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 = ˜ where, for all x F , (1) ˜(x) = ν(σ−1(x)) The induced action on the valuations rings is given by: (2) = σ(Rν) and = σ(mν) Hence every σ in G induces a K-isomorphism on the residue fields: (3) ¯ : −→ (This last arrow determines the action of G on the corresponding places.) Remark 7. For any such G-action, two conjugate valuations ν and ˜ 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 ˜ 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 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) = 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. 7
Previous Page Next Page