2 GUILLAUME DUVAL 1.2. Valuation theory. Let F be a field and (Γ, +, ) be a totally ordered Abelian group. A valuation of F is a map ν : F −→ Γ ∪ {∞} such that: • ν(x) = ∞ if and only if x = 0 • ν(xy) = ν(x) + ν(y) • ν(x + y) inf{ν(x),ν(y)}. The Abelian group Γ = Γν is called the value group of ν. Moreover, to any such ν, we associate its valuation ring Rν, its maximal ideal mν, its group of units U(Rν) and its residue field kν. These objects are defined by • Rν = {x ∈ F |ν(x) 0} • mν = {x ∈ F |ν(x) 0} • U(Rν) = {x ∈ F |ν(x) = 0} • kν = Rν/mν. When ν(x) = 0 for all x ∈ F \{0}, we will say that the valuation is trivial. The place ℘ associated to ν is the map ℘ : F → kν ∪ {∞} given by the reduction morphism Rν → kν. A place of a field is the functional counterpart of a valuation. It generalises the notion of evaluating a function at a point. Therefore, an element f ∈ F ∗ is said to have a zero at ℘ if and only if ν(f) 0. If ν(f) 0, f is said to have a pole at ℘. These notions show the local nature of valuation in their power of measuring local phenomenon. Given a relative extension F/K, the Riemann-Zariski variety S∗(F/K) is the set of equivalent classes of valuations which are trivial on K and non-trivial on F . That ν is trivial on K is equivalent to K ⊂ Rν and implies that kν is naturally a field extension of K. Here, we say that two valuations ν : F −→ Γ ∪ {∞} and ν : F −→ Γ ∪ {∞} are equivalent if and only if there is an increasing group isomorphism σ : Γ → Γ such that σ ◦ ν = ν . Two valuations are equivalent if and only if they share the same valuation ring. When F/K is an algebraic function field in one variable with K algebraically closed, there exists up to isomorphism a unique smooth algebraic projective curve C such that F coincides with the field of rational functions on C. The local rings at the points of C range over the valuation rings of F/K. This gives a natural bijection between S∗(F/K) and the points of C. The name Riemann-Zariski variety comes from this geometric results originally due to Dedekind, Weber and Ostrowski (see [9] Chap 1, Th 6.9, p. 21). 1.3. Content of the present work. Let (F/K, ∂) be as above. In abstract differential algebra, the derivation ∂ is an operator satisfying the Leibniz rule, but in contrast to what happens in classical analysis, it does not have any apparent infinitesimal meaning. On the other hand, any valuation ν of F/K or of F/C is a local object able to measure infinitesimal behaviour. It is therefore susceptible to describe the infinitesimal contents of the derivation. For example, when F = ((t)) and ν = ordt is the valuation of t-adic order. For the derivation ∂ = d dt of F , for all f ∈ F , we have: ordt(∂f) ordt(f) − 1. This inequality expresses the continuity of the derivation ∂ w.r.t. the ν-adic topol- ogy, (see Defintion 27 below for the precise setting). This notion justifies the usual rule of derivation of formal power series by which the derivation of a sum coincides with the sum of the derivatives.

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