = \emptyset$ and $V' \subset V$. We deduce that $U' \cap V' = \emptyset$. $\square$

If $Y$ is an Hausdorff topological space we have the adjunction formula

$$\mathrm{Hom}(X, Y) = \mathrm{Hom}(X^B, Y)$$

and if $X$ is analytic $X^B$ is then the biggest Hausdorff quotient of $X$.

### 1.10.2. Overconvergent sheaves and their cohomology. —
**1.10.2.1.** *Overconvergent sheaves.* — Suppose $X$ is analytic. Let $\beta : X \to X^B$ be the quotient map to the Berkovich spectrum.

**Definition 1.10.5.** — A sheaf $\mathcal{F}$ on $X$ is overconvergent if for all $x, y \in X$ satisfying $x \ge y$, $\mathcal{F}_y \xrightarrow{\sim} \mathcal{F}_x$.

The basic properties of overconvergent sheaves are the following.

**Proposition 1.10.6.** — Let $\mathcal{F}$ be a sheaf of sets on $X$. The following are equivalent:

1. $\mathcal{F}$ is overconvergent,

2. for any maximal point $x$ of $X$, $\mathcal{F}|_{\overline{\{x\}}}$ is constant,

3. for any quasi-compact open subset $U$ of $X$, if $i : \bar{U} \hookrightarrow U$,

$$ \Gamma(\bar{U}, i^*\mathcal{F}) \xrightarrow{\sim} \Gamma(U, \mathcal{F}), $$

4. for any quasi-compact open subset $U$ of $X$,

$$ \varinjlim_{U \subset \subset V} \Gamma(V, \mathcal{F}) \xrightarrow{\sim} \Gamma(U, \mathcal{F}) $$

where $U \subset \subset V$ means $\bar{U} \subset V$.

*Proof.* — (2) $\Leftrightarrow$ (1) Is deduced from the equality for any $x$ in $X$, $(\mathcal{F}|_{\overline{\{x^{max}\}}})_x = \mathcal{F}_x$ and the fact that for a sheaf $\mathcal{G}$ on $\overline{\{x^{max}\}}$, if $i : \{x^{max}\} \hookrightarrow \overline{\{x^{max}\}}$ then $\mathcal{G}$ is constant iff $\mathcal{G} \xrightarrow{\sim} i_*i^*\mathcal{G}$.

(2) $\Leftrightarrow$ (3) Let $U$ be open $\mathrm{qc}$ and set $j : U \hookrightarrow \bar{U}$. There is a morphism

$$ i^*\mathcal{F} \to j_*j^*i^*\mathcal{F} = j_*\mathcal{F}|_U. $$

For $x \in \bar{U}$ i.e. $x^{max} \in U$, if $\mathcal{G} = \mathcal{F}|_{X_x}$ then the induced morphism on the stalk at $x$ is identified with

$$ \Gamma(X_x, \mathcal{G}) \to \Gamma(X_x \cap U, \mathcal{G}). $$

The equivalence between (2) and (2) is easily deduced.

(3) $\Leftrightarrow$ (4) Since $X$ is analytic, $\bar{U}$ is stable under generalizations. One thus has

$$ \bar{U} = \varinjlim_{U \subset \subset V} V $$

in the category of spectral spaces with $V$ $\mathrm{qc}$ open. From this we deduce that

$$ \Gamma(\bar{U}, i^*\mathcal{F}) = \varinjlim_{U \subset \subset V} \Gamma(V, \mathcal{F}), $$

see Equation (2) in Section 1.8. The result is immediately deduced. $\square$

The following is then easy.

**Proposition 1.10.7.** — The functor $\beta^*$ is fully faithful and identifies the category of sheaves on $X^B$ with the category of overconvergent sheaves on $X$.

**1.10.2.2. Cohomology.** — Next proposition says that the cohomology of overconvergent sheaves is the same as the cohomology of sheaves on the Berkovich space.

**Proposition 1.10.8.** — If $\mathcal{F}$ is an overconvergent sheaf of abelian groups on the analytic spectral space $X$ then

$$ R\Gamma(X, \mathcal{F}) \xrightarrow{\sim} R\Gamma(X^B, \beta_*\mathcal{F}). $$
*Proof.* — We have to prove that $R^i \beta_* \mathcal{F} = 0$ for $i > 0$. Let $x$ be a maximal point of $X$. If $U$ is an open neighborhood of $\overline{\{x\}}$ then $U \setminus (X \setminus U)^{gen}$, where the upperscript “gen” means we take the set of generalizations, is an overconvergent open neighborhood of $x$ contained in $U$ (Lemma 1.10.3). We deduce that $\overline{\{x\}}$ has a basis of open overconvergent neighborhoods. We thus have

$$ (R^i \beta_* \mathcal{F})_{\beta(x)} = H^i(\overline{\{x\}}, i^* \mathcal{F}) $$

where $i: \overline{\{x\}} \hookrightarrow X$. Since $\mathcal{F}$ is overconvergent the sheaf $i^*\mathcal{F}$ is constant. Now, if $M$ is any abelian group the sheaf $\overline{M}$ on $\overline{\{x\}}$ is flabby in the topological sense. More precisely, for any non-empty open subset $V$ of $\overline{\{x\}}$, $\Gamma(\overline{\{x\}}, \overline{M}) \xrightarrow{\sim} \Gamma(V, \overline{M}) = M$. It is thus acyclic and we deduce the result. $\square$

In practice, when one considers adic spaces and their cohomology, a lot of sheaves we consider are overconvergent. Last proposition says that in this case, à priori, we do not care about the adic space and we should only focus on the Berkovich spaces. This is in fact false for two reasons:

* The category of sheaves of abelian groups on a quasi-compact quasi-separated analytic adic spaces is compactly generated with some explicit compact generators, see Proposition 1.8.2. This is à priori not the case for a compact topological space like $X^B$.

* Compactly supported cohomology, already for an affinoid space, needs the introduction of Huber's canonical compactifications, see Section 2.12, where non-overconvergent sheaves show up (the extension by zero $j_1\mathcal{F}$, where $j$ is the inclusion inside the canonical compactification, is not overconvergent).

## 1.11. En résumé

Let $X$ be a spectral space. It gives rise to a diagram

<figure><img src="image_18.png" /></figure>
<figure><img src="image_19.png" /></figure>
# CHAPTER 2

## ADIC SPACES

### 2.1. Huber rings

Huber introduced the most general “reasonable” definition for a topological ring to be a “ring of non-Archimedean holomorphic functions”. This contains the “classical case” of Banach algebras over a non-archimedean field and the one of adic rings for a finitely generated ideal that are the topological rings used by Grothendieck in his theory of formal schemes. Given the recent developments of the theory it is essential to work in a context where our topological rings may look like Banach algebras but may not contain a field. Typically, in the work [28] the authors consider an object $\mathrm{Spa}(\mathbb{Z}_p)^\circ$ that interpolates between $\mathbb{F}_p$-perfectoid spaces and $\mathbb{Q}_p$-perfectoid spaces. This allows them to reduce some statements for the so-called $B_{dR}$-affine Grassmanian over $\mathbb{Q}_p$ to a more “classical one” over $\mathbb{F}_p$ using a degeneracy from $p \neq 0$ to $p = 0$. In this degeneracy process some connected perfectoid $\mathbb{Z}_p$-algebras that do not contain a field appear.

**2.1.1. Generalities** ([37, Section 1],[38]). — Recall the following terminology for a non-archimedean topological ring $A$ (here by non-archimedean we mean that $\mathbb{Z}_{1A}$ is bounded in $A$):

1. A subset $S$ of $A$ is called bounded if for any neighborhood $U$ of $0$ in $A$ there exists a neighborhood $V$ of $0$ in $A$ such that $V \cdot S \subset U$.

2. We note $A^\circ$ the subring of $A$ of power bounded elements in $A$ with its ideal $A^{\circ\circ}$ of topologically nilpotent elements. Let us note that $A^\circ$ is integrally closed in $A$ and $\sqrt{A^{\circ\circ}} = A^{\circ\circ}$. We note $\tilde{A} = A^\circ/A^{\circ\circ}$, a reduced ring.

Huber's definition is the following.

**Definition 2.1.1.** — A Huber ring or an $f$-adic ring, is a topological ring $A$ that admits an open bounded subring $A_0$ whose topology is the $I$-adic topology for an ideal $I$ of finite type. Morphisms between Huber rings are continuous morphisms.

Thus, to define a Huber ring we need:

* a ring $A$,

* a subring $A_0 \subset A$

* a finite type ideal $I = (f_1, \dots, f_n) \subset A_0$,

* coherence condition: such that for all $a \in A$ and $1 \le i \le n$, $f_i^k a \in A_0$ for $k \gg 0$.
**Remark 2.1.2.** — The coherence condition is equivalent to the existence of a factorization (that is unique if it exists)

<figure><img src="image_20.png" /></figure>

which gives some kind of feeling that $\text{Spec}(A)$ sits in between $\text{Spec}(A_0)$ and “its generic fiber” $\text{Spec}(A_0) \setminus V(I)$. See Remark 2.6.4 for a follow up on this.

**Remark 2.1.3.** — Since we will use it all the time, one of the many reasons why we use adic topologies associated to finitely generated ideals is the following. Let $R$ be a ring and $I$ an ideal of $R$. Note $\tilde{R} = \lim_{\leftarrow k \ge 0} R/I^k$ its $I$-adic completion where the topology of $\tilde{R}$ is the projective limit topology with $R/I^k$ discrete for $k \ge 0$. If $I$ is finitely generated then the topology of $\tilde{R}$ is the $\tilde{R}$.I-adic one. This is false in general when $I$ is not finitely generated, $\tilde{R}$ being only an admissible topological ring ([33, Chapitre 0-7.1]). This is for example false for the polynomial ring $A[X_j]_{j \in J}$, where $A$ is any ring, $J$ is infinite and the ideal $I$ is $(X_j)_{j \in J}$. This is the reason Grothendieck admitted more general topological rings than the adic one in the definition of a formal scheme ([33, Chapitre I-10.1]).

The following lemma tells us that this is a good well behaved definition.

**Lemma 2.1.4.** — For a Huber ring $A$ the topology of any open bounded subring of $A$ is the $I$-adic one for a finitely generated ideal $I$.

*Proof.* — Let $A_0$ be a bounded open subring whose topology is the $I$-adic one for a finitely generated ideal $I$ of $A_0$. Let $A_1$ be an open bounded subring. Since $A_1$ is open we can find $n \ge 1$ such that $I^n \subset A_1$. Since $A_1$ is bounded we can find $m \ge n$ such that $A_1 I^m \subset I$. Let $J = A_1 I^m \subset A_1$, an ideal of finite type in $A_1$. One then has for $k \ge 1$,

$$I^{km} \subset J^k \subset I^k.$$

From this we deduce that the topology on $A_1$ is the $J$-adic topology. $\square$

The preceding lemma leads to the following definition.

**Definition 2.1.5.** — A ring of definition of a Huber ring is an open bounded subring. An ideal of definition of a ring of definition is a finitely generated ideal whose associated adic topology is the topology of the ring of definition.

Since we will use it all the time let us note the following.

**Remark 2.1.6.** — • Any ring of definition of $A$ is contained in $A^0$.

• A morphism $f : A \to B$ between two Huber rings is continuous iff one can find two rings of definition $A_0 \subset A$ and $B_0 \subset B$ such that $f(A_0) \subset B_0$ and $f|_{A_0} : A_0 \to B_0$ is continuous. The later is equivalent to saying that if $A_0$ has the $I$-adic topology and $B_0$ the $J$-adic one then $f(I^n) \subset J$ for $n \gg 0$.

**Example 2.1.7.** — 1. Any discrete ring is a Huber ring.

2. If $A$ is a ring equipped with the $I$-adic topology for a finitely generated ideal $I$ then $A$ is a Huber ring and $A$ itself is a ring of definition.

3. Let $A$ be any ring and $f \in A$. Equip $A/(f^\infty$-torsion) with the $f$-adic topology. Then, any subring of $A[1/f]$ containing the image of $A$ is a Huber ring with ring of definition the image of $A$.
4. If $K$ is a complete non-Archimedean field and $A$ is an affinoid algebra in the sense of Tate, that is to say topologically finitely generated, then $A$ is a Huber ring. If $K\langle X_1, \dots, X_n \rangle \to A$ is a surjection then the image of $\mathcal{O}_K\langle X_1, \dots, X_n \rangle$ is a ring of definition.

5. If $A$ is a Huber ring and $\mathfrak{a}$ is an ideal of $A$ then $A/\mathfrak{a}$ with the quotient topology is a Huber ring. If $A_0$ is a ring of definition of $A$ then $A_0/A_0 \cap \mathfrak{a}$ is a ring of definition of $A/\mathfrak{a}$.

6. If $A$ is a Huber ring and $n \ge 1$, the topological ring $A[X_1, \dots, X_n]$, where a basis of neighborhoods of $0$ is given by $\{U[X_1, \dots, X_n]\}_U$ with $U$ going through the set of neighborhood of $0$ in $A$, is a Huber ring. If $A_0$ is a ring of definition of $A$ then $A_0[X_1, \dots, X_n]$ is a ring of definition of $A[X_1, \dots, X_n]$. This Huber rings represents the functor on $A$-Huber rings

$$B \mapsto (B^\circ)^n.$$

7. More generally, if $A$ is a Huber ring, $I$ is a set and $\mathfrak{a}$ is an ideal of $A[X_i]_{i \in I}$, the Huber ring $A[X_i]_{i \in I}/\mathfrak{a}$ represents the functor on $A$-Huber rings

$$B \mapsto \{(b_i)_{i \in I} \mid b_i \in B^\circ, \forall f \in \mathfrak{a}, f(b_i)_{i \in I} = 0\}.$$

We will distinguish two classes of Huber rings that are the one we will work with later.

**Definition 2.1.8.** — A Huber ring $A$ is called

1. *Tate* if there exists a topologically nilpotent unit in $A$, an element

$$\varpi \in A^{\circ\circ} \cap A^\times.$$

Those topologically nilpotent units are called *pseudo-uniformizers*.

2. *Uniform* if $A^0$ is bounded (and is thus a ring of definition since $A^{\circ\circ} \subset A^\circ$ is open).

Let us note the following that clarifies the structure of Tate rings.

**Lemma 2.1.9.** — If $A$ is Tate, $A_0$ is a ring of definition and $\varpi$ a pseudo-uniformizer then

$$A = A_0[\frac{1}{\varpi}]$$

and one can choose $\varpi$ such that $\varpi \in A_0$, in which case the topology on $A_0$ is the $\varpi$-adic topology.

*Proof.* — Let $I$ be a finite type ideal of $A_0$ such that the topology of $A_0$ is the $I$-adic one. Since $\varpi$ is topologically nilpotent and $I$ is open, up to replacing $\varpi$ by a positive power we can assume $\varpi \in I$. For any $f \in I$, since $f$ is topologically nilpotent, $f^n \varpi^{-1} \in A_0$ for $n \gg 0$. Since $I$ is finitely generated we deduce that for $n \gg 0$, $I^n \subset A_0\varpi$. For such an $n$, we thus have

$$I^n \subset A_0\varpi \subset I.$$

The topology of $A_0$ is thus the $\varpi$-adic one. The equality $A = A_0[\frac{1}{\varpi}]$ is immediately deduced from the topological nilpotency of $\varpi$. $\square$

Tate rings are a generalization of Banach algebras (although they may not contain a field). In fact, if $A$ is a Tate ring, fix $A_0$, $\varpi \in A_0$ as before and $\beta \in ]0, 1[$. Define

$$\| \cdot \| : A \to \mathbb{R}_+$$

by the formula

$$\|a\| = \inf\{\beta^n \mid n \in \mathbb{Z} \text{ and } a \in \varpi^n A_0\}.$$

It is easily verified that

$$\varpi^{-1} \notin A_0 \Leftrightarrow A/\overline{\{0\}} \neq 0 \Leftrightarrow \|\varpi\| = \beta \Leftrightarrow \|\varpi\| \neq 0.$$

Then, $\|\cdot\|$ is a non-Archimedean ring semi-norm:

* $\|x + y\| \le \sup\{\|x\|, \|y\|\}$,

* $\|xy\| \le \|x\|\|y\|,$
• $\|1\| = 1$ if $A/\overline{\{0\}} \neq 0$.

This semi-norm defines the topology of $A$. In fact we have just proven the following characterization of Tate rings.

**Proposition 2.1.10.** — A topological ring $A$ such that $A/\overline{\{0\}} \neq 0$ is a Tate ring if and only if there exists $\varpi \in A^{\circ\circ} \cap A^\times$ and $\|\cdot\| : A \to \mathbb{R}_+$ a ring semi-norm defining its topology satisfying

• $\|\varpi\| \in ]0, 1[$,

• $\forall a \in A \text{ and } k \in \mathbb{Z}, \|\varpi^k a\| = \|\varpi\|^k \|a\|$.

**Example 2.1.11.** — 1. In point (3) of Example 2.1.7, $A[\frac{1}{f}]$ is a Tate ring. If moreover the image of $A$ in $A[\frac{1}{f}]$ is integrally closed in $A[\frac{1}{f}]$ then

$$A[\frac{1}{f}]^\circ \subset f^{-1} \mathrm{Im}(A \to A[\frac{1}{f}])$$

and thus $A[\frac{1}{f}]$ is uniform.

2. In point (4) of Example 2.1.7, $A$ is a Tate ring. Moreover, $A$ is reduced if and only if it is uniform.

**2.1.2. Complete Huber rings.** — We will mostly be interested in the case of complete Huber rings. Here by complete we will always mean separated and complete i.e. if $A_0$ is a ring of definition of $A$ whose topology is the $I$-adic one then

$$A \xrightarrow{\sim} \lim_{\underset{n \ge 0}{\leftarrow}} A/I^n.$$

For a Huber ring $A$, with the preceding notations, set

$$\hat{A} := \lim_{\underset{n \ge 0}{\leftarrow}} A/I^n$$

equipped with the projective limit topology where $\lim_{\underset{n \ge 0}{\leftarrow}} A/I^n$ is discrete.

**Lemma 2.1.12.** — 1. The topological ring $\hat{A}$ does not depend on the choice of a ring of definition $A_0$ and the ideal $I$; in fact

$$\hat{A} = \lim_{\underset{U}{\leftarrow}} A/U$$

where $U$ goes through the set of neighborhoods of 0 in $A$ that are additive subgroups.

