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2 Algebra and Algebraic Geometry 3 Examples |
In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language.
Informally, it is an assignment of particular values to the variables in a mathematical statement or equation. So for example the statement "x=y" is satisfied by (i.e. true for) valuations in which "x" is mapped to the same value as "y", and not satisfied by (i.e. false for) all other valuations. This may seem trivial in such a simple case, but is part of the process of formalising logical arguments using mathematical symbols.
In algebra (or algebraic geometry), valuations are, in some sense, the generalization to commutative algebra of the geometrical concept of contact between two algebraic or analytic varieties.
Given a field K and a commutative ordered group (G,+,>), a valuation is a map
Example 1. Let K be the quotient field of a principal ideal domain R. Let f be any irreducible element of R(so that the ideal (f) is prime). Any element g of R belongs to some power (f)k of the ideal (f) (If g=0, it belongs to (f)k for any k, while if g is coprime with f, take then k=0). Any nonzero element s of K can then be written as
When R is Z (the integers) and p is a prime number, this called the p-adic valuation on the rational numbers.
Example 2. Let (R, μ) be a local integral ring with maximal ideal μ. Any f in R belongs to some power k of μ. Define, for any f in R
For example takes as R the ring of formal power series over a field. To be more specific, let R be C[[x,y]] the ring of formal power series in two variables over the complex numbers and μ = (x,y) its maximal ideal. The μ-adic valuation in this case is given by the difference of the orders of the power series in the numerator and the denominator:
Model Theory
Algebra and Algebraic Geometry
(where ∞ is a symbol with the property that ∞ ≥ g for any g ∈ G)
satisfying the following conditions:
Usually (and we are going to do it in the sequel), ν is required to be surjective, especially because many arguments are done using preimages of elements of G.
Examples
where p and q inR are coprime with f, and k is an integer. Defining ν(s)=k (and ν(0)=∞) gives a valuation from K to Z (the additive group of integer numbers).
and extend it to the quotient field K of R as follows:
(this is easily proved to be well-defined). Also, ν(0)=∞ as usual. This is the μ-adic valuation on K.
Example 3. (Geometrical notion of contact). For simplicity, let K be the field of rational functions in two variables over the complex nubers, K=C(x,y) and R the ring of polynomials R=C[x,y], and consider the power series