CHAPTER 1

The Berkovich unit disc

In this chapter, we recall a theorem of Berkovich which states that points

of the Berkovich unit disc D(0, 1) over K can be identified with equivalence

classes of nested sequences of closed discs {D(ai,ri)}i=1,2,... contained in the

closed unit disc D(0, 1) of K. This leads to an explicit description of the

Berkovich unit disc as an “infinitely branched tree”; more precisely, we show

that D(0, 1) is an inverse limit of finite R-trees.

1.1. Definition of D(0, 1)

Let A = K T be the ring of all formal power series with coeﬃcients

in K, converging on D(0, 1). That is, A is the ring of all power series

f(T) =

∑∞

i=0

aiT

i

∈ K[[T]] such that limi→∞ |ai| = 0. Equipped with the

Gauss norm defined by f = maxi(|ai|), A becomes a Banach algebra

over K.

A multiplicative seminorm on A is a function [ ]x : A → R≥0 such that

[0]x = 0, [1]x = 1, [f · g]x = [f]x · [g]x, and [f + g]x ≤ [f]x + [g]x for all

f, g ∈ A. It is a norm provided that [f]x = 0 if and only if f = 0.

A multiplicative seminorm [ ]x is called bounded if there is a constant Cx

such that [f]x ≤ Cx f for all f ∈ A. It is well known (see [49, Proposition

5.2]) that boundedness is equivalent to continuity relative to the Banach

norm topology on A. The reason for writing the x in [ ]x is that we will be

considering the space of all bounded multiplicative seminorms on A, and we

will identify the seminorm [ ]x with a point x in this space.

It can be deduced from the definition that a bounded multiplicative

seminorm [ ]x on A behaves just like a non-Archimedean absolute value,

except that its kernel may be nontrivial. For example, [ ]x satisfies the

following properties:

Lemma 1.1. Let [ ]x be a bounded multiplicative seminorm on A. Then

for all f, g ∈ A,

(A) [f]x ≤ f .

(B) [c]x = |c| for all c ∈ K.

(C) [f + g]x ≤ max([f]x, [g]x), with equality if [f]x = [g]x.

Proof. (A) For each n,

([f]x)n

= [f

n]x

≤ Cx f

n

= Cx f

n,

so [f]x ≤

Cx/n 1

f , and letting n → ∞ gives the desired inequality.

(B) By the definition of the Gauss norm, c = |c|. If c = 0, then

trivially [c]x = 0; otherwise, [c]x ≤ c = |c| and

[c−1]x

≤

c−1

=

|c−1|,

1

http://dx.doi.org/10.1090/surv/159/01