2. $\hat{A}$ is a complete Huber ring and the functor

$$A \mapsto \hat{A}$$

is a left adjoint to the inclusion

$$\{\text{complete Huber rings}\} \hookrightarrow \{\text{Huber rings}\}.$$

3. If $A_0$ is a ring of definition of $A$, $\hat{A}_0$ is a ring of definition of $\hat{A}$ and there is an isomorphism

$$\hat{A}_0 \otimes_{A_0} A \xrightarrow{\sim} \hat{A}.$$

4. We have $\hat{A}^\circ = \hat{A}^\circ$ and $\hat{A}^{\circ\circ} = \hat{A}^{\circ\circ}$.

*Proof.* — Point (1) is clear. For point (2) we use the fact that if $R$ is a ring and $J$ a finite type ideal of $R$ then its $J$-adic completion $\hat{R}$ is a $\hat{R}J$-adic ring and $R/J \xrightarrow{\sim} \hat{R}/\hat{R}J$.

Let us verify point (3). Fix $A_0$ a ring of definition and $I$ an ideal of definition. There is a canonical morphism $\hat{A}_0 \to \hat{A}$ that induces

$$f: \hat{A}_0 \otimes_{A_0} A \to \hat{A}.$$

Let $a = (a_n)_{n \ge 0} \in \hat{A}$ with $a_n \in A$ and $a_{n+1} - a_n \in I^n$ for $n \ge 0$. Then,

$$\alpha = (a_n - a_0)_{n \ge 0} \in \hat{A}_0$$
and

$$a = f(\alpha \otimes 1 + 1 \otimes a_0).$$

This proves the surjectivity of $f$.

Consider now

$$x = \sum_i a_i \otimes b_i \in \ker f.$$

We can choose $l \ge 0$ such that for all $i$, $b_i I^l \subset A_0$. Write $a_i = \alpha_i \cdot 1_{\widehat{A_0}} + \beta_i$ with $\alpha_i \in A_0$ and $\beta_i \in I^l \widehat{A_0}$. Since $f(x) = 0$,

$$z = \sum_i \alpha_i b_i \in A_0.$$

We then have

$$x = z \cdot 1 \otimes 1 + \sum_i \beta_i \otimes b_i.$$

Now, write $\beta_i = u_i \gamma_i$ with $u_i \in I^l$ and $\gamma_i \in \widehat{A_0}$. We have

$$\sum_i \beta_i \otimes b_i = \sum_i (u_i \gamma_i) \otimes b_i = \sum_i \gamma_i \otimes u_i b_i = \sum_i (u_i b_i) \cdot \gamma_i \otimes 1.$$

At the end we obtain

$$x = (\sum_i \alpha_i b_i) \otimes 1 + \sum_i (u_i b_i) \cdot \gamma_i \otimes 1.$$

But

$$\sum_i (\alpha_i b_i + (u_i b_i) \cdot \gamma_i) \in \widehat{A_0}$$

is sent to 0 via the injection $\widehat{A_0} \hookrightarrow \widehat{A}$ and we conclude.

Point (4) is left to the reader.

□

In the preceding lemma, point (3) tells us that the computation of the completion of a Huber ring is reduced to the computation of a “usual” adic completion with respect to an ideal and a scalar extension. This simplifies the abstract operation of completion of a general Huber ring.

**Example 2.1.13.** — With the notations of example 2.1.7:

1. The completion of $A[X_1, \dots, X_n]$ is $\widehat{A}\langle X_1, \dots, X_n \rangle$, the subring of $\widehat{A}\llbracket X_1, \dots, X_n \rrbracket$ of power series $\sum_{\alpha=(\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n} a_\alpha X_1^{\alpha_1} \dots X_n^{\alpha_n}$ such that $a_\alpha \xrightarrow{|\alpha| \to +\infty} 0$

2. The completion of $A/\mathfrak{a}$ is $\widehat{A}/\widehat{\mathfrak{a}}$ where $\widehat{\mathfrak{a}}$ is the closure of the image of $\mathfrak{a}$ in $\widehat{A}$.

**2.1.3. Uniform Tate rings and the spectral norm after Berkovich.** — Let $A$ be a Tate ring and fix a pseudo-uniformizer $\varpi$ in $A$. We will always suppose that $A/\{0\} \neq 0$. Next proposition is an upgrade of Proposition 2.1.10.

**Proposition 2.1.14.** — The Tate ring $A$ is uniform if and only its topology is defined by a non-Archimedean ring semi-norm $\|\cdot\| : A \to \mathbb{R}_+$ satisfying

$$\|a^n\| = \|a\|^n$$

for $a \in A$ and $n \in \mathbb{N}$. If this is the case then $A^\circ = \{a \in A \mid \|a\| \le 1\}$ and $A^{\circ\circ} = \{a \in A \mid \|a\| < 1\}$.
*Proof.* — Only one direction is non-evident. Suppose $A$ is Tate uniform. Fix a pseudo-uniformizer $\varpi$ and $\beta \in ]0, 1[$. For $a \in A$ set

$$ \|a\| = \inf\{\beta^{-N} \mid \varpi^N a \in A^\circ\}. $$

This is a non-Archimedean ring semi-norm defining the topology of $A$ and satisfying $\|\varpi^k a\| = \beta^k \|a\|$ for all $k \in \mathbb{Z}$ and $a \in A$. Now set

$$ \|a\|_{\infty} = \lim_{n \to +\infty} \|a^n\|^{1/n} $$

(the limit exists since the sequence $(\|a^n\|)_{n \ge 1}$ is submultiplicative). For any $a \in A$ one has

$$ \|a\|_{\infty} \le \|a\|. $$

Now for a given $a \in A \setminus \{0\}$, choose $N$ such that $\varpi^N a \in A^\circ$ and $\varpi^{N-1} a \notin A^\circ$. Since $\varpi^{N-1} a \notin A^\circ$, there exists an infinite number of integers $n$ such that $\|(\varpi^{N-1} a)^n\| \ge 1$ that is to say $\|a^n\| \ge \beta^{n(1-N)}$. From this we deduce $\|a\|_{\infty} \ge \beta^{1-N}$. We have $\|a\| = \beta^{-N}$. Thus, $\beta \cdot \|a\| \le \|a\|_{\infty} \le \|a\|$ and the power multiplicative norm $\|\cdot\|_{\infty}$ defines the topology of $A$. $\square$

In fact we have a uniqueness statement once we fix some simple constraints.

**Proposition 2.1.15.** — Let $A$ be a uniform Tate ring. Fix $\varpi$ a pseudo-uniformizer and $\beta \in ]0, 1[$. Then there exists a unique power multiplicative ring semi-norm defining the topology of $A$ satisfying $\|\varpi\| = \beta$, $\|\varpi^{-1}\| = \beta^{-1}$.

*Proof.* — We already saw the existence in the proof of Proposition 2.1.14. Let $\|\cdot\| : A \to \mathbb{R}_+$ be a semi-norm satisfying the assumptions of the statement. From the inequalities $\|\varpi a\| \le \|\varpi\| \|a\|$ and $\|a\| \le \|\varpi^{-1}\| \|a\|$ for $a \in A$, we deduce that for any $a \in A$ one has $\|\varpi a\| = \|\varpi\| \|a\|$ and thus for any $k \in \mathbb{Z}$ and $a \in A$,

$$ \|\varpi^k a\| = \beta^k \|a\|. $$

Let $\|\cdot\|'$ be another semi-norm satisfying the assumptions of the statement. For $a \in A \setminus \{0\}$, if $a \in \varpi^k A^\circ \setminus \varpi^{k+1} A^\circ$ then

$$ \beta^{k+1} \le \|a\| \le \beta^k \text{ and } \beta^{k+1} \le \|a\|' \le \beta^k. $$

From this we deduce that for any $a \in A$ one has

$$ \beta \|a\|' \le \|a\| \le \beta^{-1} \|a\|'. $$

Replacing $a$ by $a^n$ for all $n$ and taking $1/n$-nth roots we obtain

$$ \beta^{1/n} \|a\|' \le \|a\| \le \beta^{-1/n} \|a\|'. $$

Taking the limit when $n \to +\infty$ we obtain $\|a\| = \|a\|'$. $\square$

This unique semi-norm is unique for a good reason: it has a description as a *spectral norm* in the following sense. Let

$$ \mathcal{M}(A) = \{|\cdot| : A \to \mathbb{R}_+ \text{ continuous multiplicative semi-norms satisfying } |\varpi| = \beta\} $$

as in [6, Chapter 1]. With the notations of sections 2.3 and 1.10 one has $\mathcal{M}(A) = |\text{Spa}(A, A^\circ)|^B$ and this is a compact topological space, see Sections 1.10 and 2.8 for more details. Any $a \in A$ defines a continuous function

$$ \begin{aligned} \mathcal{M}(A) & \to \mathbb{R}_+ \\ x & \mapsto |a(x)|. \end{aligned} $$
**Theorem 2.1.16 (Berkovich [6, Theorem 1.3.1])**. — Let A be a uniform Tate ring and $\varpi$ a pseudo-uniformizer. For the unique power multiplicative ring semi-norm $\|\cdot\|$ defining the topology of A and satisfying $\|\varpi\| = \beta$ and $\|\varpi^{-1}\| = \beta^{-1}$, we have

$$ \forall a \in A, \quad \|a\| = \sup_{x \in \mathcal{M}(A)} |a(x)|. $$

*Proof.* — If $x \in \mathcal{M}(A)$, the continuity hypothesis together with the formulas $|(\varpi^k a)(x)| = \beta^k |a(x)|$ and $\|\varpi^k a\| = \beta^k \|a\|$ for $k \in \mathbb{Z}$ implies that there exists $C > 0$ such that

$$ |a(x)| \le C \|a(x)\|. $$

Using the power-multiplicativity of $\|\cdot\|$ we deduce that we can take $C = 1$. We thus have the inequality $\sup_{x \in \mathcal{M}(A)} |a(x)| \le \|a\|$.

Let $a \in A$ and chose

$$ \rho > \sup_{x \in \mathcal{M}(A)} |a(x)|. $$

We can suppose that A is complete. Let

$$ A\langle\rho T\rangle = \left\{ \sum_{n \ge 0} a_n T^n \mid a_n \in A, \lim_{n \to +\infty} \|a_n\|\rho^{-n} = 0 \right\}. $$

For $f = \sum_{n \ge 0} a_n T^n \in A\langle\rho T\rangle$ we put

$$ \|f\| = \sup_{n \ge 0} \|a_n\|\rho^{-n}. $$

This is a non-archimedean ring norm. Equipped with this, $A\langle\rho T\rangle$ is a complete Tate ring. It is moreover power-multiplicative. In fact, if

$$ f = \sum_{n \ge 0} a_n T^n \in A\langle\rho T\rangle, $$

let

$$ n_0 = \inf\{n \ge 0 \mid \|a_n\|\rho^{-n} = \|f\|\}. $$

One can write

$$ f = a_{n_0} T^{n_0} + g + T^{n_0+1}h $$

where $\|g\| < \|f\|$ and $\|T^{n_0+1}h\| \le \|f\|$. Then for $k \ge 1$ one has

$$ f^k = a_{n_0}^k T^{kn_0} + \underbrace{u}_{\|-\|f\|\|^k} + \underbrace{T^{kn_0+1}}_{\|-\|f\|\|^k \rho^{kn_0+1}} + \underbrace{v}_{\|-\|f\|\|^k \rho^{kn_0+1}}. $$

One deduces that $\|f^k\| = \|f\|^k$.

Let now $x \in \mathcal{M}(A\langle\rho T\rangle)$. One has

$$ |aT(x)| = |a(x)|.|T(x)| \underset{\text{since } \|\cdot\|_{\text{power-mult. on } A\langle\rho T\rangle}}{\le} |a(x)|\rho^{-1} < 1. $$

We thus have

$$ |(1 - aT)(x)| \ne 0. $$

From this we deduce that

$$ 1 - aT \in A\langle\rho T\rangle^\times, $$

see Proposition 2.3.9 and Remark 2.3.10. Since $A\langle\rho T\rangle \subset A\llcorner T$,

$$ (1 - aT)^{-1} = \sum_{n \ge 0} a^n T^n $$

and we thus have

$$ \lim_{n \to +\infty} \|a^n\|\rho^{-n} = 0. $$
In particular, for $n \gg 0$, $\|a^n\|\rho^{-n} \le 1$ and thus

$$ \|a\| \le \rho. $$

This being true for all $\rho > \sup_{x \in \mathcal{M}(A)} |a(x)|$, we deduce that

$$ \|a\| \le \sup_{x \in \mathcal{M}(A)} |a(x)|. \quad \square $$

**Remark 2.1.17.** — The proof is similar to the one that says that for $(A, \|\cdot\|)$ a Banach $\mathbb{C}$-algebra, the spectral radius $\rho(a)$ is given by $\lim_{n \to +\infty} \|a^n\|^{1/n}$. In fact, if $\sigma(a)$ is the spectrum of $a$ then the resolvent $\mathbb{C} \setminus \sigma(a) \ni z \mapsto (\mathrm{Id} - za)^{-1}$ is holomorphic on $\{z \in \mathbb{C} \mid |z| < \rho(a)^{-1}\}$ and its power series expansion as an holomorphic function of $z$ gives the result.

## 2.2. Affinoid rings

**2.2.1. Generalities.** — The following definition may seem strange at first sight: where does this $A^+$ come from? Why is it there? In classical rigid geometry we only consider one ring, Banach algebra like $A$ but there is no $A^+$. The fact is that we implicitly choose $A^+ = A^\circ$ in classical Tate rigid geometry. There are in fact deeper reasons, even when one looks at $\mathrm{Spa}(A, A^\circ)$:

* The local structure around a point of an analytic adic space, even one like $\mathrm{Spa}(A, A^\circ)$, is given by spectra of affinoid fields $\mathrm{Spa}(K, K^+)$ where $K^+$ may be different from $K^\circ$, see Proposition 2.6.5 for example.

* Huber's canonical compactifications, see Section 2.12, are an essential tool in étale cohomology of adic spaces.

**Definition 2.2.1.** — An affinoid ring or a Huber pair is a couple $(A, A^+)$ where $A$ is a Huber ring and $A^+ \subset A^\circ$ is an open subring integrally closed in $A$. Moreover,

1. Such a subring $A^+$ of $A$ is called a ring of integral element.

2. The affinoid ring $(A, A^+)$ is called Tate if $A$ is a Tate ring.

3. It is called complete if $A$ is complete.

4. A morphism $(A, A^+) \to (B, B^+)$ of affinoid rings is a continuous morphism $f : A \to B$ satisfying $f(A^+) \subset B^+$.

**Remark 2.2.2.** — 
* Since $A^\circ$ is integrally closed in $A$ it is equivalent to ask that $A^+$ is integrally closed in $A^\circ$.

* Since $A^+$ is open integrally closed, $A^{\circ\circ} \subset A^+$ and thus at the end

$$ A^{\circ\circ} \subset A^+ \subset A^\circ. $$

* Once $A$ is fixed the choice of $A^+$ a ring of integral elements is then equivalent to the choice of an integrally closed subring of the reduced ring $\tilde{A} := A^\circ/A^{\circ\circ}$.

Let us now remark the following easy lemma.

**Lemma 2.2.3.** — Let $(A, A^+)$ be an affinoid ring. The topological closure of the image of $A^+$ in $\tilde{A}$ is integrally closed in $\tilde{A}$.

This leads to the following evident definition.

**Definition 2.2.4.** — For $(A, A^+)$ an affinoid ring we define its completion $(\tilde{A}, \tilde{A}^+)$ where $\tilde{A}^+ = \tilde{A}^+$ is the closure of the image of $A^+$ in $\tilde{A}$.
Let us finally remark that if $A$ is a Huber ring then the choice of a ring of integral element in $A$ is equivalent to the choice of a ring of integral elements in $\hat{A}$ via the equality

$$A^{\circ}/A^{\circ\circ} = \hat{A}^{\circ}/\hat{A}^{\circ\circ}.$$

## 2.3. The adic spectrum

### 2.3.1. Continuous valuations. —

**2.3.1.1. Valuations ([16, Chapitre 6]).** — Recall some basic definitions and facts about valuations. Let $R$ be a ring and $\Gamma$ be an ordered abelian group. Here we mean that the order on $\Gamma$ is total (and thus $\Gamma$ is torsion free) and compatible with the group law:

$$x \le y \Rightarrow x + z \le y + z.$$

We consider valuations $v : R \to \Gamma \cup \{\infty\}$ i.e. applications that satisfy

* $v(xy) = v(x) + v(y)$,

* $v(1) = 0$, $v(0) = +\infty$,

* $v(x + y) \ge \inf\{v(x), v(y)\}$.

We can always suppose that $v(R) \setminus \{\infty\}$ generates $\Gamma$. The reciprocal image of $+\infty$ is a prime ideal of $R$, the support of $v$. To give oneself a valuation with support $\mathfrak{p}$ is the same as to give oneself a valuation on $\text{Frac}(R/\mathfrak{p})$. This is the same as the datum of a valuation subring $V \subset K = \text{Frac}(R/\mathfrak{p})$.

Recall the following equivalent characterizations of such a $V \subset K$:

* $V$ is such that $\forall x \in K^\times, x \in V$ or $x^{-1} \in V$.

* $V$ is maximal for the following order relation on local rings inside $K$ whose field of fractions is $K$: $(R, \mathfrak{m}) \le (R', \mathfrak{m}')$ is $R \subset R'$ and $\mathfrak{m}' \cap R = \mathfrak{m}$.

* For $x, y \in V$ there exists $\lambda \in V$ such that $x = \lambda y$ or $y = \lambda x$.

* $V$ is a Bezout local ring.

For such a $V$ the ordered group $\Gamma$ is identified with the group of fractional ideals of $V$ with the order relation given by the inclusion.

**The prime ideals of $V$ form a chain.** This is in bijection with the set of convex subgroups of $\Gamma$ (that form a chain too), to $\mathfrak{p} \subset V$ one associates the convex subgroup

$$H = v(V \setminus \mathfrak{p}) \cup -v(V \setminus \mathfrak{p}).$$

The localization $V_{\mathfrak{p}}$ is a valuation ring with associated ordered group $\Gamma/H$. If $v$ is the valuation defined by $V$ we note $v/H$ the corresponding valuation.

By definition, the rank of $v$ is the height of $V$. We call this the rank of $\Gamma$ too: the length of the chain of convex subgroups. One has the inequality

$$\text{rk}(\Gamma) \le \dim_{\mathbb{Q}} \Gamma \otimes_{\mathbb{Z}} \mathbb{Q}.$$

Moreover, let us remark that for an ideal $I$ of $V$, $I$ is a prime ideal if and only if $\sqrt{I} = I$. Ideals of $V$ correspond to subsets $S$ of $\Gamma_+$ satisfying $x \in S \Rightarrow \forall y, y \le x, y \in S$. If $S$ corresponds to $I$, the convex subgroup $H$ associated to $\sqrt{I}$ is such that $H_+ = \{\gamma \in \Gamma | \forall n \ge 1, n\gamma \notin S\}$.

