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 1/n 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
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