Sec. 4] the adelic picard group 29

4.16. Remark. It is convenient to consider two adelically metrized invert-

ible sheaves with different underlying sheaves as of distance infinity. Then the

distance δ is no longer a metric but only a separated écart in the sense of N. Bour-

baki [Bou-T, §1].

4.17. Lemma. Let f : X → Y be a morphism of projective varieties over .

i) Then the homomorphism f ∗ : Pic(Y ) → Pic(X) is continuous with respect to the

metric topology.

ii) Even more,

δ(f

∗

L1,f

∗

L2) ≤ δ(L1, L2)

for arbitrary adelically metrized invertible sheaves L1, L2 ∈ Pic(Y ).

Proof. ii) is obvious. i) follows immediately from ii).

iii. Adelic heights.

4.18. Example. Let K be a number field. Then there is an isomorphism

l : Pic(Spec K)

∼

=

w∈Val(K)

im λ ,

where λ is the mapping

λ:

K∗

−→

w∈Val(K)

,

t → (− log |t|w)w∈Val(K) .

Proof. We have Pic(Spec K) = 0. Thus, only the metrizations of the

trivial invertible sheaf OSpec

K

have to be considered. We choose a section

0 = s ∈ Γ(Spec K, OSpec K) = K.

For every ν ∈ Val( ), one has K⊗

ν

∼

=

w

Kw [Cas67, formula (10.2)], where

w runs through the valuations of K, extending ν. Hence,

Spec K ×Spec Spec

ν

∼

=

w

Spec Kw,

and accordingly for the structure sheaves. Thus, the section s induces the homo-

morphism

ι: Pic(Spec K) −→

w∈Val(K)

,

given by (L , { . ν}w∈Val(K)) → (− log s w)w∈Val(K). Here, condition ii) of Defi-

nition 4.11.a) ensures that all the valuations of s, except finitely many, are actually

equal to 1. Therefore, the image of ι is indeed contained in the direct sum.

Furthermore, ι is a surjection, as the existence of an appropriate model is required

only outside a finite number of primes.