Recall finally that $\Gamma$ has rank 1 i.e. the only convex subgroups of $\Gamma$ are $(0)$ and $\Gamma$ if and only if $\Gamma$ is isomorphic to a subgroup of $(\mathbb{R}, +)$ as an ordered group. Thus, up to equivalence, rank 1 valuations are given by valuations $v : R \to \mathbb{R} \cup \{\infty\}$. Phrased in another way, those are the same as absolute values i.e. multiplicative semi-norms $|\cdot| : R \to \mathbb{R}_+$.
**Remark 2.3.1.** — More generally, the rank of $\Gamma$ is $\le n$ if and only if there is an embedding of $\Gamma$ inside $\mathbb{R}^n$ equipped with the lexicographic order, see [18].

**Example 2.3.2.** — 1. Let $v$ be a valuation on the ring $R$. Equip $\Gamma_v \times \mathbb{Z}$ with the lexicographic order. Then the formulas

$$R[T] \longrightarrow \Gamma_v \times \mathbb{Z} \\ \sum_{n \ge 0} a_n T^n \longmapsto \begin{cases} \inf_{n \in \mathbb{N}} \{(v(a_n), n)\} \\ \inf_{n \in \mathbb{N}} \{(v(a_n), -n)\} \end{cases}$$

define two valuations $w_1$ and $w_2$ on $R[T]$ of rank rk($v$) + 1. The subgroup $H = \{0\} \times \mathbb{Z}$ is convex and $w_1/H = w_2/H$ that is the so called Gauss valuation

$$\sum_{n \ge 0} a_n T^n \mapsto \inf_{n \in \mathbb{N}} \{v(a_n)\}.$$

2. Let $\Gamma$ be an ordered abelian group and $R$ a ring. The ring of Hahn series $R((T^\Gamma))$ is

$$R((T^\Gamma)) = \{ \sum_{\gamma \in \Gamma} a_\gamma T^\gamma \mid \{\gamma \mid a_\gamma \neq 0\} \text{ is well ordered} \}$$

(here $T^\gamma$ is a formal symbol, we simply consider collections of elements in $R^\Gamma$ whose support is well ordered). Then if we put

$$v(\sum_{\gamma} a_{\gamma} T^{\gamma}) = \inf\{\gamma \mid a_{\gamma} \neq 0\}$$

for such a non-zero element of $R((T^\Gamma))$, this defines a valuation

$$v : R((T^\Gamma)) \to \Gamma \cup \{+\infty\}.$$

3. Let $(R, \mathfrak{m})$ be a regular local ring of dimension $n$. Fix a regular sequence of parameters $\mathfrak{m} = (x_1, \dots, x_n)$. Note $k = R/\mathfrak{m}$. Equip $\mathbb{Z}^n$ with the lexicographic order. Then

$$k[X_1, \dots, X_n] \xrightarrow{\sim} \mathrm{gr}_{\mathfrak{m}}(R)$$

via $X_i \mapsto x_i \mod \mathfrak{m}^2$. There is a valuation

$$v : k[X_1, \dots, X_n] \to \mathbb{Z}^n \cup \{+\infty\}$$

defined by

$$v : \sum_{\alpha \in \mathbb{N}^n} a_\alpha T_1^{\alpha_1} \cdots T_n^{\alpha_n} \mapsto \begin{cases} \inf\{\alpha \mid a_\alpha \neq 0\} & \text{if } \exists \alpha, a_\alpha \neq 0 \\ +\infty & \text{otherwise} \end{cases}.$$

For $x \in R$ non-zero let $k \in \mathbb{N}$ be the integer such that $x \in \mathfrak{m}^k/\mathfrak{m}^{k+1}$ and note $\bar{x} \in \mathfrak{m}^k/\mathfrak{m}^{k+1}$. Then if we set $w(x) = v(\bar{x})$ this defines a valuation

$$w : R \to \mathbb{Z}^n \cup \{+\infty\}.$$

The subgroup $H = (0) \times \mathbb{Z}^{n-1}$ of $\mathbb{Z}^n$ is convex and

$$w/H : R \to \mathbb{Z}$$

is the discrete valuation associated to the regular Cartier divisor $V(x_1) \subset \mathrm{Spec}(R)$. Those are the type of valuations that show up in the theory of Okounkov bodies, see [15, Section 2].
**2.3.1.2. Continuous valuations.** — Let $A$ be a Huber ring. Consider a valuation $v: A \to \Gamma \cup \{\infty\}$. We say that $v$ is continuous if for any $\gamma \in \Gamma$, $\{a \in A \mid v(a) \ge \gamma\}$ is open in $A$. If $A_0$ is a ring of definition of $A$ whose topology is the $(f_1, \dots, f_n)$-adic one, set

$$ \gamma = \inf\{v(f_1), \dots, v(f_n)\} \in \Gamma \cup \{+\infty\}. $$

Then, $v$ is continuous if and only if

$$ n\gamma \xrightarrow{n \to +\infty} +\infty $$

in the sens that for any $\gamma' \in \Gamma$, $n\gamma \ge \gamma'$ for $n \gg 0$.

**2.3.2. The adic spectrum.** — Let us recall the following definition.

**Definition 2.3.3.** — Let $(A, A^+)$ be a Huber pair.

1. The set $\text{Spa}(A, A^+)$ is the set of equivalence classes of continuous valuations $x$ on $A$ satisfying $|f(x)| \le 1$ for all $f \in A^+$.

2. We equip it with the topology generated by the subsets

$$ \{x \in \text{Spa}(A, A^+) \mid |f(x)| \le |g(x)| \ne 0\} $$

with $f, g \in A$.

Those open subsets that define the topology are not the one we will use.

**Definition 2.3.4.** — A rational subset of $X = \text{Spa}(A, A^+)$ is an open subset of the form

$$ X\left(\frac{f_1, \dots, f_n}{g}\right) := \{x \in \text{Spa}(A, A^+) \mid |f_1(x)| \le |g(x)| \ne 0, \dots, |f_n(x)| \le |g(x)| \ne 0\} $$

where $f_1, \dots, f_n \in A$ generate an open ideal of $A$ and $g \in A$.

**Remark 2.3.5.** — If $A$ is Tate the only open ideal is $A$ itself and thus

$$ X\left(\frac{f_1, \dots, f_n}{g}\right) := \{x \in \text{Spa}(A, A^+) \mid |f_1(x)| \le |g(x)|, \dots, |f_n(x)| \le |g(x)|\}. $$

**Lemma 2.3.6.** — A base of the topology of $\text{Spa}(A, A^+)$ is given by the rational open subsets.

*Proof.* — Let $f, g \in A$ be such that $|f(x)| \le |g(x)| \ne 0$. Let $I$ be a finitely generated ideal of a ring of definition defining its topology. Since $I$ is finitely generated, using the continuity of the valuation defined by $x$, there exists $n \ge 1$ such that for any $h \in I^n$, $|h(x)| \le |g(x)|$. If $I^n = (h_1, \dots, h_k)$ then $x$ is contained in the rational subset $\text{Spa}(A, A^+)(\frac{h_1, \dots, h_k}{g})$. $\square$

All usual statements for rational subsets hold in this context:

* they are stable under finite intersections,

* a rational subset of a rational subset is a rational subset,

* $X\left(\frac{f_1, \dots, f_n}{g}\right) = X\left(\frac{f'_1, \dots, f'_n}{g'}\right)$ when the collection $(f'_1, \dots, f'_n, g')$ is sufficiently near from $(f_1, \dots, f_n, g)$ ([37, Lemma 3.10]).

One of the main results of Huber is the following.

**Theorem 2.3.7 ([37], Theorem 2.13.1).** — The topological space $\text{Spa}(A, A^+)$ is spectral.

We will give a proof of this theorem in Section 2.13 using Zariski-Riemann spaces “à la Raynaud” that is different from Huber’s one and more geometric.

The proof by Huber is done in four steps:
1. First Huber proves ([37, Proposition 2.2]) that the valuation spectrum of any ring $R$, $\text{Spv}(R) = \{\text{valuations on } R\} / \sim$ equipped with the topology generated by the subsets $\{x \mid |f(x)| \le |g(x)| \ne 0\}$, is spectral. This is an easy verification.

2. Then Huber defines for an ideal $I$ of a ring $R$ a subset $\text{Spv}(R, I) \subset \text{Spv}(R)$ that is a retract of $\text{Spv}(R)$ and deduces $\text{Spv}(R, I)$ is spectral ([37, Proposition 2.6]).

3. Suppose now that $A$ is a Huber ring. Huber proves then ([37, Theorem 3.1]) that the set of continuous valuations $\text{Cont}(A)$ is a closed subset of $\text{Spv}(A, AA^{\circ\circ})$.

4. The result is finally easily deduced since $\text{Spa}(A, A^+) = \{x \in \text{Cont}(A) \mid \forall f \in A^+, |f(x)| \le 1\}$ is pro-constructible.

**Example 2.3.8.** — Let $K$ be a field equipped with a non-trivial valuation $v$ that is microbial in the sense that there exists $\gamma$ in the valuation group such that

$$n\gamma \xrightarrow{n \to +\infty} +\infty.$$

Let $K^+ \subset K$ be the valuation subring. Equip $K$ with the valuation topology. Then $(K, K^+)$ is a uniform Tate Huber ring. If $\varpi \in K$ is such that

$$nv(\varpi) \xrightarrow{n \to +\infty} +\infty$$

then $\varpi$ is a pseudo-uniformizer. One has $K^\circ$ is a rank 1 valuation ring obtained by localizing $K^+$ at the prime ideal that corresponds to the convex subgroup $H$ whose positive part is

$$\{\gamma \mid \forall n \ge 1, n\gamma < v(\varpi)\}.$$

Moreover $\text{Spa}(K, K^+)$ is a chain (for the specialization order) identified with the open prime ideals of $K^+$ and homeomorphic to $\text{Spec}(K^+/K^{\circ\circ})$. The maximal point of this chain is $\text{Spa}(K, K^\circ)$, the rank 1 valuation. The closed point is the valuation $v$.

Let us point that we will give proofs of point (1) and (2) and (4) in Section 2.13 of the next proposition using Zariski-Riemann spaces that are different from Huber's one. Point (3) is an easy exercise.

**Proposition 2.3.9 ([37, Proposition 3.3 (i), Proposition 3.6, Proposition 3.9], Corollary 2.13.5, 2.13.8 and 2.13.10)**

1. We have $A^+ = \{a \in A \mid \forall x \in \text{Spa}(A, A^+), |a(x)| \le 1\}$.

2. We have $\text{Spa}(A, A^+) = \emptyset$ iff $\hat{A} = 0$ that is to say $\overline{\{0\}} = A$.

3. The morphism $(A, A^+) \to (\hat{A}, \hat{A}^+)$ induces an homeomorphism $\text{Spa}(\hat{A}, \hat{A}^+) \xrightarrow{\sim} \text{Spa}(A, A^+)$.

4. We have $A^\times = \{a \in A \mid \forall x \in \text{Spa}(A, A^+), |a(x)| \ne 0\}$.

Looking at point (3) of this proposition and the definition of Huber's presheaf one can ask why we did not start with a complete Huber pair? Outside of the fact that the theory works well like that, one of the reasons is that one falls sometimes on non-complete Huber pairs naturally in the theory (for example Henselian Huber pairs). For example if $x \in \text{Spa}(A, A^+)$ and $K = A/\text{supp}(x)$ with its valuation ring $K^+$ there is a morphism $(K, K^+) \to (k(x), k(x)^+)$ where $k(x)$ is the residue field of the structure presheaf at $x$. This morphism is an isomorphism on completions and thus $\text{Spa}(k(x), k(x)^+) \xrightarrow{\sim} \text{Spa}(K, K^+)$.

**Remark 2.3.10.** — When $A$ is a Tate ring, points (2) and (3) of Proposition 2.3.9 can be proven as in [6, Theorem 1.2.1, Corollary 1.2.4] using the Berkovich spectrum $\mathcal{M}(A)$ that is identified as a set with the maximal points of $\text{Spa}(A, A^+)$.
## 2.4. Affinoid fields

Before going further we need to fix some terminology.

**Definition 2.4.1.** — An affinoid ring $(K, K^+)$ is an *affinoid field* if $K$ is a field, $K^+$ is a valuation ring for $K$ and

1. (discrete case) the topology of $K$ is the discrete topology

2. (analytic case) or

* there exists $\varpi \in K^+ \setminus \{0\}$ such that $nv(\varpi) \xrightarrow{n \to +\infty} +\infty$ where $v$ is the valuation defined by $V$ and the condition means $\forall \gamma, \exists n \ge 0, nv(\varpi) \ge \gamma$,

* and the topology of $K$ is the one defined by the valuation i.e. $K^+$ is a ring of definition equipped with the $\varpi$-adic topology.

In the analytic case $(K, K^+)$ is a Tate ring. As explained in Example 2.3.8, if $\mathfrak{p} = \sqrt{(\varpi)} \in \text{Spec}(K^+)$ then $K_{\mathfrak{p}}^+ = K^\circ$ is a valuation ring of rank 1 whose valuation with values in $\mathbb{R}$ defines the topology of $K$.

## 2.5. The presheaf of holomorphic functions

Let $(A, A^+)$ be a Huber pair. Let $X = \text{Spa}(A, A^+)$ as a spectral topological space.

**Definition 2.5.1.** — Let $A_0$ be a ring of definition of $A$ whose topology is the *I*-adic one. For $f_1, \dots, f_n \in A$ generating an open ideal in $A$ and $g \in A$ we set

* $A\left\langle \frac{f_1, \dots, f_n}{g} \right\rangle$ is the completion of the Huber ring $A\left[\frac{1}{g}\right]$ with ring of definition $A_0\left[\frac{f_1}{g}, \dots, \frac{f_n}{g}\right]$ equipped with the $IA_0\left[\frac{f_1}{g}, \dots, \frac{f_n}{g}\right]$-adic topology,

* $A\left\langle \frac{f_1, \dots, f_n}{g} \right\rangle^+$ is the integral closure of the topological closure of the image of $A^+\left[\frac{f_1}{g}, \dots, \frac{f_n}{g}\right]$.

Let us remark that, in the preceding definition, the given ring is indeed a Huber ring. In fact, chose $k \ge 1$ such that $I^k \subset A f_1 + \dots + A f_n$. This implies that for $r \ge 1$,

$$I^{kr} \cdot g^{-r} A \subset A \left[ \frac{f_1}{g}, \dots, \frac{f_n}{g} \right].$$

The coherence condition for $A\left[\frac{1}{g}\right]$, $A_0\left[\frac{f_1}{g}, \dots, \frac{f_n}{g}\right]$ and $IA_0\left[\frac{f_1}{g}, \dots, \frac{f_n}{g}\right]$ given after Definition 2.1.1 is thus deduced from the one for $A, A_0$ and $I$.

The couple

$$(6) \qquad \left( A \left\langle \frac{f_1, \dots, f_n}{g} \right\rangle, A \left\langle \frac{f_1, \dots, f_n}{g} \right\rangle^+ \right)$$

is a complete affinoid ring. We can rewrite this ring as

$$A \left\langle \frac{f_1, \dots, f_n}{g} \right\rangle = \hat{A} \langle T_1, \dots, T_n \rangle / \overline{(T_1 g - f_1, \dots, T_n g - f_n)} \left[ \frac{1}{g} \right]$$

and $A\left\langle \frac{f_1, \dots, f_n}{g} \right\rangle^+$ is the integral closure of the image of $\hat{A}^+\langle T_1, \dots, T_n \rangle$.

**Remark 2.5.2.** — (Follow up to Remark 2.3.5) If $A$ is Tate then

$$A \left\langle \frac{f_1, \dots, f_n}{g} \right\rangle = \hat{A} \langle T_1, \dots, T_n \rangle / \overline{(T_1 g - f_1, \dots, T_n g - f_n)}.$$
**Remark 2.5.3.** — If $R$ is a ring and $I$ an ideal in $R$ we can form the blow-up

$$B_I(R) = \mathrm{Proj}(\oplus_{k \ge 0} I^k) \xrightarrow{\pi} \mathrm{Spec}(R).$$

If $I = (f_\alpha)_{\alpha \in A}$ this is a union

$$B_I(R) = \cup_{\alpha} U_{\alpha}$$

where $U_\alpha$ is the open subset where $\pi^{-1}(f_\alpha)$ generates the exceptional Cartier divisor. One has

$$U_\alpha = \mathrm{Spec}(R[\frac{I}{f_\alpha}])$$

with $R[\frac{I}{f_\alpha}] \subset R[\frac{1}{f_\alpha}]$ is the sub-$R$-algebra generated by $(\frac{f_\beta}{f_\alpha})_{\beta \in A}$. The preceding Huber rings defining our rational localizations are thus obtained by completion of those type of open subsets in a blow-up. See Section 2.13 for more on this.

The following lemma is easy.

**Lemma 2.5.4.** — 1. Given a complete Huber pair $(B, B^+)$, a morphism

$$(A\langle\frac{f_1, \dots, f_n}{g}\rangle, A\langle\frac{f_1, \dots, f_n}{g}\rangle^+) \to (B, B^+),$$

is the same as a morphism $u : (A, A^+) \to (B, B^+)$ such that the image of

$$u^* : \mathrm{Spa}(B, B^+) \to \mathrm{Spa}(A, A^+),$$

is contained in the rational subset

$$\mathrm{Spa}(A, A^+)(\frac{f_1, \dots, f_n}{g}).$$

2. The morphism

$$(A, A^+) \to (A\langle\frac{f_1, \dots, f_n}{g}\rangle, A\langle\frac{f_1, \dots, f_n}{g}\rangle^+)$$

induces an homeomorphism

$$\mathrm{Spa}\left(A\left\langle\frac{f_1, \dots, f_n}{g}\right\rangle, A\left\langle\frac{f_1, \dots, f_n}{g}\right\rangle^+\right) \xrightarrow{\sim} \mathrm{Spa}(A, A^+)(\frac{f_1, \dots, f_n}{g}).$$

*Proof.* — This is an application of points (1) and (4) of Proposition 2.3.9. In fact, let $u : (A, A^+) \to (B, B^+)$ be such that the image of $u^*$ is contained in $\mathrm{Spa}(A, A^+)(\frac{f_1, \dots, f_n}{g})$. For all $y \in \mathrm{Spa}(B, B^+)$, $u(g)(y) = g(u^*(y)) \neq 0$ since $u^*(y) \in \mathrm{Spa}(A, A^+)(\frac{f_1, \dots, f_n}{g})$. From this we deduce that

$$u(g) \in B^*.$$

The same type of computation shows that for $i = 1, \dots, n$,

$$\frac{u(f_i)}{u(g)} \in B^+.$$

The morphism $u : (A, A^+) \to (B, B^+)$ thus extends to a morphism of Huber pairs

$$(A[\frac{f_1}{g}, \dots, \frac{f_n}{g}], A[\frac{f_1}{g}, \dots, \frac{f_n}{g}]^+) \to (B, B^+)$$

where $A[\frac{f_1}{g}, \dots, \frac{f_n}{g}]^+$ is the integral closure of $A^+[\frac{f_1}{g}, \dots, \frac{f_n}{g}]$ in $A[\frac{f_1}{g}, \dots, \frac{f_n}{g}] \subset A[\frac{1}{g}]$. This extends canonically to the completions and we deduce point (1) of the lemma. Point (2) is deduced by looking at morphisms toward affinoid fields $(K, K^+)$. $\square$

Let us note the following immediate corollary.

