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Berkovich spaces, Problem List 6

Let (k, | · |) be a non-Archimedean complete algebraically closed field.

1. Show that the set {P a

i

T

i

| sup |a

i

|r

i

< ∞} is a k-subalgebra of k JT K.

2. Show that there is an isomorphism (

∞

X

i=0

a

i

T

i

| sup |a

i

|r

i

< ∞ )

∼ = k{r

−1

T }

of normed k[T ]-algebras.

3. Let R be a normed ring. Show that the restriction map M( b R) → M(R)

is a bijection.

4. Show that the bijection from the lecture ψ

r

: M(K{r

−1

T }) → U

r

is a homeomorphism.

5. Let A, B be Banach rings and f : A → B be admissible epimorphism.

Show that f is bounded.

6. Let V ⊆ A

nk

be an affine algebraic variety. Show that the image of the natural map

V

an

→ A

nBerk

is closed.

7. Show that the completion of a perfect field is perfect.

1

Cytaty

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