Valuation (algebra)

Template:Short description In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.

Definition

One starts with the following objects:

• a field Template:Mvar and its multiplicative group K^×,
• an abelian totally ordered group Template:Math.

The ordering and group law on Template:Math are extended to the set Template:Math}Template:Efn by the rules

• Template:Math for all Template:Mvar ∈ Template:Math,
• Template:Math for all Template:Mvar ∈ Template:Math.

Then a valuation of Template:Mvar is any map

Template:Math

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

• Template:Math if and only if Template:Math,
• Template:Math,
• Template:Math, with equality if v(a) ≠ v(b).

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

The second property asserts that any valuation is a group homomorphism on K^×. The third property is a version of the triangle inequality on metric spaces adapted to an arbitrary Γ (see Multiplicative notation below). 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 valuation can be interpreted as the order of the leading-order term.Template:Efn The third property then corresponds to the order of a sum being the order of the larger term,Template:Efn unless the two terms have the same order, in which case they may cancel and the sum may have larger order.

For many applications, Template:Math is an additive subgroup of the real numbers <math>\R</math>Template:Efn in which case ∞ can be interpreted as +∞ in the extended real numbers; note that <math>\min(a, +\infty) = \min(+\infty, a) = a</math> for any real number a, and thus +∞ is the unit under the binary operation of minimum. The real numbers (extended by +∞) with the operations of minimum and addition form a semiring, called the min tropical semiring,Template:Efn and a valuation v is almost a semiring homomorphism from K to the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together.

Multiplicative notation and absolute values

The concept was developed by Emil Artin in his book Geometric Algebra writing the group in multiplicative notation as Template:Math:<ref>Emil Artin Geometric Algebra, pages 47 to 49, via Internet Archive</ref>

Instead of ∞, we adjoin a formal symbol O to Γ, with the ordering and group law extended by the rules

• Template:Math for all Template:Mvar ∈ Template:Math,
• Template:Math for all Template:Mvar ∈ Template:Math.

Then a valuation of Template:Math is any map

Template:Math

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

• Template:Math if and only if Template:Math,
• Template:Math,
• Template:Math, with equality if Template:Math.

(Note that the directions of the inequalities are reversed from those in the additive notation.)

If Template:Math is a subgroup of the positive real numbers under multiplication, the last condition is the ultrametric inequality, a stronger form of the triangle inequality Template:Math, and Template:Math is an absolute value. In this case, we may pass to the additive notation with value group <math>\Gamma_+ \subseteq (\R, +)</math> by taking Template:Math.

Each valuation on Template:Math defines a corresponding linear preorder: Template:Math. Conversely, given a "Template:Math" satisfying the required properties, we can define valuation Template:Math}, with multiplication and ordering based on Template:Math and Template:Math.

Terminology

In this article, we use the terms defined above, in the additive notation. However, some authors use alternative terms:

• our "valuation" (satisfying the ultrametric inequality) is called an "exponential valuation" or "non-Archimedean absolute value" or "ultrametric absolute value";
• our "absolute value" (satisfying the triangle inequality) is called a "valuation" or an "Archimedean absolute value".

Associated objects

There are several objects defined from a given valuation Template:Math;

• the value group or valuation group Template:Math = v(K^×), a subgroup of Template:Math (though v is usually surjective so that Template:Math = Template:Math);
• the valuation ring R_v is the set of a ∈ Template:Mvar with v(a) ≥ 0,
• the prime ideal m_v is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of R_v),
• the residue field k_v = R_v/m_v,
• the place of Template:Mvar associated to v, the class of v under the equivalence defined below.

Basic properties

Template:Anchor Equivalence of valuations

Two valuations v₁ and v₂ of Template:Mvar with valuation group Γ₁ and Γ₂, respectively, are said to be equivalent if there is an order-preserving group isomorphism Template:Math such that v₂(a) = φ(v₁(a)) for all a in K^×. This is an equivalence relation.