**Corollary 2.5.5.** — Any rational subset is quasi-compact.
From this we deduce that the complete Huber pair (6) depends only canonically on the rational open subset $\text{Spa}(A, A^+)(\frac{f_1, \dots, f_n}{g})$ and not on the choice of $f_1, \dots, f_n, g$. The following definition thus makes sense.

**Definition 2.5.6.** — For $X = \text{Spa}(A, A^+)$ we define presheaves $\mathcal{O}_X$ and $\mathcal{O}_X^+$ on rational open subsets of $X$ by the formulas

$$ \bullet \Gamma(X(\frac{f_1, \dots, f_n}{g}), \mathcal{O}_X) = A\langle\frac{f_1, \dots, f_n}{g}\rangle, $$

$$ \bullet \Gamma(X(\frac{f_1, \dots, f_n}{g}), \mathcal{O}_X^+) = A\langle\frac{f_1, \dots, f_n}{g}\rangle^+. $$

The question wether or not $\mathcal{O}_X$ defines a sheaf will be discussed in section 2.9. Let us remark nevertheless the following property.

**Remark 2.5.7.** — One has (Proposition 2.3.9)

$$ \mathcal{O}_X^+(U) = \{f \in \mathcal{O}_X(U) \mid \forall x \in U, |f(x)| \le 1\} $$

and thus if $\mathcal{O}_X$ is a sheaf it is immediate that $\mathcal{O}_X^+$ is a sheaf.

## 2.6. Analytic points

**2.6.1. Analytic points and their generalizations.** — We will mainly be interested in the so-called analytic adic spaces in this text.

**Definition 2.6.1.** — A point $x \in \text{Spa}(A, A^+)$ is analytic if its support $\{f \in A \mid |f(x)| = 0\}$ in $\text{Spec}(A)$ is not an open ideal. We note $\text{Spa}(A, A^+)_a$ for the set of analytic points.

This forms an open subset of $\text{Spa}(A, A^+)$. In fact, if $A_0$ is a ring of definition whose topology is the $I$-adic one

$$ \text{Spa}(A, A^+)_a = \text{Spa}(A, A^+) \setminus V(AI) $$

and this is thus Zariski open. If $I = (f_1, \dots, f_n)$ one has

$$ \text{Spa}(A, A^+)_a = \bigcup_{i=1}^n \text{Spa}(A, A^+) \left( \frac{f_1, \dots, f_n}{f_i} \right). $$

which gives a natural expression of the open subset of analytic points as a finite union of rational domains. Let us note this in a lemma.

**Lemma 2.6.2.** — The open subset of analytic points $\text{Spa}(A, A^+)_a$ is quasi-compact, a finite union of rational subsets that are spectra of Tate affinoid rings.

**Example 2.6.3.** — 1. Let $A$ be an $I$-adic ring with $I$ finitely generated. Then the open subset of analytic points of $\text{Spa}(A, A)$ is

$$ \text{Spa}(A, A)_a = \text{Spa}(A, A) \setminus V(I) $$

which gives a meaning to the fact that $\text{Spa}(A, A)_a$ is the “generic fiber” of the formal scheme $\text{Spf}(A)$.
2. For example, if $k$ is a field and $k[[x_1, \dots, x_n]]$ is equipped with the $(x_1, \dots, x_n)$-adic topology then

$$
\begin{align*}
\mathrm{Spa}(k[[x_1, \dots, x_n]], k[[x_1, \dots, x_n]])_a &= \mathrm{Spa}(k[[x_1, \dots, x_n]], k[[x_1, \dots, x_n]]) \setminus V(x_1, \dots, x_n) \\
&= \bigcup_{i=1}^n \mathrm{Spa}(k[[x_1, \dots, x_n]], k[[x_1, \dots, x_n]])(\frac{x_1, \dots, x_n}{x_i}) \\
&= \bigcup_{i=1}^n \underbrace{\mathrm{Spa}(k((x_i)), \langle \frac{x_1}{x_i}, \dots, \frac{x_n}{x_i} \rangle, k[[x_i]], \langle \frac{x_1}{x_i}, \dots, \frac{x_n}{x_i} \rangle))}_{\mathbb{B}_{k((x_i))}^{n-1}}
\end{align*}
$$

a union of $n$ closed balls over the non-Archimedean fields $k((x_i))$, $1 \le i \le n$.

3. Consider $\mathbb{Z}_p[[x_1, \dots, x_n]]$ equipped with the $(p, x_1, \dots, x_n)$-adic topology and let $X = \mathrm{Spa}(\mathbb{Z}_p[[x_1, \dots, x_n]], \mathbb{Z}_p[[x_1, \dots, x_n]])_a$. One has

$$
X = \underbrace{\mathring{\mathbb{B}}_{\mathbb{Q}_p}^n}_{\{p \neq 0\}} \cup \underbrace{\mathrm{Spa}(\mathbb{F}_p[[x_1, \dots, x_n]], \mathbb{F}_p[[x_1, \dots, x_n]])_a}_{\{p=0\}}
$$

and one can think of the quasi-compact $X$ as a “compactification” of the open ball $\mathring{\mathbb{B}}_{\mathbb{Q}_p}^n$ in
characteristic zero by adding $n$ closed ball of characteristic $p$ over $\mathbb{F}_p((x_1)), \dots, \mathbb{F}_p((x_n))$,

$$
\partial \mathring{\mathbb{B}}_{\mathbb{Q}_p}^n = \bigcup_{i=1}^n \mathbb{B}_{\mathbb{F}_p((x_i))}^{n-1}.
$$

**Remark 2.6.4 (Follow up to Remark 2.1.2).** — One has $\mathrm{Spa}(A, A^+)_a = \mathrm{Spa}(A^+, A^+)_a = \mathrm{Spa}(A^+, A^+) \setminus V(A^+I) = \mathrm{Spa}(A^+, A^+) \setminus V(A^{\circ\circ})$ where $I$ is an ideal of definition of a ring of definition contained in $A^+$.

Let $x$ be an analytic point of $\mathrm{Spa}(A, A^+)$. We note $\Gamma$ the group of the valuation, $K = \mathrm{frac}(A/\mathrm{supp}(x))$ and $K^+ \subset K$ the valuation ring. Let $A_0$ be a ring of definition of $A$ equipped with the $I$-adic topology, $I = (f_1, \dots, f_n)$. The image of $I$ in $K^+$ generates a non-zero (since $x$ is analytic) ideal $(\bar{\omega})$, $\bar{\omega} \in K^+$. Since

$$
nv(\bar{\omega}(x)) \xrightarrow{n \to +\infty} +\infty,
$$

$K = K^+ [\frac{1}{\bar{\omega}}]$. We equip $K^+$ with the $\bar{\omega}$-adic topology and $K = \lim_{\underset{n \to \infty}{\longrightarrow}} \bar{\omega}^{-n} K^+$ with the inductive limit topology. Then, $(K, K^+)$ is a uniform Tate Huber pair and there is a morphism

$$
(A, A^{+}) \to (K, K^{+}).
$$

The valuation ring $K^+$ has a smallest open prime ideal $q$ that corresponds to the convex subgroup whose intersection with $\Gamma_{\ge 0}$ is

$$
\{\gamma \in \Gamma_{\ge 0} \mid \forall n \ge 1, n\gamma < v(\bar{\omega})\}.
$$

One then has

$$
K^{\circ} = \{x \in K \mid \exists N \ge 1, \forall n \ge 1, v(x^n) \ge v(\bar{\omega}^{-N})\}.
$$

But now, for an element $\gamma \in \Gamma$, the inequality $n(-\gamma) \le Nv(\bar{\omega})$ for all $n \ge 1$ is equivalent to
$n(-\gamma) < v(\bar{\omega})$ for all $n \ge 1$. From this we deduce that

$K^{\circ} = K_{q}^{+}$

is a rank 1 valuation ring. From this considerations we deduce the following result. We note
$\kappa(x) := K$.
**Proposition 2.6.5.** — Let $x$ be an analytic point of $\text{Spa}(A, A^+)$. The set of generalizations of $x$ is a chain that is identified with $\text{Spa}(\kappa(x), \kappa(x)^+)$ where $\text{Spa}(\kappa(x), \kappa(x)^\circ)$ is the maximal generalization of $x$, $\kappa(x)^\circ = \mathcal{O}_{\kappa(x)}$ is a rank 1 valuation ring.

Let us remark the following characterization of analytic points of $\text{Spa}(A, A^+)$.

**Lemma 2.6.6.** — A point $x \in \text{Spa}(A, A^+)$ is analytic if and only if it has an open rational neighborhood that is the spectrum of a Huber Tate ring.

*Proof.* — Let $A_0$ be a ring of definition of $A$ and $I$ an ideal of $A_0$ defining its topology. If $x$ is analytic then $\exists f \in I$ such that $|f(x)| \neq 0$. Since $I$ is of finite type, by continuity of the valuation associated to $x$, there exists $n \geq 1$ such that $I^n = (g_1, \dots, g_k)$ and $x \in \text{Spa}(A, A^+)(\frac{g_1, \dots, g_k}{f}) = \text{Spa}(A(\frac{g_1, \dots, g_k}{f}), A(\frac{g_1, \dots, g_k}{f})^+)$. One easily conclude. $\square$

Finally let us note the following.

**Proposition 2.6.7 ([37]).** — One has $\text{Spa}(A, A^+)_a = \emptyset$ iff $A/\{0\}$ is discrete.

We will give a proof of this in Section 2.13 that is different from Huber's proof, see Corollary 2.13.7.

**2.6.2. Some remark about specializations in valuation spectra.** — Let $A$ be a ring and $\text{Spv}(A)$ be its valuation spectrum ([37], see point (1) after Theorem 2.3.7). We have seen one tool to construct generalizations in $\text{Spv}(A)$. Namely, for $v \in \text{Spv}(A)$, any convex subgroup of $\Gamma_v$ defines a generalization

$$v/H \geq v$$

with $\Gamma_{v/H} = \Gamma_v/H$ and the simple formula $v_{v/H}(a) = v(a)$ modulo $H$. At the level of the valuations rings this corresponds to a localization with respect to the prime ideal defined by the convex subgroup.

There is another tool used by Huber to construct specializations this time. Let $H$ be a convex subgroup of $\Gamma_v$. Define

$v|_H$ by the formula $v|_H(a) = \begin{cases} v(a) & \text{if } v(a) \in H \\ +\infty & \text{if } v(a) \notin H \end{cases}$. Let $V$ be the valuation ring of $v$ inside $\text{Frac}(A/\text{supp}(v))$. There is a biggest prime ideal $\mathfrak{p} \in \text{Spec}(V)$ such that $A \to V_{\mathfrak{p}}$ i.e. the image of $A$ in $\text{Frac}(V)$ is contained in $V_{\mathfrak{p}}$. This prime ideal corresponds to the convex subgroup $c\Gamma_v$ of $\Gamma_v$ generated by $\{v(a) \mid a \in A \text{ and } v(a) \leq 0\}$. Then $c\Gamma_v \subset H \Leftrightarrow v|H$ is a valuation. If this is the case then the support of $v|_H$ is changed, contrary to $v_{v/H}$, and the valuation ring is $V/\mathfrak{p}$ with the valuation on $A$ given by the composite $A \to V_{\mathfrak{p}} \to \text{Frac}(V/\mathfrak{p})$. Moreover one has

$$v|_H \leq v.$$

Via the surjective map

<figure><img src="image_21.png" /></figure>

the first construction produces a vertical specialization $v_H \geq v$ (vertical= in a fiber of the support map). The second construction produces an horizontal specialization $v \geq v_H$ (horizontal= we make a specialization in the base of the fibration $\text{supp}: \text{Spv}(A) \to \text{Spec}(A)$).
**Proposition 2.6.8 (Proposition 1.2.4,**[40]).** — Any specialization in Spv(A) is an horizontal specialization of a vertical specialization.

In this text we focus on analytic adic spaces where as we saw all specializations are vertical. We did not speak about the horizontal one. Nevertheless, they are essential to understand the proofs of Theorem 2.3.7 or Proposition 2.3.9 given by Huber.

## 2.7. Properties of the local rings and the residue fields

**2.7.1. Basic properties.** — Let $(A, A^+)$ be a Huber pair and set $X = \text{Spa}(A, A^+)$.

**Lemma 2.7.1.** — For any $x \in X$, the stalk $\mathcal{O}_{X,x}$ of the structure presheaf at $x$ is a local ring with maximal ideal

$$ \mathfrak{m}_x = \varinjlim_{\substack{U \ni x \\ U \text{ rat. domain}}} \text{supp}_{\mathcal{O}_X(U)}(x). $$

*Proof.* — Let $U \subset X$ be a rational domain containing $x$ and $g \in \mathcal{O}_X(U)$ such that $|g(x)| \neq 0$. Let $I$ be a finite type ideal of a ring of definition of $\mathcal{O}_X(U)$ defining its topology. Using the continuity of the valuation defined by $x$ one deduces that there exists $k \ge 1$ such that for all $f \in I^k$, $|f(x)| \le |g(x)|$. Now, if $I^k = (f_1, \dots, f_n)$ one has $x \in U(\frac{f_1 \dots f_n}{g})$ with

$$ g|_{U(\frac{f_1 \dots f_n}{g})} \in \mathcal{O}_X \left( U \left( \frac{f_1 \dots f_n}{g} \right) \right)^\times. $$

From this we deduce that $\mathcal{O}_{X,x} \setminus \mathfrak{m}_x = \mathcal{O}_{X,x}^\times$ and thus $\mathcal{O}_{X,x}$ is local with maximal ideal $\mathfrak{m}_x$. $\square$

**Definition 2.7.2.** — For $x \in X$ we note

1. $\kappa(x) = \text{Frac}(A/\text{supp}(x))$ and $\kappa(x)^+$ its valuation subring,

2. $k(x) = \mathcal{O}_{X,x}/\mathfrak{m}_x$ is the residue field of the structure presheaf at $x$ and $k(x)^+$ is its valuation subring,

3. $(K(x), K(x)^+) = (\widehat{k(x)}, \widehat{k(x)}^+)$,

where the Huber pair $(\kappa(x), \kappa(x)^+)$, resp. $(k(x), k(x)^+)$, is such that $\kappa(x)^+$, resp. $k(x)^+$, is a ring of definition of $\kappa(x)$, resp. $k(x)$, equipped with the $I.\kappa(x)^+$-adic, resp. $I.k(x)^+$-adic, topology where $I$ is an ideal defining the topology of a ring of definition of $A$ contained in $A^+$.

Those three affinoid rings are affinoid fields in the sense of Definition 2.4.1. There are two cases:

* If $x$ is not analytic then $(\kappa(x), \kappa(x)^+) = (k(x), k(x)^+) = (K(x), K(x)^+)$ are discrete affinoid fields.

* If $x$ is analytic then those are analytic affinoid fields and one verifies easily that

$$ (\widehat{\kappa(x)}, \widehat{\kappa(x)}^+) \xrightarrow{\sim} (\widehat{k(x)}, \widehat{k(x)}^+) = (K(x), K(x)^+). $$

**2.7.2. Henselian properties.** — For any complex analytic space $X$, the local rings $\mathcal{O}_{X,x}$, $x \in X$, are Henselian rings with algebraically closed residue field. If $X = \mathcal{X}^{an}$ where $\mathcal{X}$ is a $\mathbb{C}$-scheme locally of finite type, for any $x \in X = \mathcal{X}(\mathbb{C})$ the local morphism $\mathcal{O}_{\mathcal{X},x} \to \mathcal{O}_{X,x}$ induces an isomorphism $\widehat{\mathcal{O}}_{\mathcal{X},x} \xrightarrow{\sim} \widehat{\mathcal{O}}_{X,x}$. If $Y \to X$ is an étale covering of complex analytic spaces, using that the local rings are Henselian with algebraically closed residue fields, the morphism $Y \to X$ has a section over a topological covering i.e. a covering of $|X|$. Thus, the étale topos of $X$ is the same as the topos of $|X|$.

For schemes the situation is the opposite: the local rings are arbitrary with arbitrary residue fields.
For adic spaces the situation is intermediate between schemes and complex analytic spaces: the local rings are Henselian but the residue fields are arbitrary. Thus, the obstruction to split étale coverings comes from the residue fields.

**Example 2.7.3.** — Let $Y \to X$ be an étale covering of $\mathbb{C}_p$-rigid spaces “à la Tate”. If $x$ is a classical point of $X$, $x \in X(\mathbb{C}_p)$, then $Y \to X$ splits in a neighborhood of $x$. But the obstruction to split the covering over an admissible covering of $X$ comes from the residue fields of the adic space $X^{\mathrm{ad}}$ that are not algebraically closed in general.

Let's come to the basic structure results for local rings and residue fields of adic spaces.

**Proposition 2.7.4.** — For any $x \in X$,

1. The local ring $(\mathcal{O}_{X,x}, \mathfrak{m}_x)$ is Henselian.

2. If $x$ is analytic then $(k(x)^+, (\varpi))$ is Henselian.

*Proof.* — Let $P \in \mathcal{O}_{X,x}[T]$ be a unitary polynomial and $\alpha \in \mathcal{O}_{X,x}$ satisfying $|P(\alpha)(x)| = 0$ and $|P'(\alpha)(x)| \neq 0$. Then one can find a rational subset $U$ containing $x$ such that $P$ is the image of $Q \in \mathcal{O}_X(U)[T]$ and $\alpha$ the image of $\beta \in \mathcal{O}_X(U)$ such that

$$ \frac{Q(\beta)}{Q'(\beta)} \in \mathcal{O}_X(U)^{\circ\circ}. $$

Newton's method then shows that $Q$ has a root in $\mathcal{O}_X(U)$ lifting $\alpha$.

Let $x$ be analytic. The morphism $\mathcal{O}_{X,x}^+ \to k(x)^+$ is surjective. In fact let $U$ be a rational domain containing $x$ and $f \in \mathcal{O}_X(U)$ satisfying $|f(x)| \le 1$. Then, if $V = U(\frac{f,1}{1})$, $x \in V$ and $f|_V \in \mathcal{O}_X(V)^+$. Thus, $(k(x)^+, (\varpi))$ is a quotient of the filtered colimit of Henselian couples (see Lemma 2.7.5)

$$ \lim_{\substack{\longrightarrow \\ U \ni x}} (\mathcal{O}_X(U)^+, (\varpi)). \\ \text{rat. domain} $$

It is thus Henselian.

