32 the concept of a height [Chap. I

For x ∈ X(K), we now have

h(L

, . 1)

(x) − h(L

, . 2)

(x) =

1

[K: ]

deg

(

x∗L , x∗ .

1

)

−

1

[K: ]

deg

(

x∗L , x∗ .

2

)

=

1

[K: ]

deg

(

OSpec

K,x∗(

.

1

⊗ .

−1)

2

)

.

Working with the non-zero section 1 ∈ Γ(Spec K, OSpec K), we see that the latter

expression is equal to

1

[K: ]

w∈Val(K)

− log 1

w

=

1

[K: ]

k

i=1 w|νi

− log 1

w

for .

w

the extension of ( .

1

⊗ .

−1)ν

2

from

ν

to Kw. This includes a raise to

the [Kw :

ν

]-th power [Cas67, Sec. 11]. Hence,

| h(L

, .

1)(x) − h(L

, .

2)(x)| ≤

1

[K: ]

k

i=1

w|νi

[Kw : ν]log D = k log D ,

when we observe the fact that

∑

w|νi

[Kw :

ν

] = [K : ]. This is the assertion.

b) Clearly, for every x ∈ X (K), one has

h(L1⊗L2)(x) =

1

[K: ]

deg

(

x∗(L1

⊗L2)

)

=

1

[K: ]

deg

(

(x∗L1)⊗(x∗L2)

)

=

1

[K: ]

deg

(x∗L1)

+

1

[K: ]

deg

(x∗L2)

= hL1 (x) + hL2 (x) .

c) There is some k ∈ such that L ⊗k is very ample. Part b) shows that it suﬃces

to verify the assertion for L

⊗k.

Thus, we may assume that L is very ample.

Let i: X → PN be the closed embedding induced by L . Then L = i∗O(1).

Tietze’s Theorem shows that there exists a hermitian metric . on O(1)PN such

that .

L ,∞

= i∗ . .

Further, there is the model (X , L , n) over some Spec [

1

m

], which induces the

adelic metric .

L

outside the primes dividing m. The morphism i: X →

PN

extends to a rational map i : X

❴ ❴

PN

[

1

m

]

, the locus of indeterminacy of which is a

closed subset of X not meeting the generic fiber. As X is proper, this shows that

i is actually a morphism over Spec [

1

mm

], for a certain m = 0.

Now, by [EGA, Chapitre III, Proposition (4.4.1)] together with [EGA, Chapit-

re III, Corollaire (4.4.9)], there is an open subset U ⊆ X containing the generic

fiber such that i |U is an embedding. Again as X is proper, we see that i is an

embedding over Spec [

1

mm

] for some m = 0.

Define on O(1) an adelic metric in such a way that, at the finite primes ν dividing

mm , it extends the metric defined by .

L ,ν

on i(X) ⊆

PN

and, at the remaining

finite primes, it is induced by the model

(PN

[

1

mm

]

, O(1), 1). Further, at the infinite

prime, we let it agree with the hermitian metric from above.

Then i∗O(1) = L . Consequently, hL (x) = h

O(1)

(i(x)). It is, therefore, suﬃcient

to show the assertion for the height function hO(1) on

PN

.

But part a) together with Example 4.23 shows that h

O(1)

differs from hnaive by a

bounded summand. Thus, Proposition 2.6 implies the assertion.