Valuation (algebra) - Definition

Definition

To define the algebraic concept of valuation, the following objects are needed:

• a field K and its multiplicative subgroup K×,
• an abelian totally ordered group (Γ, +, ≥) (which could also be given in multiplicative notation as (Γ, ·, ≥)).

The ordering and group law on Γ are extended to the set Γ∪{∞} by the rules

• ∞ ≥ α for all α in Γ,
• ∞ + α = α + ∞ = ∞ for all α in Γ.

Then a valuation of K is any map

v : K → Γ∪{∞}

which satisfies the following properties for all a, b in K:

• v(a) = ∞ if, and only if, a = 0,
• v(ab) = v(a) + v(b),
• v(a + b) ≥ min(v(a), v(b)).

Some authors use the term exponential valuation rather than "valuation". In this case the term "valuation" means "absolute value".

A valuation v is called trivial (or the trivial valuation of K) if v(a) = 0 for all a in K×, otherwise it is called non-trivial.

For valuations used in geometric applications, the first property implies that any non-empty germ of an analytic variety near a point contains that point. The second property asserts that any valuation is a group homomorphism, while the third property is a translation of the triangle inequality from metric spaces to ordered groups.

It is possible to give a dual definition of the same concept using the multiplicative notation for Γ: if, instead of ∞, an element O is given and the ordering and group law on Γ are extended by the rules

• O ≤ α for all α in Γ,
• O · α = α · O = O for all α in Γ,

then a valuation of K is any map

v : K → Γ∪{O}

satisfying the following properties for all a, b in K:

• v(a) = O if, and only if, a = 0,
• v(ab) = v(a) · v(b),
• v(a + b) ≤ max(v(a), v(b)).

(Note that in this definition, the directions of the inequalities are reversed.)

A valuation is commonly assumed to be surjective, since many arguments used in ordinary mathematical research involving those objects use preimages of unspecified elements of the ordered group contained in its codomain. Also, the first definition of valuation given is more frequently encountered in ordinary mathematical research, thus it is the only one used in the following considerations and examples.