□

**Lemma 2.7.5.** — For $(A, A^+)$ a complete affinoid Tate ring, the couple $(A^+, (\varpi))$ is Henselian.

*Proof.* — Let $A_0$ be a ring of definition of $A$ contained in $A^+$. Since $A^+ \subset A^\circ$, for any finite collection $x_1, \dots, x_n \in A^+, A_0[x_1, \dots, x_n]$ is again a ring of definition. Thus,

$$ \lim_{\substack{\longrightarrow \\ S \subset A^+ \\ \text{finite}}} (A_0[x]_{x \in S}, (\varpi)) \xrightarrow{\sim} (A^+, (\varpi)) $$

and thus $(A^+, (\varpi))$ is Henselian as a filtered colimit of complete (and thus Henselian) couples. □

**Corollary 2.7.6.** — For any $x \in X$ there is an equivalence of categories

$$ 2 - \lim_{\substack{\longrightarrow \\ U \ni x}} \{\text{finite étale } \mathcal{O}_X(U)\text{-algebras}\} \xrightarrow{\sim} \{\text{finite étale } k(x)\text{-algebras}\} \xrightarrow{\sim} \{\text{finite étale } K(x)\text{-algebras}\}. $$

*Proof.* — This is simply an application of Elkik's approximation ([25, II Theorem 5] for the Noetherian case, [3, Theorem 1.16.23] for the general case). □
## 2.8. Overconvergent open subsets: the Berkovich spectrum

Let $(A, A^+)$ be a Tate ring with $\varpi$ a pseudo-uniformizer. Thus, all points of $X = \text{Spa}(A, A^+)$ are analytic. Our goal is to compute the compact topological space

$$X^B$$

that is the Berkovich quotient of $X$, see Section 1.10. Since all points of $X$ are analytic the generalizations of a point form a chain and this is identified as a set with $X^{max}$, the set of maximal points of $X$. Thus, if we fix $\beta \in ]0, 1[$, as a set

$$X^B = X^{max} = \{|\cdot| : A \to \mathbb{R}_+ \mid \text{continuous valuations s.t. } |\varpi| = \beta\}.$$

**2.8.1. The closure of rational subsets.** — Let us begin with a lemma.

**Lemma 2.8.1.** — Let $f_1, \dots, f_n \in A$ generate $A$ and $g \in A$. Then,

$$\overline{X\left(\frac{f_1, \dots, f_n}{g}\right)} = \bigcap_{k \ge 0} X\left(\frac{\varpi f_1^k, \dots, \varpi f_n^k}{g^k}\right).$$

*Proof.* — Since such a rational subset is constructible, its closure is the set of its specializations (Corollary 1.3.9).

Let $x$ be in the closure of our rational subset. Let $h_i \in K(x)$ be the image of $f_i$ divided by the image of $g$ in $K(x)$. There exists $y \ge x$ such that $h_i \in K(y)^+$ where we have

$$K(x)^{\circ\circ} \subset K(x)^{+} \subset K(y)^{+} \subset K(x)^{\circ}.$$

Since $\varpi \in K(x)^{\circ\circ}$ this implies that for all $k \ge 0$, $\varpi h_i^k \in K(x)^{\circ\circ} \subset K(x)^{+}$. This proves that the left hand set is contained in the right hand one.

Let now $x$ be such that, with the preceding notations, for all $k \ge 0$, $\varpi h_i^k \in K(x)^{+}$. Thus, for all $k$, $\varpi h_i^k \in K(x)^{\circ}$. Since $x^{max} = \text{Spa}(K(x), K(x)^{\circ})$ is a rank one valuation this implies $h_i \in K(x)^{\circ}$ and thus $x^{max} \ge x$ with $x^{max}$ in the rational subset. This proves that the right hand set is contained in the left hand one and finishes the proof. $\square$

**Example 2.8.2.** — The interior of the closed subset $\{|x| < 1\} \subset \text{Spa}(K\langle x \rangle, \mathcal{O}_X\langle x \rangle) = \mathbb{B}_K^1$ is the open ball $\bigcup_{k \ge 0} \{|x^k| \le |\varpi|\}$.

**Corollary 2.8.3.** — A basis of neighborhoods of $\overline{X\left(\frac{f_1, \dots, f_n}{g}\right)}$ is given by the rational subsets $X\left(\frac{\varpi f_1^k, \dots, \varpi f_n^k}{g^k}\right)$, $k \ge 1$.

*Proof.* — Let $U$ be an open subset of $X$ containing $\overline{X\left(\frac{f_1, \dots, f_n}{g}\right)}$. One has

$$\bigcap_{k \ge 0} \left( (X \setminus U) \cap X\left(\frac{\varpi f_1^k, \dots, \varpi f_n^k}{g^k}\right) \right) = \emptyset.$$

Using the compactness of $(X \setminus U)_{\text{cons}}$ we conclude that for $k \gg 0$,

$$(X \setminus U) \cap X\left(\frac{\varpi f_1^k, \dots, \varpi f_n^k}{g^k}\right) = \emptyset.$$

$\square$
**2.8.2. The Berkovich topology.** — The open subsets of the quotient $X^B$ are in bijection with the partially proper open subsets of $X$ that is to say open subsets stable under specialization. We call them the overconvergent open subsets.

**Lemma 2.8.4.** — For an open subset $U$ of $X$ the following are equivalent:

1. $U$ is overconvergent,

2. if $V \subset U$ is quasi-compact open then $\bar{V} \subset U$,

3. if $X(\frac{f_1, \dots, f_n}{g}) \subset U$ then there exists $k \ge 0$ such that $X(\frac{\varpi f_1^k, \dots, \varpi f_n^k}{g^k}) \subset U$.

*Proof.* — The equivalence between (1) and (2) is deduced from the fact that $V$ quasi-compact, $\bar{V}$ is the set of specializations of $V$. The equivalence between (2) and (3) is deduced from Corollary 2.8.3. $\square$

**Remark 2.8.5.** — From point (2) of the preceding lemma we deduce that the quasi-compact overconvergent open subsets are the open and closed subsets of $X$.

**Proposition 2.8.6.** — The topology of $X^B$ is the one defined by Berkovich on $X^{max} = \mathcal{M}(A)$, that is to say the one generated by $\{x \in X^{max} | |f(x)| < |g(x)|\}$ for $f, g \in A$.

*Proof.* — Let $f_1, \dots, f_n \in A$ that generate $A$ as an ideal. We have for $x \in X$,

$$ 
\begin{align*}
\forall i, |f_i(x^{max})| < |g(x^{max})| &\iff \exists l \ge 1, x^{max} \in X\left(\frac{f_1^l, \dots, f_n^l}{\varpi g^l}\right) \\
&\iff \exists l \ge 1, x \in X\left(\frac{f_1^l, \dots, f_n^l}{\varpi g^l}\right) \\
&\iff \exists l \ge 1, \forall k \ge 0, x \in X\left(\frac{\varpi f_1^{kl}, \dots, \varpi f_n^{kl}}{\varpi^k g^{kl}}\right) \\
&\iff x \in \left(\bigcup_{m \ge 0} X\left(\frac{f_1^m, \dots, f_n^m}{\varpi g^m}\right)\right) . \\
&\qquad \operatorname{Int}\left(\{y \in X \mid \forall i, |f_i(y)| < |g_i(y)|\}\right)
\end{align*}
$$

From this and Lemma 2.8.4 we conclude. $\square$

**2.8.3. Overconvergent sheaves.** — Let $\mathcal{F}$ be a sheaf on $X$. Recall that we say $\mathcal{F}$ is overconvergent if for $x, y \in X$ with $x \le y$, $\mathcal{F}_x \xrightarrow{\sim} \mathcal{F}_y$. From the preceding and Proposition 1.10.6 we deduce the following are equivalent for $\mathcal{F}$ a sheaf on $X$:

1. $\mathcal{F}$ is overconvergent,

2. $\mathcal{F}$ comes via pullback of a sheaf on the Berkovich quotient $X \to X^B$,

3. if $U \subset X$ is open $\mathrm{qc}$ and $i: \bar{U} \hookrightarrow X$,

$$ \Gamma(\bar{U}, i^*\mathcal{F}) \xrightarrow{\sim} \Gamma(U, \mathcal{F}), $$

4. if $U \subset X$ is open $\mathrm{qc}$ then

$$ \varinjlim_{U \subset \subset V} \Gamma(V, \mathcal{F}) \xrightarrow{\sim} \Gamma(U, \mathcal{F}) $$

(where by definition $U \subset \subset V$ means $\bar{U} \subset V$),

5. if $f_1, \dots, f_n$ generate $A$ as an ideal and $g \in A$,

$$ \varinjlim_{k \ge 0} \Gamma\left(X\left(\frac{\varpi f_1^k, \dots, \varpi f_n^k}{g^k}\right), \mathcal{F}\right) \xrightarrow{\sim} \Gamma\left(X\left(\frac{f_1, \dots, f_n}{g}\right), \mathcal{F}\right). $$
## 2.9. The sheaf property

**2.9.1. General results.** — Let us put the following definition. The category of rational open subsets form a site and we can speak about sheaves on this site.

**Definition 2.9.1.** — We say the pair $(A, A^+)$ is sheafy if

1. the preceding presheaf $\mathcal{O}_X$ on the rational open subsets of $X = \text{Spa}(A, A^+)$ is a sheaf,

2. if $U$ is a rational subset of $X$ covered by a collection $(V_i)_i$ of rational subsets then the morphism

$$ \mathcal{O}_X(U) \to \prod_{i \in I} \mathcal{O}_X(V_i) $$

is strict.

Since the rational subsets form a base of the topology stable under finite intersections this is equivalent to the fact that $\mathcal{O}_X$ extends to a sheaf of topological rings on $\text{Spa}(A, A^+)$ ([1, Exposé III-Théorème 4.1]). More precisely, for any open subset $U$, let us define

$$ \mathcal{O}_X(U) = \varinjlim_{V \subset U, \text{rational subset}} \mathcal{O}_X(V) $$

equipped with the projective limit topology. Then, the following are equivalent:

1. $(A, A^+)$ is sheafy,

2. $\mathcal{O}_X$ is a sheaf of topological rings on $X$.

Here by a sheaf of topological rings we mean that the correspondence $U \mapsto \mathcal{O}_X(U)$ is a functor from open subsets of $X$ to topological rings

* that is a sheaf of rings after forgetting the topological structure,

* such that if $U = \cup_{i \in I} V_i$ is an open cover then the morphism

$$ \mathcal{O}_X(U) \to \prod_{i \in I} \mathcal{O}_X(V_i) $$

is strict.

Here are some cases when this is known:

1. When $A$ is Tate strongly Noetherian ([38]) in the sense that for all $n \ge 1$, $\hat{A}\langle X_1, \dots, X_n \rangle$ is Noetherian. This contains the case of classical rigid spaces à la Tate i.e. adic spaces locally of finite type over $\text{Spa}(K)$ with $K$ a non-archimedean field.

2. When $\hat{A}$ has a Noetherian ring of definition ([38]).

3. A case more general than the two preceding one is treated in [55].

4. When $A$ is perfectoid ([49]) and more generally sous-perfectoid.

5. When $A$ is Tate stably uniform ([17]). This means that for all $U \subset \text{Spa}(A, A^+)$ a rational open subset, the Huber ring $\mathcal{O}_X(U)$ is uniform. This contains the perfectoid case.

6. In the “discrete case” case: if $A$ is a ring equipped with the discrete topology then $(A, A^+)$ is sheafy for any $A^+$.

**Remark 2.9.2.** — (Follow up to remark 2.5.2) When $A$ is Tate stably uniform or Tate strongly Noetherian one has the following simpler formula for holomorphic functions on a rational open subset: $A\langle f_1, \dots, f_n \rangle = \hat{A}\langle T_1, \dots, T_n \rangle / (T_1 g - f_1, \dots, T_n g - f_n)$ i.e. there is no need to take the closure of the ideal defining the quotient, it is already closed.
**Example 2.9.3 (Discrete case).** — Let $(A, A^+)$ be an affinoid ring with $A$ discrete. The continuous map

$$ \mathrm{supp} : \mathrm{Spa}(A, A^+) \to \mathrm{Spec}(A) $$

is open and surjective,

$$ \mathrm{supp}(\{x \mid |f(x)| \le |g(x)| \ne 0\}) = D(g). $$

Moreover for any rational subset $U \subset \mathrm{Spa}(A, A)$, $\Gamma(U, \mathcal{O}_{\mathrm{Spa}(A, A)}) = \Gamma(\mathrm{supp}(U), \mathcal{O}_{\mathrm{Spec}(A)})$. We deduce that $(A, A^+)$ is sheafy with

$$ \mathcal{O}_{\mathrm{Spa}(A, A^+)} = \mathrm{supp}^{-1}\mathcal{O}_{\mathrm{Spec}(A)}. $$

We will explain the strongly Noetherian and the stable uniform case in the following sections.

**2.9.2. A general dévissage.** — In this section we explain a general common strategy that allows one to prove sheafiness for Tate Huber pairs. More precisely, we seek to prove the following result.

**Theorem 2.9.4.** — Let $C$ be a class of complete Tate Huber pairs that is stable under rational localizations. Suppose that for all $(A, A^+)$ in $C$ and $f \in A$ the sequence

$$ 0 \to A \to A\langle f \rangle \oplus A\langle f^{-1} \rangle \to A\langle f, f^{-1} \rangle \to 0 $$

is exact. Then any $(A, A^+)$ in $C$ is sheafy and $\mathcal{O}_{\mathrm{Spa}(A, A^+)}$ is acyclic.

We will give a proof of this theorem in the following subsections.

**2.9.2.1. About the strictness condition.** — We did not speak about the topology of our rings in the preceding for the following reason. Let us first take a definition.

**Definition 2.9.5.** — Let $A$ be a complete Tate ring. A Banach $A$-module is a complete topological $A$-module $M$ such that there exists $A_0$ a ring of definition of $A$ equipped with a pseudo-uniformizer $\varpi \in A_0$ such that $M$ admits an open bounded sub-$A_0$-module whose topology is the $\varpi$-adic one.

The preceding definition is well behaved in the sense that for $M$ a Banach $A$-module and $A_0$ any ring of definition of $A$ equipped with $\varpi \in A_0$, there exists an open bounded sub-$A_0$-module of $M$ whose topology is the $\varpi$-adic one. For such a sub-$A_0$-module $M_0$,

$$ M = M_0[\frac{1}{\varpi}] $$

since $M_0$ is open. Moreover, since $M_0$ is bounded the topology of $M$ is the inductive limit topology via the formula $M = \lim_{\underset{\varpi}{\longrightarrow}} M_0$.

We have the following elementary result.

**Proposition 2.9.6.** — Let $A$ be a complete Tate ring.

1. Any closed sub-module of a Banach $A$-module is a Banach $A$-module.

2. If $N$ is a closed sub-module of the Banach $A$-module $M$ then $M/N$ is a Banach $A$-module.

3. Banach open mapping theorem holds: any continuous surjective map of Banach $A$-modules is open.

**Proposition 2.9.7.** — Let $A$ be a complete Tate ring and $C^\bullet$ a complex of Banach $A$-modules. Suppose it is exact as a complex of $A$-modules. Then any boundary map in $C^\bullet$ is strict.

*Proof.* — For any index $i \in \mathbb{Z}$, the continuous map of Banach $A$-modules $\partial^i : C^i / \ker \partial^i \to \ker \partial^{i+1}$ is bijective and thus an homeomorphism according to Banach open mapping theorem. $\square$
**Corollary 2.9.8.** — Let $(A, A^+)$ be a complete Tate affinoid ring. Then $(A, A^+)$ is sheafy if and only if $U \mapsto \mathcal{O}_X(U)$ is a sheaf of rings i.e. if $\mathcal{O}_X$ is a sheaf of rings it is automatically a sheaf of topological rings.

We thus get rid of the strictness condition in the definition of a sheafy pair: it is automatic. We can focus on the purely algebraic part of the result.

**2.9.2.2. A general lemma: reduction to a sub-covering.** — Let us begin by devising a general strategy that allows us to replace coverings by finer coverings.

**Lemma 2.9.9.** — Let $\mathcal{C}$ be a class of complete Huber pairs that is stable under rational localizations. Suppose that for any $(A, A^+) \in \mathcal{C}$ and any finite rational cover $\mathcal{U} = (U_i)_i$ of $\mathrm{Spa}(A, A^+)$ there exists a rational cover $\mathcal{V} = (V_j)_j$ such that $\forall j, \exists i, V_j \subset U_i$ and $A \to \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{O}_X)$ is a resolution. Then any $(A, A^+) \in \mathcal{C}$ is sheafy.

*Proof.* — Let $(A, A^+)$ be in the category $\mathcal{C}$. According to the hypothesis, for any rational open subset $U$ of $\mathrm{Spa}(A, A^+)$, if

$$ \check{H}^\bullet(U, \mathcal{O}_X) = \lim_{\substack{\longrightarrow \\ \mathcal{U}' \\ \text{finite rational cover of } U}} \check{H}^\bullet(\mathcal{U}, \mathcal{O}_X) $$

then

$$ (7) \qquad \mathcal{O}_X(U) \xrightarrow{\sim} \check{H}^0(U, \mathcal{O}_X) $$

and

$$ (8) \qquad \check{H}^i(U, \mathcal{O}_X) = 0 $$

for $i > 0$. Equip the sets of rational subsets of $\mathrm{Spa}(A, A^+)$ with the Grothendieck topology generated by the covers that are the finite rational covers. The sheaf associated to the presheaf $\mathcal{O}_X$ on this site is obtained by applying the functor $\check{H}^0$ two times to $\mathcal{O}_X$ ([1, Exposé II-Théorème 3.4]). Equation (7) thus shows that $\mathcal{O}_X$ is a sheaf on this site. Equation (8) then shows that $\mathcal{O}_X$ is acyclic on any rational open subset (usual argument by induction on the cohomological degree based on the Cech-cohomology spectral sequence $E_2^{pq} = \check{H}^p(U, \mathcal{H}^q(\mathcal{O}_X)) \Rightarrow H^{p+q}(U, \mathcal{O}_X)$, see [1, Exposé V-Proposition 4.3]). We can now apply the Čech spectral sequence for a fixed cover to obtain that $A \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{O}_X)$ is a resolution for any finite rational cover $\mathcal{U}$ of $\mathrm{Spa}(A, A^+)$. $\square$

**2.9.2.3. Refining a cover by a standard rational cover.** — We now begin to specialize to the case of Tate rings. The following lemma is a key tool.

**Lemma 2.9.10.** — Let $(A, A^+)$ be a Tate affinoid ring. For any open cover $(U_i)_{i \in I}$ of $X = \mathrm{Spa}(A, A^+)$, there exists an integer $n \ge 1$ and $f_1, \dots, f_n \in A$ generating $A$ as an ideal such that for all $k \in \{1, \dots, n\}$ there exists $i \in I$ such that $X(\frac{f_1 \cdots f_n}{f_k}) \subset U_i$.

