26 the concept of a height [Chap. I

We will denote the valuation of x ∈ K by |x|. In the case K = p, we assume that

| .| is normalized by |p| =

1

p

. We also write ν(x) := − log |x|.

4.2. Definition. Let X be a K-scheme. Then, by a metric on an invertible

sheaf L ∈ Pic(X), we mean a system of K-norms on the K-vector spaces L (x)

for x ∈ X(K).

This means, to every point x ∈ X(K) there is associated a function

. : L (x) →

+

such that

i) ∀x ∈ X(K) ∀y ∈ L (x): y = 0 ⇔ y = 0,

ii) ∀x ∈ X(K) ∀y ∈ L (x) ∀t ∈ K : ty = |t|y .

4.3. Remark. If K = , then a metric on L is the same as a (possibly

discontinuous) hermitian metric.

4.4. Definition. Assume K to be non-Archimedean, and let OK be the ring

of integers in K. Further, let X be a K-scheme, and let L ∈ Pic(X).

Then, by a model of (X, L ), we mean a triple (X , L , n) consisting of a natural

number n, a flat projective scheme π : X → OK such that XK

∼

= X, and an

invertible sheaf L ∈ Pic(X ) fulfilling L |X

∼

=

L ⊗n.

4.5. Example. Assume K to be non-Archimedean, let OK be the ring of

integers in K, and let X be a K-scheme equipped with an invertible sheaf L .

Then a model (X , L , n) of (X, L ) induces a metric . on L as follows.

x ∈ X(K) has a unique extension x: Spec OK → X . Then

x∗L

is a projective

OK-module of rank one. Each l ∈ L (x) induces l⊗n ∈ L ⊗n(x) and, therefore, a

rational section of

x∗L

. Put

l (x) := inf |a| | a ∈ K, l ∈ a ·

x∗L

1

n

. (∗)

4.6. Definition. The metric . given by (∗) is called the metric on L

induced by the model (X , L , n).

4.7. Remark. Note here that OK is, in general, a non-discrete valua-

tion ring. In particular, OK will usually be non-Noetherian.

Nevertheless, projectivity includes being of finite type [EGA, Chapitre II, Défini-

tion (5.5.2)]. This means, for the description of X , only a finite number of elements

from OK are needed.

In the particular case K =

p

, the group ν(K) is isomorphic to ( , +). Thus, for

any finite set {a1, . . . , as} ⊂ O

p

, there exists a discrete valuation ring O ⊆ O

p

containing a1, . . . , as.

By consequence, X is the base change of some scheme that is projective over a

discrete valuation ring.