10 GUILLAUME DUVAL
The set of F/K valuations composed with ν0 is {ν0}={ν S(F/K)∗|Rν
Rν0 }
There is a one-to-one correspondence:
(4) {ν0} ←→ S(kν0
/K)∗,ν
←→ ¯ ν
The following is an exact sequence of multiplicative groups:
1 −→
U(Rν0 )
U(Rν )
−→
F

U(Rν )
−→
F

U(Rν0 )
−→ 1
This sequence identifies with the following exact sequence of ordered
Abelian groups:
(5) 0 −→ Γ¯
ν
−→ Γν −→ Γν0 −→ 0
Proposition 13. Let F/K be a field extension and G a subgroup of Aut(F/K).
Let ν0 be a non-trivial G-invariant valuation of F/K. Denoting by
¯
G the natural
image of G in Aut(kν0 /K).
i. There is a one-to-one correspondence between the set {ν0}
G
of G-invariant
valuations composed with ν0, and the set S(kν0 /K)
¯
G
of
¯-invariant
G valuations
of kν0 /K.
ii. If G has locally-finite action on
S(F/K)∗
so does
¯
G when acting on S(kν0
/K)∗.
Proof. Since ν0 is G-invariant, Gmaps the set {ν0} onto itself. Since correspon-
dence (4) is one-to-one, (i) obviously follows.
Let ν be a valuation composed with ν0. (5) gives a natural identification
Γ¯
ν
U(Rν0 )
U(Rν )
. For each ¯ x kν0 , let
¯
S = {¯(¯(¯)|¯ ν σ x σ
¯}
G Γ¯
ν
U(Rν0 )
U(Rν )
. Let
x be a pull back of ¯ x in U(Rν0 ), and set S = {σ(x)|σ G} U(Rν0 ), we have
¯=π(S),
S where π : U(Rν0 ) −→
U(Rν0 )
U(Rν )
is the natural map.
Since Ghas locally-finite action on
S(F/K)∗,
ν(S) = {ν(σ(x))|σ G} is a finite
subset of Γν F

U(Rν )
. This means that there exists a finite set {x1,....,xn} F
such that, for all σ G, there exists {1,....,n} and U(Rν) with σ(x) =
xiσ .uσ.
Since σ(x) U(Rν0 ), xiσ U(Rν0 ). Therefore
¯=π(S)
S π{x1,...,xn} is a
finite set of Γ¯.
ν
Propositions 12 and 13 can be illustrated as follows. Let’s denote by f an
irreducible element of [x, y], and by C, a smooth projective curve birational to
the affine curve f = 0. Let q be any particular point on C. Let ν0 be the f-adic
valuation on (x, y). Its corresponding place is the residue map to the field of
regular functions on C:
P0 : (x, y) −→ (C) {∞}.
Now let
¯
P : (C) −→ {∞} be the place that evaluates functions at q. The
composition place P =
¯
P◦P0 provides a valuation ν of rank two on (x, y). Setting
R = Rν, we get
Oq Rν0 = RQ,
where Oq is the local ring of q in C, and Q is the height one prime ideal of the
elements of R vanishing at q.
Now for any subgroup G of Aut( (x, y)/ ), Proposition 12 asserts that if ν is
strongly G-invariant, G must fix the point q and leave the curve C globally invariant.
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