28 the concept of a height [Chap. I
4.13. Notation. The models of X, together with all birational mor-
phisms between them, form an inverse system of schemes. This is a filtered
inverse system since, for two models X and X , the closure of the diagonal
Δ X ×Spec X X ×Spec X projects to both of them.
Thus, the arithmetic Picard groups Pic(X ) for all models X of X form a filtered
direct system. The injections Pic(X ) Pic(X) fit together to yield an injection
ιX : lim
−→
Pic(X ) Pic(X) .
Similarly, the usual Picard groups Pic(X ) form a filtered direct system, too. We get
a homomorphism
αX : lim
−→
ker(Pic(X ) Pic(X)) −→ Pic(X) .
4.14. Definition (Metric on
v−1(L
) Pic(X)). Let X be a projective va-
riety over . On X, let (L , . ) and (L , . ) be two adelically metrized invertible
sheaves with the same underlying sheaf.
Then the distance between (L , . ) and (L , . ) is given by
δ((L , . ), (L , . )) :=
ν∈Val( )
δν( . ν, .
ν
)
for
δν( . ν, .
ν
) := sup
x∈X(
ν
)
log
.
ν
(x)
. ν(x)
.
4.15. Lemma. δ is a metric on the set
v−1(L
) of all metrizations of L .
Proof. We have to show that the sum is always finite.
For this, we note first that the metrics .
ν
and .
ν
are bounded by definition.
Therefore, each summand is finite.
We may thus ignore a finite set S of primes and assume that . and . are
given by triples (X , L , n) and (X , L , n ), respectively, in the sense of Defini-
tion 4.11.a.ii).
The isomorphism X

=
−→ X

=
−→ X may be extended to an open neigh-
bourhood of the generic fiber. Therefore, enlarging S if necessary, we have an
isomorphism X

=
−→ X of schemes over Spec \S. Further, the triple (X , L , n)
may be replaced by (X , L
⊗n
, nn ) without any change of the induced metric.
Thus, without restriction, n = n .
To summarize, we are reduced to the case that . and . are given by (X , L , n)
and (X , L , n). We have an isomorphism
L |X

=
−→ L
⊗n

=
−→ L |X ,
which may be extended to an open neighbourhood of the generic fiber. There-
fore, in the definition of δ((L , . ), (L , . )), all the summands vanish, except
finitely many.
Positivity, symmetry, and the triangle inequality are clear.
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