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
= {x F |ν(x) 0};
= {x F |ν(x) 0};
U(Rν) = {x F |ν(x) = 0};
= 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 {∞} given by the
reduction morphism 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 and implies that 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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