*Proof.* — Using the quasi-compacity of $\mathrm{Spa}(A, A^+)$ and the fact that the rational subsets form a base of the topology, we can suppose that $I$ is finite and $U_i$ is a rational subset. Write

$$ U_i = X(\frac{T_i}{g_i}) $$

where $T_i \subset A$ is finite generating $A$ as an ideal. We can suppose, up to replacing $T_i$ by $T_i \cup \{g_i\}$, that

$$ g_i \in T_i. $$

Fix some $i \in I$. Let $\varpi$ be a pseudo-uniformizer of $A$. Choose $N \ge 0$ such that

$$ \varpi^N (g_{i|U_i})^{-1} \in \mathcal{O}_X(U_i)^+. $$

Up to replacing $T_i$ by $\varpi^{-N}T_i$ and $g_i$ by $\varpi^{-N}g_i$ we can suppose that $1 \in T_i$.
We now set

$$S = \left\{ \prod_{i \in I} f_i \mid f_i \in T_i, \exists j \in I, f_j = g_j \right\}.$$

Since $1 \in T_i$ for all $i \in I$, $g_i \in S$ for all $i$. Since $X = \cup_{i \in I} X(\frac{T_i}{g_i})$, for all $x \in X$ there exists $i \in I$ such that $|g_i(x)| \neq 0$. From this we deduce that the ideal generated by $S$ is $A$. We have

$$X = \bigcup_{s \in S} X(\frac{S}{s}).$$

Let now $s = \prod_{i \in I} f_i \in S$ with $f_i \in T_i$ for all $i \in I$ and $f_j = g_j$. One has

$$X(\frac{S}{s}) \subset \bigcap_{f \in T_j} \left\{ x \in X \mid \left| \prod_{i \neq j} f_i \cdot f(x) \right| \leq |s(x)| \right\} = X(\frac{T_j}{g_j}).$$

This proves the result.

<figure><img src="image_22.png" /></figure>

For $f_1, \dots, f_n$ as before we call the cover $(X(\frac{f_1 \dots f_n}{f_i}))_{1 \leq i \leq n}$ of $X$ the standard rational cover generated by $f_1, \dots, f_n$.

**2.9.2.4. Intersection with a cover.** — Let $(A, A^+)$ be a Huber pair and $X = \text{Spa}(A, A^+)$.

**Lemma 2.9.11.** — Let $\mathcal{U} = (U_i)_i$ and $\mathcal{V} = (V_j)_j$ be finite open covers of $X$ by rational subsets and let $\mathcal{U} \times_X \mathcal{V} = (U_i \cap V_j)_{i,j}$. Suppose that

1. $A \to \check{C}^\bullet(\mathcal{U}, \mathcal{O}_X)$ is a resolution,

2. For all $i_1, \dots, i_r, \mathcal{O}_X(U_{i_1} \cap \dots \cap U_{i_r}) \to \check{C}^\bullet(\{U_{i_1} \cap \dots \cap U_{i_r}\} \times_X \mathcal{V}, \mathcal{O}_X)$ is a resolution.

Then,

$$A \to \check{C}^\bullet(\mathcal{U} \times_X \mathcal{V}, \mathcal{O}_X)$$

is a resolution.

*Proof.* — This is deduced from the spectral sequence associated to a double complex.

<figure><img src="image_23.png" /></figure>

**2.9.2.5. Proof of Theorem 2.9.4.** — We finally give the proof of Theorem 2.9.4. Let $(A, A^+)$ be a complete Tate Huber pair. We note $X = \text{Spa}(A, A^+)$. For $f_1, \dots, f_n \in A$ we call the cover

$$(X(\frac{1}{f_1}), X(\frac{1}{f_1})) \times_X \dots \times_X (X(\frac{1}{f_n}), X(\frac{1}{f_n}))$$

of $\text{Spa}(A, A^+)$ the *Laurent cover associated to* $f_1, \dots, f_n \in A$.

**Lemma 2.9.12.** — Under the assumptions of Theorem 2.9.4, for any Laurent cover of $\text{Spa}(A, A^+)$ the associated *Čech complex is a resolution of A*.

*Proof.* — This is proved by induction on $n$ using Lemma 2.9.11.

<figure><img src="image_24.png" /></figure>

**Lemma 2.9.13.** — For any $f_1, \dots, f_n \in A$ generating $A$ as an ideal, there exists a Laurent cover $\mathcal{V}$ of $\text{Spa}(A, A^+)$ such that for any $V \in \mathcal{V}$ the intersection of the rational cover generated by $f_1, \dots, f_n$ with $V$ is a rational cover generated by units of $\mathcal{O}_X(V)$.

*Proof.* — Let $\varpi$ be a pseudo-uniformizer. There exists $N > 0$ such that $\varpi^N \in A^+f_1 + \dots + A^+f_n$. From this we deduce that for all $x \in X$, there exists $i \in \{1, \dots, n\}$ such that $|\varpi^{N+1}(x)| < |f_i(x)|$. Let now $\mathcal{V}$ be the Laurent cover associated to

$$\varpi^{-N-1} f_1, \dots, \varpi^{-N-1} f_n.$$

For each element $V$ of $\mathcal{V}$, the rational cover $(V \cap U_i)_{1 \leq i \leq n}$ of $V$ is generated by units. In fact, if

$$V = \bigcap_{i \notin I} X\left(\frac{\varpi^{-N-1} f_i}{1}\right) \cap \bigcap_{i \in I} X\left(\frac{1}{\varpi^{-N-1} f_i}\right)$$

with $I \subset \{1, \dots, n\}$, $V \neq \emptyset$ implies $I \neq \emptyset$ and moreover $(V \cap U_i)_{1 \leq i \leq n}$ is generated by the elements $(f_i)_{i \in I}$ that are in $\mathcal{O}_X(V)^\times$.

<figure><img src="image_25.png" /></figure>
We will conclude the proof of Theorem 2.9.4 using the following result.

**Lemma 2.9.14.** — Any rational cover generated by units of A can be refined to a Laurent cover.

*Proof.* — In fact, the rational cover generated by $f_1, \dots, f_n \in A^\times$ is refined to the Laurent cover associated to $(f_i f_j^{-1})_{1 \le i < j \le n}$. $\square$

*Proof of Theorem 2.9.4.* According to Lemmas 2.9.9 and 2.9.10 we are reduced to proving that $A \to \check{C}^\bullet(U, \mathcal{O}_X)$ is a resolution for $U$ a standard rational cover. Using Lemmas 2.9.11, 2.9.12 and 2.9.13 we can suppose that our standard rational cover $U$ is generated by units. According to Lemma 2.9.14 there exists a Laurent cover $V$ refining $U$. We can apply Lemmas 2.9.11 and 2.9.12 to deduce that $A \to \check{C}^\bullet(U \times_X V, \mathcal{O}_X)$ is a resolution. Moreover, for any $U_1, \dots, U_r$ in $U$, Lemma 2.9.12 shows that $\mathcal{O}_X(U_1 \cap \dots \cap U_r) \to \check{C}^\bullet(\{U_1 \cap \dots \cap U_r\} \times_X V, \mathcal{O}_X)$ is a resolution. The spectral sequence associated to a double complex then shows that $\check{C}^\bullet(U, \mathcal{O}_X) \to \check{C}^\bullet(U \times_X V, \mathcal{O}_X)$ is a quasi-isomorphism. We conclude. $\square$

**2.9.3. An example: the strongly Noetherian case.** — We are going to prove the following theorem using Theorem 2.9.4.

**Theorem 2.9.15 ([38, Theorem 2.5]).** — If $(A, A^+)$ is an affinoid Tate ring with A strongly Noetherian then $(A, A^+)$ is sheafy and moreover if $X = \text{Spa}(A, A^+)$, $H^i(X, \mathcal{O}_X) = 0$ for $i > 0$.

This result contains as a particular case the one of topologically of finite type affinoid algebras over a non-archimedean field i.e. the case of “classical” Tate rigid spaces.

**2.9.3.1. Background on Noetherian Banach modules.** — Let us recall the following. Here A is a complete Tate ring.

**Proposition 2.9.16.** — Let M be a Banach-A-module. The following are equivalent:

1. M is a Noetherian A-module.

2. Any sub-module of M is closed.

*Proof.* — Suppose (1) is verified. Let $N \subset M$ be a sub-module. Since $\bar{N}$ is of finite type, there exists a surjection

$$f: A^n \to \bar{N}$$

for some $n \ge 0$. Such a morphism is automatically continuous and we can apply Banach's open mapping theorem to deduce that $W = f(A_0^n)$ is a bounded open sub-A$_0$-module of $\bar{N}$. Moreover,

$$W = (W \cap N) + \varpi W$$

by density of $N$ in $\bar{N}$. We deduce that the A$_0$-module of finite type $P = W/W \cap N$ satisfies

$$P = \varpi P.$$

Since $\varpi \in \text{Rad}(A_0)$, Nakayama lemma implies that $P = 0$ and thus $N = \bar{N}$.

Reciprocally, suppose (2) is verified. Let $(M_i)_{i \ge 0}$ be a growing chain of sub-A-modules and note

$$M_\infty = \bigcup_{i \ge 0} M_i.$$

Bair's theorem implies that there exists $i \ge 0$ such that the interior of $M_i$ in $M_\infty$ is non-empty. This implies, using a translations argument, that $M_i$ is open in $M_\infty$ and thus $M_\infty = M_i$. $\square$

Any finite type A-module M has a canonical topology: the quotient topology defined by a surjective morphism $A^n \to M$. One verifies immediately that this topology does not depend on the choice of such a surjection. For this canonical topology, any morphism between finite type A-modules is continuous.

**Proposition 2.9.17.** — Suppose A is Noetherian.
1. Any finite type A-module is a Banach A-module.

2. Any morphism between finite type A-modules is strict.

3. Any ideal of A is closed.

As a consequence of the preceding proposition, if A is a complete Tate strongly Noetherian ring then for $f_1, \dots, f_n \in A$ generating A as an ideal and $g \in A$,

$$A\left\langle \frac{f_1, \dots, f_n}{g} \right\rangle = A\langle T_1, \dots, T_n \rangle / (gT_1 - f_1, \dots, gT_n - f_n)$$

and this ring is strongly Noetherian.

**2.9.3.2.** Flatness of $A\langle X \rangle$, $A\langle f^{-1} \rangle$ and $A\langle f \rangle$. — Let A be a strongly Noetherian complete Tate ring. For M a Banach A-module,

$$(9) \qquad M \hat{\otimes}_A A\langle X \rangle = \left\{ \sum_{i \ge 0} m_i X^i \mid m_i \in M, \lim_{i \to +\infty} m_i = 0 \right\}.$$

From Proposition 2.9.17 we deduce the following Lemma.

**Lemma 2.9.18.** — If M is a finite type A-module then $M \otimes_A A\langle X \rangle = M \hat{\otimes}_A A\langle X \rangle$.

We can now prove our result.

**Proposition 2.9.19.** — For any $f \in A$ the following rings are flat over A:

$$A\langle X \rangle, A\langle X \rangle/(fX - 1), A\langle X \rangle/(X - f).$$

*Proof.* — Let

$$0 \to M' \to M \to M'' \to 0$$

be an exact sequence of finite type A-modules. Using Lemma 2.9.18 the sequence obtained by applying $- \otimes_A A\langle X \rangle$ is an exact sequence

$$M' \hat{\otimes}_A A\langle X \rangle \to M \hat{\otimes}_A A\langle X \rangle \to M'' \hat{\otimes}_A A\langle X \rangle \to 0.$$

But now, using the explicit description (9), one verifies immediately that in fact the sequence

$$0 \to M' \hat{\otimes}_A A\langle X \rangle \to M \hat{\otimes}_A A\langle X \rangle \to M'' \hat{\otimes}_A A\langle X \rangle \to 0$$

is exact. We deduce from this that $A\langle X \rangle$ is flat over A.

For $P \in \{fX - 1, X - f\}$ and M a finite type A-module there is an exact sequence

$$M \hat{\otimes}_A A\langle X \rangle \xrightarrow{\times P} M \hat{\otimes}_A A\langle X \rangle \to M \hat{\otimes}_A (A\langle X \rangle / (P)) \to 0.$$

Let

$$\sum_{i \ge 0} m_i X^i$$

be in the kernel of the left hand map. We use the Equation (9). If $P = fX - 1$ we get $m_0 = 0$ and the induction relation $fm_i = m_{i+1}$ for $i \ge 0$. We deduce from this that our element is zero. For $P = X - f$ we get $fm_0 = 0$ and $fm_{i+1} = m_i$ for $i \ge 0$. Let N be the sub-module of M generated by $(m_i)_{i \ge 0}$. It is finitely generated, let's say

$$N = Am_0 + \dots + Am_d.$$

For any $i \ge 0$ we can write

$$m_{i+d+1} = \sum_{k=0}^{d} a_k m_k.$$

From this we deduce that

$$m_i = f^{d+1} m_{i+d+1} = 0.$$

Our element in the kernel is thus zero.
Thus, for all $M$ of finite type we have an exact sequence

$$0 \to M \hat{\otimes}_A A \langle X \rangle \xrightarrow{\times P} M \hat{\otimes}_A A \langle X \rangle \to M \hat{\otimes}_A (A \langle X \rangle / (P)) \to 0.$$

From this we deduce that

$$\mathrm{Tor}_1^A(M, A\langle X \rangle/(P)) = 0$$

for all such $M$ and thus $A\langle X \rangle/(P)$ is $A$-flat.

□

**2.9.3.3. The sheaf property.** — The complete Tate ring $A$ is assumed to be strongly Noetherian.

**Proposition 2.9.20.** — For any $f \in A$, the Cech complex

$$0 \to A \to A\langle f \rangle \oplus A\langle f^{-1} \rangle \to A\langle f, f^{-1} \rangle \to 0.$$

is exact.

*Proof.* — According to Proposition 2.9.19, the morphism $A \to A\langle f \rangle \times A\langle f^{-1} \rangle$ is flat. According to Corollary 2.13.9, there is a diagram

$$\begin{tikzcd}
\mathrm{Spa}(A, A^+)(\frac{1}{f}) \arrow[r, "\times fT-1"] & \mathrm{Spa}(A, A^+)(\frac{f}{1}) \arrow[r, "\mathrm{supp}"] & \mathrm{Spec}(A\langle\frac{1}{f}\rangle) \arrow[l, "\mathrm{supp}"] \arrow[r, "\mathrm{supp}"] & \mathrm{Spec}(A) \\
\mathrm{Spa}(A, A^+) & & &
\end{tikzcd}$$

We deduce that $A \to A\langle f \rangle \times A\langle f^{-1} \rangle$ is faithfully flat and thus injective. The exactness in the middle and on the right are elementary computations. □

Using Theorem 2.9.4 we thus have proven Theorem 2.9.15.

**2.9.4. Another example: the stably uniform case ([17]).** — We are now going to prove the following result using Theorem 2.9.4.

**Theorem 2.9.21.** — Let $(A, A^+)$ be a stably uniform affinoid Tate ring. Then $(A, A^+)$ is sheafy.

**2.9.4.1. Laurent domains computations.** — Let us begin by exploiting the spectral norm of Theorem 2.1.16.

**Lemma 2.9.22.** — Let $A$ be a uniform complete Tate ring. For any $f \in A$, the morphisms

$$A\langle T \rangle \xrightarrow{\times fT-1} A\langle T \rangle$$

$$A\langle T \rangle \xrightarrow{\times T-f} A\langle T \rangle.$$

are strict injections. In particular, the ideals $(fT - 1)$ and $(T - f)$ of $A\langle T \rangle$ are closed and $A\langle f \rangle = A\langle T \rangle / (fT - 1)$, $A\langle f^{-1} \rangle = A\langle T \rangle / (T - f)$.

*Proof.* — Fix $\varpi$ a pseudo-uniformizer in $A$ and let $\|\cdot\|$ be a power multiplicative norm defining the topology of $A$ and such that

$$\|\varpi a\| = \|\varpi\| \|a\| \text{ and } \|\varpi^{-1} a\| = \|\varpi\|^{-1} \|a\|.$$

Since $A$ is uniform, $A\langle T \rangle$ is uniform. In fact, the Gauss norm on $A\langle T \rangle$ associated to $\|\cdot\|$ on $A$ is power multiplicative (see the proof of Theorem 2.1.16) and defines its topology. We can now use the interpretation of the Gauss norm on $A\langle T \rangle$ as a spectral norm, see Theorem 2.1.16. We use the projection map

$$\mathcal{M}(A\langle T \rangle) \to \mathcal{M}(A).$$

In fact, if $K(x)$ is the completed residue field associated to $x \in \mathcal{M}(A)$, for $P \in A\langle T \rangle$ we have

$$\|P\| = \sup_{x \in \mathcal{M}(A)} \sup_{y \in \mathcal{M}(K(x)\langle T \rangle)} |P_x(y)|$$
where $P_x$ is the image of $P$ in $K(x)\langle T\rangle$. Now we use that the Gauss norm of $K(x)\langle T\rangle$ is multiplicative. We thus have for $P \in A\langle T\rangle$

$$ \|(fT - 1)P\| = \sup_{x \in \mathcal{M}(A)} \left[ \sup\{|f(x)|, 1\} \cdot \|P_x\| \right] $$

and thus

$$ \|P\| \le \|(fT - 1)P\| \le \sup\{\|f\|, 1\} \cdot \|P\|. $$

The same formula holds for $T - f$ instead of $fT - 1$ and this implies the result. $\square$

**Lemma 2.9.23.** — Let $(U_i)_{i \in I}$ be a finite rational cover of $X = \text{Spa}(A, A^+)$ with $A$ a complete stably uniform Tate ring. Then the application

$$ A \to \bigoplus_{i \in I} \mathcal{O}_X(U_i) $$

is a strict embedding.

*Proof.* — Fix $\varpi$ and $\beta \in ]0, 1[$ as in Theorem 2.1.16. For each $i$, the topology of $\mathcal{O}_X(U_i)$ is induced by the supremum norm on $\mathcal{M}(\mathcal{O}_X(U_i)) = U_i^B$. The same goes on for $A$. The result is therefore deduced from the covering $\mathcal{M}(A) = \text{Spa}(A, A^+)^B = \bigcup_{i \in I} U_i^B$. $\square$

**Proposition 2.9.24.** — Let $A$ be a stably uniform complete Tate ring and $f \in A$. The Čech complex

$$ 0 \to A \to A\langle X\rangle/(fX - 1) \oplus A\langle Y\rangle/(f - Y) \to A\langle X, Y\rangle/(fX - 1, f - Y) \to 0 $$

associated to the Laurent cover $\{|f| \le 1\}, \{|f| \ge 1\}$ is exact and all the maps are strict.

