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Berkovich spaces, Problem List 3

Let (k, | · |) be an algebraically closed complete non-Archimedean non-trivial normed field and let A 1 Berk denote the Berkovich affine line over k.

1. Show that

O Q

p

∼ = lim ←− n Z/p n Z (an isomorphism of topological rings).

2. Let D, D 0 be closed balls in k. Show that | · | D 6 | · | D

0

if and only if D ⊆ D 0 .

3. Let | · | 0 > | · | 1 > | · | 2 > . . . be semi-norms on a ring R. For r ∈ R we define

|r| := inf

i |r| i . Show that | · | is a semi-norm on R.

4. Let ξ be a nested set of balls, B := T ξ and assume that B 6= ∅. Show that:

(a) B is a closed ball, (b) | · | ξ = | · | B .

5. Let x ∈ A 1 Berk . Show that x is of type (1) if and only if the radius of x is 0.

6. Let a ∈ k and r ∈ R >0 . Show that

ζ a,r (T − a) = r.

7. Let r < r 0 be non-negative real numbers and a ∈ k. Show that [ζ a,r , ζ a,r

0

] (an interval in A 1 Berk ) is homeomorphic to [r, r 0 ] (an interval in R).

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