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.
