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,


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)

im λ ,
where λ is the mapping
t (− log |t|w)w∈Val(K) .
Proof. We have Pic(Spec K) = 0. Thus, only the metrizations of the
trivial invertible sheaf OSpec
have to be considered. We choose a section
0 = s Γ(Spec K, OSpec K) = K.
For every ν Val( ), one has K⊗

Kw [Cas67, formula (10.2)], where
w runs through the valuations of K, extending ν. Hence,
Spec K ×Spec Spec

Spec Kw,
and accordingly for the structure sheaves. Thus, the section s induces the homo-
ι: Pic(Spec 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.
Previous Page Next Page