*Proof.* — Consider the exact sequence of Tate rings

$$ 0 \to A \xrightarrow{u} A[X]/(fX - 1) \oplus A[Y]/(f - Y) \xrightarrow{v} A[X, Y]/(fX - 1, f - Y) \to 0. $$

According to Lemma 2.9.23 the map $u$ is strict. The map $v$ is open for evident reasons and thus strict. The associated sequence obtained by completion is thus exact. $\square$

**2.9.4.2. Sheafiness.** — According to Theorem 2.9.4 we thus have proven Theorem 2.9.21. We can go even further.

**Proposition 2.9.25.** — If $A$ is a stably uniform complete Tate ring then for $f_1, \dots, f_n \in A$ generating the unit ideal and $g \in A$ we have

$$ A\left\langle \frac{f_1, \dots, f_n}{g} \right\rangle = A\langle T_1, \dots, T_n \rangle / (gT_1 - f_1, \dots, gT_n - f_n). $$

*Proof.* — One can find $\lambda_1, \dots, \lambda_n \in A^\circ$ such that $\sum_{i=1}^n \lambda_i f_i = \varpi$ for a pseudo-uniformizer $\varpi$. If $U$ is our rational subset we then have $\varpi g|_{U}^{-1} \in \mathcal{O}(U)^+$. We then have

$$ A\left\langle \frac{f_1, \dots, f_n}{g} \right\rangle = A\left\langle \frac{1}{\varpi^{-1}g} \right\rangle \left\langle \frac{g^{-1}f_1}{1} \right\rangle \dots \left\langle \frac{g^{-1}f_n}{1} \right\rangle. $$

From Lemma 2.9.22 we deduce that

$$ A\left\langle \frac{f_1, \dots, f_n}{g} \right\rangle = A\langle X, Y_1, \dots, Y_n \rangle / (\varpi^{-1}gX - 1, Y_1 - \varpi^{-1}f_1X, \dots, Y_n - \varpi^{-1}f_nX). $$
There are two morphisms

$$ \begin{gathered} A\langle T_1, \dots, T_n \rangle / (gT_1 - f_1, \dots, gT_n - f_n) \\ X \mapsto \left. \begin{array}{r@{\,}c@{\,}l@{\quad}l} Y_i & \mapsto & T_i & \\ \multicolumn{1}{@{}c@{}}{\left. \begin{array}{l} Y_i \\ X \end{array} \right|} & & \lambda_i T_i & \\ \multicolumn{1}{@{}c@{}}{\downarrow} & & & \\ T_i & \mapsto & Y_i & \end{array} \right\} \\ A\langle X, Y_1, \dots, Y_n \rangle / (\varpi^{-1}gX - 1, Y_1 - \varpi^{-1}f_1X, \dots, Y_n - \varpi^{-1}f_nX) \end{gathered} $$

that are clearly inverse to each other.

□

### 2.9.5. The universally adhesive case. —

#### 2.10. Adic spaces

We refer to the beginning of Section 2.9.1 for the definition of a sheaf of topological rings.

**Definition 2.10.1.** — The category of adic spaces is the subcategory of the category of triplets $(X, \mathcal{O}_X, (v_x)_{x \in X})$ that are locally isomorphic to $\text{Spa}(A, A^+)$ with $(A, A^+)$ a sheafy Huber pair and where

* $X$ is a topological space,

* $\mathcal{O}_X$ a sheaf of topological rings such that $\forall x \in X, \mathcal{O}_{X,x}$ is a local ring,

* $v_x$ is a valuation on the residue field $k(x)$ of $\mathcal{O}_{X,x}$,

with morphisms the morphisms of locally ringed spaces $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ satisfying:

* for any open subset $V$ of $Y$ the morphism $f^* : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}(V))$ is continuous,

* for any $x \in X$ the morphism $k(f(x)) \to k(x)$ is compatible with $v_x$ and $v_{f(x)}$.

We need to verify the following elementary result so that our definition is a good one. This is the analog of the basic result in scheme theory that says that the functor $\text{Spec}(-)$ from rings to locally ringed spaces is fully faithful.

**Lemma 2.10.2.** — The functor $(A, A^+) \mapsto \text{Spa}(A, A^+)$ from sheafy Huber pairs to adic spaces is fully faithful.

The underlying topological space of an adic space is locally spectral. We will mainly be interested in analytic adic spaces.

**Definition 2.10.3.** — An adic space $X$ is analytic if any point has a neighborhood of the form $\text{Spa}(A, A^+)$ with $(A, A^+)$ a sheafy Tate Huber pair.

This definition is justified by the fact that one verifies immediately that, with the terminology of section 2.6, a point of $\text{Spa}(A, A^+)$ is analytic if and only if it has a neighborhood of the form $\text{Spa}(A, A^+)$ with $A$ a Tate Huber ring. We thus have in particular the following.

**Proposition 2.10.4.** — For $x \in X$ an analytic adic space, the set of generalizations of $x$, $X_x$, is a chain that is identified with $\text{Spa}(K(x), K(x)^+)$ where the topology on $K(x)$ is the one deduced from the maximal generalization $\text{Spa}(K(x), K(x)^\circ)$ of $x$, a rank 1 valuation.
## 2.11. Analytic adic spaces

Analytic adic spaces have other nice properties.

**Lemma 2.11.1.** — For $(A, A^+)$ a Tate Huber ring, $\varpi \in A$ a pseudo-uniformizer, and $x \in \text{Spa}(A, A^+)$ one has

$$O_{X,x}^{+}/\varpi \xrightarrow{\sim} K(x)^{+}/\varpi.$$

*Proof.* — For $f \in K(x)^+$ we can lift it to an element $g \in O_X(U)$ where $U$ is an affinoid neighborhood of $x$. Now, let $V = U(\frac{1}{g})$. Then $g|_V \in O_X^+(V)$. Its image in $O_{X,x}^+$ is sent to $f$. Thus, $O_{X,x}^+ \to K(x)^+$ is surjective. Now, if $f \in \varpi K(x)^+$ this means $|g(x)| \le |\varpi(x)|$. We can then shrink $V$ to $W = V(\frac{\varpi}{g})$. One has $g|_W \in \varpi O_X^+(W)$ and we conclude for the injectivity. $\square$

Thus, the $\varpi$-adic completion of $O_{X,x}^+$ is $K(x)^+$. From this we deduce the following.

**Proposition 2.11.2.** — For $x \in X$ an analytic adic space

$$\text{Spa}(K(x), K(x)^+) \xrightarrow{\sim} \lim_{\underset{U \ni x}{\leftarrow}} U$$

in the category of adic spaces.

This sets analytic adic spaces apart from schemes. In fact, if $X$ is a scheme then $\lim_{\underset{U \ni x}{\leftarrow}} U = \text{Spec}(O_{X,x})$, the spectrum of a local ring. Here what shows up is not any local ring but a valuation ring. Let us give an application of this phenomenon that we will use all the time.

**Proposition 2.11.3.** — A morphism between analytic adic spaces is generalizing.

*Proof.* — Let $f: X \to Y$ be such a morphism with $f(x) = y$. This gives rise to a morphism $\text{Spa}(K(x), K(x)^+) \to \text{Spa}(K(y), K(y)^+)$. Since $K(y)^+$ is a valuation and $K(x)^+$ without torsion as a $K(y)^+$-module, $\text{Spec}(K(x)^+) \to \text{Spec}(K(y)^+)$ is flat and thus generalizing. $\square$

**Remark 2.11.4.** — Thus, is $f: X \to Y$ a morphism between qcqs analytic adic spaces then $\text{Im}(f) \subset |Y|$ is pro-constructible stable under generalizations. For example, if $Y$ is a strongly Noetherian adic space and $X \subset Y$ Zariski closed then $|X| \subset |Y|$ is closed stable under generalizations! We are in a very different situation from the scheme case.

This is an important result because of the following.

**Corollary 2.11.5.** — Let $f: X \to Y$ be a qcqs surjective morphism between analytic adic spaces. Then $|f|: |X| \to |Y|$ is a quotient map.

*Proof.* — Apply Lemma 2.11.6. $\square$

This last corollary is one of the starting points of the $v$-topology on perfectoid spaces.

**Lemma 2.11.6.** — If $f: X \to Y$ is a surjective generalizing qc map between spectral spaces then $f$ is a quotient map i.e. the topology on $Y$ is the quotient topology on $X$ by the equivalence relation $X \times_Y X \subset X \times X$.

*Proof.* — Let $V \subset Y$ be a subset such that $f^{-1}(V)$ is open. One has

$$Y \setminus V = f(X \setminus f^{-1}(V))$$

which is thus pro-constructible. Moreover since $V = f(f^{-1}(V))$ with $f^{-1}(V)$ open, $V$ is stable under generalizations. Thus, $Y \setminus V$ is pro-constructible stable under generalizations and thus closed. $\square$
## 2.12. Canonical compactifications

Let $f: X = \text{Spa}(B, B^+) \to \text{Spa}(A, A^+) = Y$ be a morphism of affinoid adic spaces where we suppose that our Huber pairs are sheafy. Define a new Huber pair $(C, C^+)$ by setting $C = B$ and $C^+ = f(A^+) + B^{\circ\circ}$. Set $X^Y = \text{Spa}(C, C^+)$. We then have a diagram

<figure><img src="image_26.png" /></figure>

where the inclusion $j$ is a pro-constructible immersion. In general this is not an open immersion but it happens to be the case in a lot of “standard” situations. For example, if $A$ and $B$ are topologically of finite type over the non-Archimedean field $K$ and $A^+ = A^0$, $B^+ = B^0$ (i.e. we are working with affinoid adic spaces associated to “classical” affinoid Tate spaces). Then, if $f_1, \dots, f_n \in B^\circ$ generate the image of $\tilde{A} \to \tilde{B}$, $X = X^Y(\frac{f_1, \dots, f_n}{1})$ and $j$ is an open immersion.

The fact now is that $X^Y \to Y$ is proper and when the preceding immersion is open we have thus constructed a canonical compactification of $f$ by changing $B^+$. In the classical case of affinoid adic spaces associated to affinoid Tate rigid spaces this compactification does not exist in the world of Tate, it exists only in the more general category of adic spaces. This is an occurrence where the power of considering rings of integral elements $A^+$ more general than the case $A^+ = A^\circ$ shows up in the definition of an adic space.

**Remark 2.12.1.** — The reason why $\text{Spa}(A, A^\circ)$ is smaller than $\text{Spa}(A, A^+)$ is that in general, in a totally ordered abelian group $\Gamma$, an element $\gamma$ is such that $\mathbb{N}\cdot\gamma$ is bounded does not imply $\gamma \ge 0$ unless $\Gamma$ has rank 1. For example, for $\Gamma = \mathbb{Z} \times \mathbb{Z}$ with the lexicographic order, $\mathbb{N}\cdot(0, -1) > (-1, 0)$.

**Example 2.12.2.** — Take $\mathbb{B}_K^1 = \text{Spa}(K\langle T \rangle, \mathcal{O}_K\langle T \rangle)$ the classical 1-dimensional ball over $K$. It represents the functor $(R, R^+) \mapsto R^+$ on $K$-affinoid rings. Its canonical compactification $\mathbb{B}_K^{1,c} = \text{Spa}(K\langle T \rangle, \mathcal{O}_K + K^{\circ\circ}\langle T \rangle)$ represents the functor $(R, R^+) \mapsto R^\circ$. This is in fact a “one point compactification”,

$$ \mathbb{B}_K^{1,c} \setminus \mathbb{B}_K^1 = \{x\} $$

with $x$ being the valuation with value group $\mathbb{R} \times \mathbb{Z}$ equipped with the lexicographic order given by

$$ v\left(\left(\sum_{n \ge 0} a_n T^n\right)(x)\right) = \inf_{n \ge 0} \{(v(a_n), -n)\}. $$

## 2.13. The Zariski-Riemann-Raynaud point of view

The point of view we develop here on adic spectra is the one due to Raynaud. For anyone who really wants to understand well adic spaces, all points of view are important. This part is new and original, not available in the literature.

**2.13.1. The adic spectrum as a Zariski-Riemann space.** — Suppose $(A, A^+)$ is Tate Huber pair and let $\varpi$ be a pseudo-uniformizer. Consider the diagram

<figure><img src="image_27.png" /></figure>

Consider the topological space

$$ \lim_{X' \to X} |X'_s| $$
where the indexing category is the one of proper morphisms $X' \to X$ that are isomorphisms outside $\varpi$ i.e.

$$X'_{\eta} := X' \times_X X_{\eta} \xrightarrow{\sim} X_{\eta},$$

and $X'_s := X' \times_X X_s$. For such an $X' \to X$, its image is closed and contains the dense subset $X_{\eta}$. We deduce that $|X'| \to |X|$ is surjective and thus $|X'_s| \to |X_s|$ is surjective too. The indexing category is cofiltered. In fact for $X' \to X$ and $X'' \to X$ as before, $X' \times_X X''$ is again in this category. If moreover we have a diagram

$$\begin{tikzcd}
  X'' \arrow[r, "\sim"] \arrow[d, "\sim"] & X' \\
  X \arrow[u, "\sim"] & X
\end{tikzcd}$$

we can form

$$X''' = X'' \times_{X' \times_X X', \Delta_{X'}/X} X',$$

the equalizer of our two morphisms, that is again in our indexing category. Let us finally remark that, since we can replace $X' \to X$ by the schematical closure of $X'_{\eta} = X_{\eta}$ inside $X'$, the subcategory of $X' \to X$ such that $X'_{\eta}$ is schematically dense in $X'$ is cofinal. This last category is equivalent to a small category and the projective limit makes sense. This projective limit is then a spectral space equipped with a surjective qc map to $|X_s|$. We will always assume from now on that $X'_{\eta}$ is schematically dense in $X'$.

We now define a map

$$\mathrm{Spa}(A, A^+) \to \lim_{X' \xrightarrow{\sim} X} |X'_s|.$$

Let $x \in \mathrm{Spa}(A, A^+)$. It is given by a valuation ring $V$ and a morphism $f : \mathrm{Spec}(V) \to X$ such that $f^*\varpi \in V \setminus \{0\}$. We have $\mathrm{Frac}(V) = V[\frac{1}{f^*\varpi}]$ by continuity of the valuation $x$ and after inverting $\varpi$ the morphism $f$ litts to a morphism $\mathrm{Spec}(\mathrm{Frac}(V)) \to X_{\eta} = X'_{\eta} \hookrightarrow X'$. We thus have a diagram

$$\begin{tikzcd}
  \mathrm{Spec}(\mathrm{Frac}(V)) \arrow[r] & X' \\
  \downarrow \arrow[d, "\sim"] & \downarrow \\
  \mathrm{Spec}(V) \arrow[r, "\sim"] & X.
\end{tikzcd}$$

The valuation criterion of properness gives a unique morphism $\mathrm{Spec}(V) \to X'$ as in the preceding diagram. This induces a morphism $|\mathrm{Spec}(V/\varpi)| \to |X'_s|$. The image of $x$ is then defined to be the image of the closed point of $\mathrm{Spec}(V/\varpi)$. When $X' \to X$ varies, this defines an element of the projective limit.

**Theorem 2.13.1.** — The preceding map is an homeomorphism

$$\mathrm{Spa}(A, A^+) \xrightarrow{\sim} \lim_{X' \xrightarrow{\sim} X} |X'_s|$$

and thus $\mathrm{Spa}(A, A^+)$ is spectral.

*Proof.* — Let us begin with the injectivity. Let $v$ and $w$ be two distinct elements of $\mathrm{Spa}(A, A^+)$. There exists $a, b \in A$ such that $v(a) \ge v(b)$ and $w(a) < w(b)$. Let us choose $n \ge 1$ such that $v(\varpi^n) > w(a)$, which is possible since $w(a) \ne +\infty$. Let

$$\tilde{X} \to X$$

be the blow-up of the ideal $(a, b, \varpi^n)$. The morphism $\tilde{X} \to X$ is in our category, $\tilde{X}_{\eta} \xrightarrow{\sim} X_{\eta}$. Let $U \subset X_{\eta}$ be the open subset where $(b, \varpi^n)$ generates the exceptional divisor. The morphism $\mathrm{Spec}(V) \to \tilde{X}$ defined by $v$ has its image in $U$. Thus, the image of the closed point of $\mathrm{Spec}(V)$ lies in $U \cap \tilde{X}_s$. The image of the morphism $\mathrm{Spec}(W) \to \tilde{X}$ defined by $w$ is not contained in $U$. Since all points of $\mathrm{Spec}(W)$ are generalizations of its closed point, the image of the closed point
of Spec($W$) lies in $\tilde{X}_s \setminus U$. This prove the injectivity statement.

For the surjectivity, let $\mathcal{X}$ be the ringed space

$$ \lim_{X' \to X} X'. $$

Let $x \in \mathcal{X}_s$ and consider the local ring $\mathcal{O}_{\mathcal{X},x}$. For $I \subset \mathcal{O}_{\mathcal{X},x}$ a finite type ideal that contains a power of $\varpi$, we can find some $X' \to X$ in our category of modifications, an affine open subset $U \subset X'$ such that $x$ maps to an element of $U$, and a finite type ideal $J \subset \mathcal{O}_{X'}(U)$ that contains a power of $\varpi$ that generates $I$ when pulled back to $\mathcal{O}_{\mathcal{X},x}$. We can extend $J$ to a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_{\mathcal{X}',x}$ such that $\varpi^n \mathcal{O}_{\mathcal{X}',x} \subset \mathcal{J}$ when $n \gg 0$. Considering the blow up of $\mathcal{J}$ and pulling back the situation to $\mathcal{X}$ we deduce that $\mathcal{O}_{\mathcal{X},x}$ satisfies the hypothesis of Lemma 2.13.3. Thus,

$$ V = \mathcal{O}_{\mathcal{X},x} / \cap_{n \ge 0} (\varpi^n), $$

which is non-zero since $x \in \mathcal{X}_s$, is a valuation ring in which $v(\varpi^n) \xrightarrow{n \to +\infty} +\infty$. The morphism $A^+ \to V$ defines thus a point of $\mathrm{Spa}(A, A^+)$. This proves that the map of the statement is surjective.

We now have to check that this is an homomorphism. For this, consider the subcategory of the indexing category of modifications of $X$ in our projective limit formed by the admissible blow ups. By definition, those are the blow ups $\tilde{X} \to X$ of a finite type ideal $I$ of $A^+$ containing a power of $\varpi$. They have the following property:

* If $\tilde{X} \to X$ and $\tilde{\tilde{X}} \to X$ are two admissible blow ups then $\tilde{X} \times_X \tilde{\tilde{X}} \to X$ is an admissible blow up.

* If $\mathcal{I} \subset \mathcal{O}_{\tilde{X}}$ is a finite type quasi-coherent sheaf of ideals with $\tilde{X} \to X$ an admissible blow up then the blow up of $\tilde{X}$, $\tilde{\tilde{X}} \to \tilde{X}$ is such that the composite $\tilde{\tilde{X}} \to \tilde{X} \to X$ is an admissible blow up.

