CHAPTER 2

Invariant valuations and solutions of l.d.e.

2.1. Group actions on the Riemann-Zariski variety. Let K → F denote

an arbitrary field extension of finite type and G ⊂ Aut(F/K) be a subgroup of the

automorphism group of the field extension. We define a natural action of G on

S∗

= S(F/K) by permutation of the valuations in the following way:

G ×

S∗

−→

S∗

(σ, ν) −→ σ · ν = ˜ ν

where, for all x ∈ F ,

(1) ˜(x) ν =

ν(σ−1(x))

The induced action on the valuations rings is given by:

(2) R˜

ν

= σ(Rν) and m˜

ν

= σ(mν)

Hence every σ in G induces a K-isomorphism on the residue fields:

(3) ¯ σ : kν −→ k˜

ν

(This last arrow determines the action of G on the corresponding places.)

Remark 7. For any such G-action, two conjugate valuations ν and ˜ ν share the

same value groups by (1); they have K-isomorphic valuation rings by (2), and K-

isomorphic residue fields by (3). Since ν and ˜ ν have the same value group, they have

the same rank ([33] p 10). As a consequence, the rank, rational rank, transcendence

degree of the residue fields are invariant for this action. It is, therefore, impossible

to have a transitive action on

S(F/K)∗

as soon as tr . deg(F/K) 2.

Definition 8. With the notations above, we would say that ν is G-invariant

if and only if the valuation ring Rν is mapped onto itself by all σ in G. Similarly,

ν is strongly invariant if and only if ν(σ(x)) = ν(x) for all x in F and all σ in G.

Strong invariance implies invariance; but the converse is false as shown in the

following example suggested by M.Spivakovsky: let F = K((t )) be the field of a

formal power series expansion with rational exponents and well-ordered support.

Let ν = νt be the t-adic valuation. Then Γν = . Let g be the F/K-automorphism

defined by: g(ϕ(t)) =

ϕ(t3)

for all ϕ ∈ F . The element g generates a group G

isomorphic to and the following two relations hold: σ(Rν) = Rν for all σ in G,

and ν(g(ϕ)) = 3ν(ϕ) for all ϕ in F . Therefore, this valuation is G-invariant but

not strongly invariant.

The following definition and proposition describe a context where these two

notions of invariance are going to coincide.

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