2.2. Berkovich spaces of schemes [02IM]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

2.2. Berkovich spaces of schemes

In this section we recall Berkovich’s theory of analytic spaces. We will not present the most general theory developed in [Ber90] but we will content ourselves with the analytic spaces associated to algebraic varieties, that are simpler to define and enough for our purposes.

Let $K$ be a field complete with respect to a nontrivial non-Archimedean absolute value $|\cdot|$ . Such fields will be called non-Archimedean fields. Let $K^{\circ}=\{\alpha\in K\mid|\alpha|\leq 1\}$ be the valuation ring, $K^{\circ\circ}=\{\alpha\in K\mid|\alpha|<1\}$ the maximal ideal and $k=K^{\circ}/K^{\circ\circ}$ the residue field.

Let $X$ be a scheme of finite type over $K$ . Following [Ber90, §1 and Remark 3.4.2], we can associate an analytic space $X^{{\text{\rm an}}}$ to the scheme $X$ as follows. First assume that $X=\operatorname{Spec}(A)$ , where $A$ is a finitely generated $K$ -algebra. Then, the points of $X^{{\text{\rm an}}}$ are the multiplicative seminorms of $A$ that extend the absolute value of $K$ , see [Ber90, §1.1]. Every element $a$ of $A$ defines a function $|a(\cdot)|\colon X^{{\text{\rm an}}}\to\mathbb{R}_{\geq 0}$ given by evaluation of the seminorm. The topology of $X^{{\text{\rm an}}}$ is the coarsest topology that makes the functions $|a(\cdot)|$ continuous for all $a\in A$ .

To each point $p\in X^{{\text{\rm an}}}$ we attach a prime ideal

\mathfrak{p}_{p}=\{a\in A\mid|a(p)|=0\}.

This induces a map $\pi\colon X^{{\text{\rm an}}}\to X$ defined as $\pi(p)=\mathfrak{p}_{p}$ . The point $p$ is a multiplicative seminorm on $A$ and so it induces a non-Archimedean absolute value on the field of fractions of $A/\mathfrak{p}_{p}$ . We denote by $\mathscr{H}(p)$ the completion of this field with respect to that absolute value.

Let $U$ be an open subset of $X^{{\text{\rm an}}}$ . An analytic function on $U$ is a function

f\colon U\longrightarrow\coprod_{p\in{U}}\mathscr{H}(p)

such that, for each $p\in U$ , $f(p)\in\mathscr{H}(p)$ and there is an open neigborhood $U^{\prime}\subset U$ of $p$ with the property that, for all $\varepsilon>0$ , there are elements $a,b\in A$ with $b\not\in\mathfrak{p}_{q}$ and $|f(q)-a(q)/b(q)|<\varepsilon$ for all $q\in U^{\prime}$ . The analytic functions form a sheaf, denoted $\mathcal{O}_{X^{{\text{\rm an}}}}$ , and $(X^{{\text{\rm an}}},\mathcal{O}_{X^{{\text{\rm an}}}})$ is a locally ringed space [Ber90, §1.5 and Remark 3.4.2]. In particular, every element $a\in A$ determines an analytic function on $X^{{\text{\rm an}}}$ , also denoted $a$ . The function $|a(\cdot)|$ can then be obtained by composing $a$ with the absolute value map

|\cdot|\colon\coprod_{p\in X^{{\text{\rm an}}}}\mathscr{H}(p)\longrightarrow\mathbb{R}_{\geq 0},

which justifies its notation.

Now, if $X$ is a scheme of finite type over $K$ , the analytic space $X^{{\text{\rm an}}}$ is defined by gluing together the affine analytic spaces obtained from an affine open cover of $X$ . If we want to stress the base field we will denote $X^{{\text{\rm an}}}$ by $X_{K}^{{\text{\rm an}}}$ .

Let $K^{\prime}$ be a complete extension of $K$ and $X^{\text{\rm an}}_{K^{\prime}}$ the analytic space associated to the scheme $X_{K^{\prime}}$ . There is a natural map $X_{K^{\prime}}^{{\text{\rm an}}}\to X_{K}^{{\text{\rm an}}}$ defined locally by restricting seminorms.

Definition 2.6.