The preceding proof works as before to prove that

$$ \mathrm{Spa}(A, A^+) \to \lim_{\tilde{X} \to X} |\tilde{X}_s| $$

is bijective, where here is the indexing category is the one of admissible blow ups. The quasi-compact continuous map of spectral spaces

$$ \lim_{X' \to X} |X'_s| \to \lim_{\tilde{X} \to X} |\tilde{X}_s| $$

is thus bijective. This morphism is moreover specializing since if we have a diagram

$$ \begin{array}{ccc} X' & \to & \tilde{X} \\ \downarrow & & \downarrow \\ X & & \end{array} $$

then the morphism $X' \to \tilde{X}$ is proper. This is thus an homeomorphism.

Thus, it suffices to prove that

$$ \mathrm{Spa}(A, A^+) \to \lim_{\tilde{X} \to X} |\tilde{X}_s| $$

is an homeomorphism. But the exceptional divisors of our blow ups are very ample. If $\tilde{X} \to X$ is the admissible blow up of the ideal $I \subset A^+$ then a basis of the topology of $\tilde{X}$ is the $D^+(f)$ where $f \in I^n$. For $f \in I^n$, the morphism $\mathrm{Spec}(V) \to \mathrm{Spec}(A^+)$ factorizes through $D^+(f)$ if and only if the image of $I^n$ in $V$ is generated by $f$. This means exactly that for all $g \in I^n$, $v(g) \ge v(f)$, which means that $v$ is in the rational subset $\mathrm{Spa}(A, A^+) \langle \frac{I^n}{f} \rangle$. The result is easily deduced. $\square$
**Remark 2.13.2.** — The proof gives in fact that the three projective limits $\lim_{\leftarrow X' \to X} |X'_s|$ where the modification $X' \to X$ is either taken proper, projective or an admissible blow up are homeomorphic. Nevertheless this can be proven directly “in finite level” without taking the projective limit using [22, Corollaire 1.4] for the projective/Noetherian case and [20, Theorem 2.11] for the general case that proves that in fact the category of admissible blow-ups is cofinal among the preceding proper modifications.

**Lemma 2.13.3 (see [29, Section 3]).** — Let $V$ be a local ring equipped with some element $\varpi$ in its maximal ideal satisfying: any finitely generated ideal of $V$ that contains a power of $\varpi$ is principal generated by a regular element i.e. is a Cartier divisor. Then the separation of $V$ for the $\varpi$-adic topology, $V/\cap_{n\ge 0}(\varpi^n)$, if non-zero, is a valuation ring in which any non-zero finite type ideal contains a power of $\varpi$.

*Proof.* — The hypothesis implies that $\varpi$ is a regular element. Now for any $a \in V$ and $n \ge 0$, if we write $(a, \varpi^n) = (\alpha)$ with $\alpha$ regular, then we can find $\lambda, \mu, u, v \in V$ such that

$$ \left\{ \begin{array}{l} \alpha = ua + v\varpi^n \\ a = \lambda\alpha \\ \varpi^n = \mu\alpha. \end{array} \right. $$

We thus have $\alpha = (u\lambda + v\mu)\alpha$. Since $\alpha$ is regular we deduce $u\lambda + v\mu = 1$. Since $V$ is local this implies either $u, \lambda \in V^\times$ or $v, \mu \in V^\times$. Thus, either $(a, \varpi^n) = (a)$ or $(a, \varpi^n) = \varpi^n$. From this we deduce that if $a \notin \cap_{n\ge 0}(\varpi^n)$ then there exists $n \ge 1$ such that $\varpi^n \in (a)$ and $a$ is regular. Thus, any non-zero finitely generated ideal of $V/\cap_{n\ge 0}(\varpi^n)$ is principal.

Now, if $a, b \notin \cap_{n\ge 0}(\varpi^n)$, write $\varpi^k = \lambda a$ and $\varpi^l = \mu b$ for $k, l \gg 0$ and $\lambda, \mu \in V$. If $ab \in \cap_{n\ge 0}(\varpi^n)$ then we can write $ab = c\varpi^{k+l+1}$ with $c \in V$. Using the regularity of $\varpi$, we deduce that $\lambda\mu c\varpi = 1$, which is a contradiction. Thus, $V/\cap_{n\ge 0}(\varpi^n)$ is an integral domain. $\square$

**Corollary 2.13.4.** — For any Huber pair $(A, A^+)$, $\text{Spa}(A, A^+)_a$ and $\text{Spa}(A, A^+)$ are spectral spaces.

*Proof.* — Since one can write

$$ \text{Spa}(A, A^+)_a = \cup_{i=1}^n U_i $$

where for all $i, j, U_i$ and $U_i \cap U_j$ are adic spectra of affinoid Tate rings, we deduce that $\text{Spa}(A, A^+)_a$ is spectral. Now, one verifies that if $X$ is a (T0) topological space that admits an open subset $U \subset X$ such that $U$ and $X \setminus U$ are spectral spaces, then $X$ is a spectral space. One thus has to verify that

$$ \text{Spa}(A/AA^{\circ\circ}, (A/AA^{\circ\circ})^{+}) $$

is spectral. We are thus reduced to proving that $\text{Spa}(R, R^+)$ is spectral for $R$ discrete. Since $\text{Spa}(R, R^+)$ is pro-constructible in $\text{Spv}(R) = \text{Spa}(R, \mathbb{Z}.1)$, it suffices to verify that $\text{Spv}(R)$ is spectral. This is [37, Proposition 2.2] whose proof is elementary. $\square$

**Corollary 2.13.5.** — For $(A, A^+)$ a Huber pair, one has

$$ A^+ = \{a \in A \mid \forall x \in \text{Spa}(A, A^+), |a(x)| \le 1\}. $$

*Proof.* — Let us first suppose that $A$ is Tate. Let $f: X' \to X$ be as in the proof of Theorem 2.13.1. Since $f$ is proper of finite presentation $f_*\mathcal{O}_{X'}$ is a quasicoherent $\mathcal{O}_X$-module of finite type. Let us write $f_*\mathcal{O}_{X'} = \widetilde{B}$ with $B$ a finite $A^+$-algebra. Since $f$ is an isomorphism outside the schematically dense open set $D(\varpi)$, we have $A^+ \subset B \subset A$. Since $A^+$ is integrally closed in $A$ we deduce that $B = A^+$.
Let

$$ \mathcal{X} = \varinjlim_{X' \to X} X' $$

as a ringed space. For $x \in |\mathcal{X}|$ that corresponds to the valuation $v_x \in \text{Spa}(A, A^+)$, an element $a \in A = \mathcal{O}_{\mathcal{X},x}[\frac{1}{\varpi}]$ satisfies $v_x(a) \ge 0$ if and only its image in

$$ \mathcal{O}_{\mathcal{X},x}[\frac{1}{\varpi}]/\cap_{n \ge 0} \mathcal{O}_{\mathcal{X},x}\varpi^n $$

is in

$$ \mathcal{O}_{\mathcal{X},x}/\cap_{n \ge 0} \mathcal{O}_{\mathcal{X},x}\varpi^n. $$

This is equivalent to saying that $a \in \mathcal{O}_{\mathcal{X},x}$. Thus, an element $a \in A$ satisfies $v_x(a) \ge 0$ if and only if there exists some $X' \to X$ and $U \subset X'$ whose pullback to $\mathcal{X}$ is a neighborhood of $x$ such that $a|_{U_n}$ extends to an element of $\mathcal{O}_{X'}(U)$. Using the quasi-compacity of $|\mathcal{X}|$ we deduce that if $a$ satisfies

$$ \forall x \in \text{Spa}(A, A^+), |a(x)| \le 1 $$

then there exists some $X' \to X$ such that $a \in \Gamma(X', \mathcal{O}_{X'})$. This finishes the proof when $A$ is a Tate ring.

When the ring $A$ is discrete the statement is immediately reduced to the fact that if $R$ is a normal integral domain then $R$ is the intersection of the valuation rings of $\text{Frac}(R)$ containing $R$.

Let now $A$ be any Huber ring. Let $A_0$ be a ring of definition contained in $A^+$ and $I$ an ideal of finite type of $A_0$ defining its topology. Let $X = \text{Spec}(A^+)$ and

$$ \pi : \widetilde{X} \to X $$

be the blow up of the ideal $A^+I$. Let

$$ \sigma : \widetilde{\widetilde{X}} \to \widetilde{X} $$

be the normalization of $\widetilde{X}$ inside the complementary of the exceptional Cartier divisor, that is to say $X \setminus V(A^+I)$. There is a diagram

<figure><img src="image_28.png" /></figure>

If $a \in A$ satisfies $|a(x)| \le 1$ for all $x \in \text{Spa}(A, A^+)a$ then according to the Tate ring case treated before,

$$ f^*a \in \Gamma(\widetilde{\overline{X}}, \mathcal{O}_{\widetilde{\overline{X}}}). $$

But now, since $\sigma$ is integral $\sigma_* \mathcal{O}_{\widetilde{\overline{X}}}$ is a filtered colimit of quasi-coherent $\mathcal{O}_{\widetilde{\overline{X}}}$-algebras of finite type. Since $\pi$ is proper of finite presentation we deduce that $(\pi\sigma)_* \mathcal{O}_{\widetilde{\overline{X}}}$ is an integral quasi-coherent sheaf of $\mathcal{O}_{\widetilde{\overline{X}}}$-algebras. From this we deduce that $f^*a$ lies in the normalization of $X$ inside $X \setminus V(A^+I)$. This means concretely that there exists $b_1, \dots, b_n \in A^+$ such that

$$ f^*(a^n + b_1 a^{n-1} + \dots + b_{n-1} a + b_n) = 0. $$

We deduce from this that $a^n + b_1 a^{n-1} + \dots + b_{n-1} a + b_n$ is killed by a power of $I$. Now, the case of a discrete Huber ring treated before shows that the image of $A$ in $A/AI$ lies in the integral closure of the image of $A^+$ in $A/AI$. There exists thus $c_1, \dots, c_m \in A^+$ such that

$$ a^m + c_1 a^{m-1} + \dots + c_{m-1} a + c_m \in AI. $$

Thus, for $k \gg 0$,

$$ (a^n + b_1 a^{n-1} + \dots + b_{n-1} a + b_n)(a^m + c_1 a^{m-1} + \dots + c_{m-1} a + c_m)^k = 0. $$

From this we deduce that $a$ is integral over $A^+$ and thus $a \in A^+$. $\square$

<figure><img src="image_29.png" /></figure>
**2.13.2. The specialization map and applications.** — Let $(A, A^+)$ be a Tate ring and $\varpi$ a pseudo-uniformizer. To any point $x \in \text{Spa}(A, A^+)$ there is associated a morphism $\text{Spec}(V_x) \to \text{Spec}(A^+)$ where $V_x$ is a valuation ring. We note

$$ \text{sp}(x) \in \text{Spec}(A^+/\varpi) $$

the image of the closed point of $\text{Spec}(V_x)$.

**Proposition 2.13.6.** — The specialization map $\text{sp}: \text{Spa}(A, A^+) \to \text{Spec}(A^+/\varpi)$ is a surjective specializing quasi-compact continuous map between spectral spaces.

*Proof.* — For $f \in A^+$ with image $\bar{f}$ in $A^+/\varpi$,

$$ \text{sp}^{-1}(D(\bar{f})) = \{x \in \text{Spa}(A, A^+) \mid |f(x)| = 1\}. $$

We deduce that $\text{sp}$ is quasi-compact continuous.

The surjectivity is deduced from Theorem 2.13.1 since the transition maps in the projective limit are surjective. Since the transition map in the projective limit are specializing we deduce that $\text{sp}$ is specializing. $\square$

**Corollary 2.13.7.** — Let $(A, A^+)$ be a Huber pair. Then, $\text{Spa}(A, A^+)_a = \emptyset$ if and only if $A/\{0\}$ is discrete.

*Proof.* — By surjectivity of the specialization map, if $A$ is Tate then $\text{Spa}(A, A^+) = \emptyset$ if and only if

$$ \text{Spec}(A^+/\varpi) = \emptyset. $$

But this is equivalent to $\varpi \in (A^+)^\times$. Since $A^+ \subset A^\circ$ this implies $\varpi^{-1} \in A^\circ$. Now, if $A_0$ is a ring of definition of $A$, there exists $N \ge 0$ such that for all $n \ge 0$, $\varpi^{-n} \in \varpi^{-N}A_0$. Taking $n = N + 1$ we find that $\varpi \in A_0^\times$. Thus, $A_0 = A = \{0\}$.

In general, if $I = (f_1, \dots, f_n)$ is an ideal of a ring of definition $A_0$ defining its topology, let

$$ \pi: \tilde{X} \to X = \text{Spec}(A_0) $$

be the blow up of $I$. If $R$ is a ring and $f \in R$ a regular element, the $f$-adic completion of $R$ is $0$ if and only if $D(f) \subset \text{Spec}(R)$ is closed. Thus, if $\text{Spa}(A, A^+)_a = \emptyset$ then

$$ \tilde{X} \setminus \pi^{-1}(V(I)) $$

is closed in $\tilde{X}$ according to the preceding Tate ring case. Since $\pi$ is proper we deduce that $\text{Spec}(A_0) \setminus V(I)$ is closed in $\text{Spec}(A_0)$. This is equivalent to saying that there exists an idempotent $a \in A_0$ such that $V(I) = V(e)$ or equivalently $\sqrt{I} = \sqrt{(e)}$. Thus, the topology of $A_0$ is the $e$-adic one. Since $e^2 = e$, $\{0\} = \cap_{n \ge 1} (e^n) = (e)$ is open and we deduce $A/\{0\}$ is discrete. $\square$

**Corollary 2.13.8.** — Let $(A, A^+)$ be a Huber pair. Then, $\text{Spa}(A, A^+) = \emptyset$ if and only if $\{0\} = A$. *Proof.* — Suppose $\text{Spa}(A, A^+) = \emptyset$. We can suppose $A$ is separated. According to Corollary 2.13.7, $A$ is discrete. If $A \ne 0$ we can choose a maximal ideal $\mathfrak{m}$ of $A$. We can moreover choose a valuation ring $V$ of $A/\mathfrak{m}$ containing $A^+/A^+ \cap \mathfrak{m}$. Then, $(A, A^+) \to (A/\mathfrak{m}, V)$ defines a point in $\text{Spa}(A, A^+)$. We thus deduce that $A = 0$. $\square$

Let us point the following immediate corollary.

**Corollary 2.13.9.** — For $(A, A^+)$ a Huber pair, the image of the support map $\text{supp}: \text{Spa}(A, A^+) \to \text{Spec}(A)$ is the set of closed prime ideals of $A$.

**Corollary 2.13.10.** — Let $(A, A^+)$ be a complete Huber pair and $a \in A$. One has $a \in A^\times$ if and only if for all $x \in \text{Spa}(A, A^+)$, $|a(x)| \ne 0$.
*Proof.* — Let $a \in A$ satisfy $|a(x)| \neq 0$ for all $x \in \text{Spa}(A, A^+)$. Since $A$ is complete, $1 + A^{\circ\circ} \subset A^\times$ that is thus open inside $A$. We deduce that any maximal ideal of $A$ is closed. Let $\mathfrak{m}$ be such a maximal ideal. Let $(A/\mathfrak{m})^+$ be the integral closure of the image of $A^+$ in $A/\mathfrak{m}$. Since $A/\mathfrak{m}$ is separated, according to Corollary 2.13.8, $\text{Spa}(A/\mathfrak{m}, (A/\mathfrak{m})^+) \neq \emptyset$. Using the map

$$ \text{Spa}(A/\mathfrak{m}, (A/\mathfrak{m})^+) \to \text{Spa}(A, A^+) $$

one can thus find $x \in \text{Spa}(A, A^+)$ such that $\text{supp}(x) = \mathfrak{m}$ and thus $a \notin \mathfrak{m}$ since $|a(x)| \neq 0$. $\square$

## 2.14. Other structure sheaves on the adic spectrum: Henselian rigid spaces

### 2.14.1. Henselization. —

**2.14.1.1. Background on Henselian pairs.** — Let $(A, I)$ be a pair where $A$ is a ring and $I$ an ideal of $A$. We note $X = \text{Spec}(A)$ and $Y = V(I) \hookrightarrow X$. Recall that we say that $(A, I)$ is Henselian if

$$ I \subset \text{Rad}(A) $$

i.e. the only neighborhood of $Y$ in $X$ is $X$ itself, and one of the following equivalent properties is satisfied:

1. For any finite $A$-algebra $B$,

$$ \{\text{open/closed subsets of } \text{Spec}(B)\} \xrightarrow{\sim} \{\text{open/closed subsets of } \text{Spec}(B/IB)\}. $$

2. For any $P \in A[T]$ unitary and $x \in A/I$ satisfying $P(x) = 0$, $P'(x) \in (A/I)^\times$, there exists $a \in A$ such that $a \equiv x \pmod I$ and $P(a) = 0$.

3. For any $U \to X$ étale with $U$ affine, the map

$$ \{\text{sections of } U \to X\} \to \{\text{sections of } U \times_X Y \to Y\} $$

is surjective.

If $(A, I)$ is Henselian then one has the following additional property: reduction modulo $I$ induces a bijection

$$ \{\text{finite étale } A\text{-algebras}\} \xrightarrow{\sim} \{\text{finite étale } A/I\text{-algebras}\}. $$

**2.14.1.2. Zariski localization around a closed subset.** — Let $(A, I)$ be a pair as before. Before looking at étale neighborhoods of $Y$ in $X$ let us look at the Zaiski neighborhoods. For this let $S = 1 + I$, a multiplicative subset of $A$. One has

$$ S^{-1}I \subset \text{Rad}(S^{-1}A) $$

since for $a, f \in I$, $1 + \frac{a}{1+f} = \frac{1+a+f}{1+f} \in (S^{-1}A)^\times$. We deduce from this that the open subsets $D(f), f \in S$, form a basis of neighborhoods of $Y$ in $X$. We have

$$ \begin{aligned} S^{-1}A &= \varinjlim_{f \in S} \Gamma(D(f), \mathcal{O}_X) \\ &= \varinjlim_{\substack{U \supset Y \\ \text{open}}} \Gamma(U, \mathcal{O}_X) \end{aligned} $$

and $\text{Spec}(S^{-1}A) = \bigcap_{U \supset Y} U = Y^{\text{gen}}$. We thus have

$$ i_* i^{-1} \mathcal{O}_X = \mathcal{S}^{-1} \mathcal{O}_X $$

with $\mathcal{S} = 1 + \tilde{I}$. For the next lemma we follow [52, Proposition B.1.4]. The statement is simple but the proof is slightly more complicated than what we could expect.