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.