A rational point of $X_{K}^{\text{\rm an}}$ is a point $p\in X^{{\text{\rm an}}}$ satisfying $\mathscr{H}(p)=K$ . We denote by $X^{\text{\rm an}}(K)$ the set of rational points of $X^{{\text{\rm an}}}$ . More generally, for a complete extension $K^{\prime}$ of $K$ , the set of $K^{\prime}$ -rational points of $X^{{\text{\rm an}}}$ is defined as $X^{\text{\rm an}}(K^{\prime})=X^{\text{\rm an}}_{K^{\prime}}(K^{\prime})$ . There is a map $X^{{\text{\rm an}}}(K^{\prime})\to X^{{\text{\rm an}}}$ , defined by the composing the inclusion $X^{{\text{\rm an}}}(K^{\prime})\hookrightarrow X_{K^{\prime}}^{{\text{\rm an}}}$ with the map $X_{K^{\prime}}^{{\text{\rm an}}}\to X_{K}^{{\text{\rm an}}}$ as above. The set of algebraic points of $X^{{\text{\rm an}}}$ is the union of $X^{{\text{\rm an}}}(K^{\prime})$ for all finite extensions $K^{\prime}$ of $K$ . Its image in $X^{{\text{\rm an}}}$ is denoted $X^{{\text{\rm an}}}_{{\text{\rm alg}}}$ . We have that $X^{{\text{\rm an}}}_{{\text{\rm alg}}}=\{p\in X|\,[\mathscr{H}(p):K]<\infty\}$ .

The basic properties of $X^{{\text{\rm an}}}$ are summarized in the following theorem.

Theorem 2.7.

Let $X$ be a scheme of finite type over $K$ and $X^{{\text{\rm an}}}$ the associated analytic space.

(1)

$X^{{\text{\rm an}}}$ is a locally compact and locally arc-connected topological space.
(2)

$X^{{\text{\rm an}}}$ is Hausdorff (respectively compact and Hausdorff, arc-connected) if and only if $X$ is separated (respectively proper, connected).
(3)

The map $\pi\colon X^{{\text{\rm an}}}\to X$ is continuous. A locally constructible subset $T\subset X$ is open (respectively closed, dense) if and only if $\pi^{-1}(T)$ is open (respectively closed, dense).
(4)

Let $\psi\colon X\longrightarrow Y$ be a morphism of schemes of finite type over $K$ and $\psi^{{\text{\rm an}}}\colon X^{{\text{\rm an}}}\longrightarrow Y^{{\text{\rm an}}}$ its analytification. Then $\psi$ is flat (respectively unramified, étale, smooth, separated, injective, surjective, open immersion, isomorphism) if and only if $\psi^{{\text{\rm an}}}$ has the same property.
(5)

Let $K^{\prime}$ be a complete extension of $K$ . Then the map $\pi_{K^{\prime}}:X^{{\text{\rm an}}}_{K^{\prime}}\to X_{K^{\prime}}$ induces a bijection between $X^{{\text{\rm an}}}(K^{\prime})$ and $X(K^{\prime})$ .
(6)

Set $X_{{\text{\rm alg}}}=\{p\in X|\,[K(p):K]<\infty\}$ . Then $\pi$ induces a bijection between $X^{{\text{\rm an}}}_{{\text{\rm alg}}}$ and $X_{{\text{\rm alg}}}$ . The subset $X^{{\text{\rm an}}}_{{\text{\rm alg}}}\subset X^{{\text{\rm an}}}$ is dense.

Proof.

The proofs can be found in [Ber90] and the next pointers are with respect to the numeration in this reference: (1) follows from Theorem 1.2.1, Corollary 2.2.8 and Theorem 3.2.1, (2) is Theorem 3.4.8, (3) is Corollary 3.4.5, (4) is Proposition 3.4.6, (5) is Theorem 3.4.1(i), while (6) follows from Theorem 3.4.1(i) and Proposition 2.1.15. ∎

Example 2.8.

Let $M$ be a finitely generated free $\mathbb{Z}$ -module of rank $n$ . Consider the associated group algebra $K[M]$ and the algebraic torus $\mathbb{T}_{M}=\operatorname{Spec}(K[M])$ . The corresponding analytic space $\mathbb{T}_{M}^{{\text{\rm an}}}$ is the set of multiplicative seminorms of $K[M]$ that extend the absolute value of $K$ . This is an analytic group. We warn the reader that the set of points of an analytic group is not an abstract group, hence some care has to be taken when speaking of actions and orbits. The precise definitions and basic properties can be found in [Ber90, §5.1].

Its analytification $\mathbb{T}_{M}^{{\text{\rm an}}}$ is an analytic torus as in [Ber90, §6.3]. The subset

\mathbb{S}^{{\text{\rm an}}}=\{p\in\mathbb{T}_{M}^{{\text{\rm an}}}|\,|\chi^{m}(p)|=1\text{ for all }m\in M\}.

is a compact subgroup, called the compact torus of $\mathbb{T}_{M}^{{\text{\rm an}}}$ .

Remark 2.9.

Not every analytic space in the sense of Berkovich can be obtained by the above procedure. The general theory is based on spectra of affinoid $K$ -algebras, that provide compact analytic spaces that are the building blocks of the more general analytic spaces.