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| =
. 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
is a projective
OK-module of rank one. Each l L (x) induces l⊗n L ⊗n(x) and, therefore, a
rational section of
. Put
l (x) := inf |a| | a K, l a ·
. (∗)
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 =
, the group ν(K) is isomorphic to ( , +). Thus, for
any finite set {a1, . . . , as} O
, there exists a discrete valuation ring O O
containing a1, . . . , as.
By consequence, X is the base change of some scheme that is projective over a
discrete valuation ring.
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