A.1 Berkovich spectrum [03XB]

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A.1 Berkovich spectrum

We refer the reader to [Be1] for the general definition of an analytic space and more details. In this Appendix we restrict ourselves to analytic spaces associated with algebraic varieties (although we use the general definition in the paper as well).

Let $R=R/K$ be a commutative unital finitely generated $K$ -algebra. The underlying set of the Berkovich spectrum $Spec^{an}(R):=Spec^{an}(R/K)$ can be defined in two ways. First one uses valuations (or, equivalently, multiplicative seminorms).

Definition 15

(Valuations) A point $x$ of $Spec^{an}(R/K)$ is an additive valuation

val_{x}:R\to{\bf R}\cup\{+\infty\}

extending $val:=val_{K}$ , i.e. it is a map satisfying the conditions

•

$val_{x}(r+r^{\prime})\leq\max(val_{x}(r),val_{x}(r^{\prime}))$ ;
•

$val_{x}(rr^{\prime})=val_{x}(r)+val_{x}(r^{\prime})$ ;
•

$val_{x}(\lambda)=val_{K}(\lambda)$

for all $r,r^{\prime}\in R$ and all $\lambda\in K$ .

Having a valuation and a real number $q_{0}\in(0,1)$ one can define the multiplicative seminorm $|a|=q_{0}^{val_{K}(a)},a\in R$ . In particular, in the previous definition one can take seminorms $|\cdot|_{x}$ instead of valuations $val_{x}(\cdot)$ . The reader has noticed that in the main body of the paper, for $R=K$ we often took $|a|=e^{-val(a)}$ . It is easy to translate the definition of Berkovich spectrum to the language of multiplicative seminorms. We use it freely in the paper.

The second way to define $X^{an}$ uses evaluations (characters).

Definition 16

(Evaluation maps) A point $x$ of $Spec^{an}(R/K)$ is an equivalence class of homomorphisms of $K$ -algebras

eval_{x}\,:\,R\to K_{x}\,\,,

where $K_{x}\supset K$ is a complete field equipped with a non-archimedean valuation, which extends the valuation $val_{K}$ , and such that $K_{x}$ is generated by the closure of the image of $eval_{x}$ .

The field $K_{x}$ is determined by $x\in X^{an}$ in a canonical way. We define for $r\in R$ and $x\in X^{an}$ the “value” $r(x)\in K_{x}$ as the image $eval_{x}(r)$ .

In order to pass from the first description of $Spec^{an}(R/K)$ to the second, starting with a valuation $val_{x}$ one defines the field $K_{x}$ as the completion of the field of fractions of $R/I_{x}$ , where $I_{x}=(val_{x})^{-1}(\{+\infty\})$ .

Definition 17

The topology on $Spec^{an}(R/K)$ is the weakest topology such that for all $r\in R$ the map

\begin{array}[]{ccc}Spec^{an}(R/K)&\to&{\bf R}\cup\{+\infty\},\\ x&\mapsto&val_{x}(r)\end{array}

is continuous.

An element $f\in R$ defines a function $f:Spec^{an}(R)\to K_{x}$ , where $K_{x}$ is the non-archimedean valuation field, which is the completion of the field of fractions of the domain $R/\ker(val_{x})$ . Since each $K_{x}$ carries a seminorm, we obtain a function $|f|:Spec^{an}\to{{\bf R}}_{\geq 0},x\mapsto|f(x)|$ .

A fundamental system of neighborhoods $U=U_{x}\subset Spec^{an}(R)$ of a point $x$ is parametrized by the following data: a finite collections of functions

(f_{i})_{i\in I},\,\,(g_{j})_{j\in J}\,\,\,\in R

and numbers

\beta^{+}_{i},\beta^{-}_{i},\gamma_{j}\in{\bf R}_{>0}

such that $\beta_{i}^{-}<|{f_{i}(x)}|<\beta_{i}^{+},\,\,|{g_{j}(x)}|=0$ , The corresponding neighborhood consists of points $x^{\prime}$ such that $\beta_{i}^{-}<|{f_{i}(x^{\prime})}|<\beta_{i}^{+},\,\,|{g_{j}(x^{\prime})}|<\gamma_{j}$ for all $i\in I,j\in J$ and $x^{\prime}\in U$ .

Let us assume that elements $(f_{i})_{i\in I},\,(g_{j})_{j\in J}$ generate $R$ , i.e.

R=K[(f_{i})_{i\in I},\,(g_{j})_{j\in J}]/I

where $I$ is an ideal. Let us consider the algebra of series

s=\sum_{n_{I}\in{\bf Z}^{I},\,m_{J}\in{\bf N}^{J}}c_{I,J}f_{I}^{n_{I}}g_{J}^{m_{J}}

with constants $c_{I,J}\in K$ , absolutely convergent when variables $(f_{i})_{i\in I},\,(g_{j})_{j\in J}$ satisfy the above inequalities. The quotient of this algebra by the topological closure of ideal $I$ is the algebra ${\cal O}_{Spec^{an}(R/K)}(U)$ .

As in the case of schemes we can glue $Spec^{an}(R/K)$ into ringed spaces called analytic spaces (or rigid analytic spaces). Moreover we get a functor

\begin{array}[]{ccc}(Schemes/K)&\to&(K-{\rm analytic}\,\,{\rm spaces})\\ X&\mapsto&(X^{an},{\cal O}_{X^{an}}).\end{array}

Proposition 8

The space $X^{an}$

a) is a locally compact Hausdorff space as long as $X$ is separated;

b) has the homotopy type of a finite $CW$ -complex;

c) is contractible if $X$ has good reduction with irreducible special fiber.

Example 1

Let $X={\bf A}^{1}=Spec(K[x])$ be the affine line. The analytic space $X^{an}$ contains, among others, points of the following types:

•

$X(K)\hookrightarrow X(\overline{K})/Gal(\overline{K}/K)\hookrightarrow X^{an}$ ;
•

for $r\in{{\bf R}}_{\geq 0}$ define

$|\sum_{j=0}^{d}c_{j}z^{j}|_{r}:=\max_{j}(|{c_{j}}|r^{j})\,\,.$

This gives an embedding ${\bf R}_{\geq 0}\hookrightarrow X^{an}$ .

We see that $X^{an}$ contains, in a sense, both $p$ -adic and real points.

Define the cone over $X^{an}$ as

C_{X^{an}}({\bf R}):=X^{an}\times{\bf R}_{>0}\,\,.

We interpret a point ${\bf x}=(x,\lambda)$ of $C_{X^{an}}({\bf R})$ as a $K_{x}$ -point of $X$ , where $K_{x}\supset K$ is a complete field with the ${\bf R}$ -valued valuation

val_{\bf x}:=\lambda\,val_{x}\,\,,

whose restriction to $K$ is proportional to $val_{K}$ . The set of points ${\bf x}\in C_{X^{an}}({\bf R})$ such that the valuation $val_{\bf x}$ is ${\bf Z}$ -valued is denoted by $C_{X^{an}}({\bf Z})$ .