Sec. 4] the adelic picard group 27
4.8. Definition. Let K be an algebraically closed valuation field. Assume
K to be non-Archimedean.
Then a metric . on L Pic(X) is called continuous, respectively bounded,
if . = f · . for . a metric induced by some model and f a function on X(K)
that is continuous or bounded, respectively.
4.9. Remark. If K = , then we adopt the concepts of bounded, con-
tinuous, and smooth metrics in their the usual meaning from complex geometry.
Note that smooth metrics are continuous and that continuous metrics are automat-
ically bounded in the case K = .
ii. The global case. Adelically metrized invertible sheaves.
4.10. Definition. Let X be a projective variety over and m .
Then, by a model of X over Spec [
1
m
], we mean a scheme X that is projective
and flat over Spec [
1
m
] such that the generic fiber of X is isomorphic to X.
4.11. Definition. Let X be a projective variety over and L Pic(X)
be an invertible sheaf.
a) Then an adelic metric on L is a system
. = { .
ν
}ν∈Val(
)
of continuous and bounded metrics on L
ν
Pic(X
ν
) such that
i) for each ν Val( ), the metric .
ν
is Gal(
ν
/
ν
)-invariant,
ii) for some m , there exist a model X of X over Spec [
1
m
], an invertible
sheaf L Pic(X ), and a natural number n such that
L |X

=
L
⊗n
and, for all prime numbers p m, the metric .
νp
is induced by (Xp, L |Xp , n).
b) An invertible sheaf equipped with an adelic metric is called an adelically metrized
invertible sheaf.
c) All adelically metrized invertible sheaves on X form an abelian group, which will
be called the adelic Picard group of X and denoted by Pic(X).
4.12. Notation. Let X be a model of X over Spec . Then taking the
induced metric yields two natural homomorphisms
iX : Pic(X ) Pic(X) ,
aX : ker(Pic(X ) Pic(X)) Pic(X) .
Further, one has the forgetful homomorphism
v : Pic(X) Pic(X) .
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