Two valuations of K are equivalent if and only if they have the same valuation ring.

An equivalence class of valuations of a field is called a place. Ostrowski's theorem gives a complete classification of places of the field of rational numbers <math>\Q:</math> these are precisely the equivalence classes of valuations for the p-adic completions of <math>\Q.</math>

Template:Anchor Extension of valuations

Let v be a valuation of Template:Mvar and let L be a field extension of Template:Mvar. An extension of v (to L) is a valuation w of L such that the restriction of w to Template:Mvar is v. The set of all such extensions is studied in the ramification theory of valuations.

Let L/K be a finite extension and let w be an extension of v to L. The index of Γ_v in Γ_w, e(w/v) = [Γ_w : Γ_v], is called the reduced ramification index of w over v. It satisfies e(w/v) ≤ [L : K] (the degree of the extension L/K). The relative degree of w over v is defined to be f(w/v) = [R_w/m_w : R_v/m_v] (the degree of the extension of residue fields). It is also less than or equal to the degree of L/K. When L/K is separable, the ramification index of w over v is defined to be e(w/v)pⁱ, where pⁱ is the inseparable degree of the extension R_w/m_w over R_v/m_v.

Complete valued fields

When the ordered abelian group Template:Math is the additive group of the integers, the associated valuation is equivalent to an absolute value, and hence induces a metric on the field Template:Mvar. If Template:Mvar is complete with respect to this metric, then it is called a complete valued field. If K is not complete, one can use the valuation to construct its completion, as in the examples below, and different valuations can define different completion fields.

In general, a valuation induces a uniform structure on Template:Mvar, and Template:Mvar is called a complete valued field if it is complete as a uniform space. There is a related property known as spherical completeness: it is equivalent to completeness if <math>\Gamma = \Z,</math> but stronger in general.

Examples

p-adic valuation

The most basic example is the [[p-adic valuation|Template:Mvar-adic valuation]] ν_p associated to a prime integer p, on the rational numbers <math>K=\Q,</math> with valuation ring <math>R=\Z_{(p)}, </math> where <math>\Z_{(p)} </math> is the localization of <math>\Z </math> at the prime ideal <math>(p) </math>. The valuation group is the additive integers <math>\Gamma = \Z.</math> For an integer <math>a \in R= \Z,</math> the valuation ν_p(a) measures the divisibility of a by powers of p:

<math> \nu_p(a) = \max\{e \in \Z \mid p^e \text{ divides } a\};</math>

and for a fraction, ν_p(a/b) = ν_p(a) − ν_p(b).

Writing this multiplicatively yields the [[p-adic absolute value|Template:Mvar-adic absolute value]], which conventionally has as base <math>1/p = p^{-1}</math>, so <math>|a|_p := p^{-\nu_p(a)}</math>.

The completion of <math>\Q</math> with respect to ν_p is the field <math>\Q_p</math> of p-adic numbers.

Order of vanishing

Let K = F(x), the rational functions on the affine line X = F¹, and take a point a ∈ X. For a polynomial <math>f(x) = a_k (x{-}a)^k + a_{k+1}(x{-}a)^{k+1}+\cdots+ a_n(x{-}a)^n</math> with <math>a_k\neq 0</math>, define v_a(f) = k, the order of vanishing at x = a; and v_a(f /g) = v_a(f) − v_a(g). Then the valuation ring R consists of rational functions with no pole at x = a, and the completion is the formal Laurent series ring F((x−a)). This can be generalized to the field of Puiseux series K{{t}} (fractional powers), the Levi-Civita field (its Cauchy completion), and the field of Hahn series, with valuation in all cases returning the smallest exponent of t appearing in the series.

Template:Mvar-adic valuation

Generalizing the previous examples, let Template:Mvar be a principal ideal domain, Template:Mvar be its field of fractions, and Template:Mvar be an irreducible element of Template:Mvar. Since every principal ideal domain is a unique factorization domain, every non-zero element a of Template:Mvar can be written (essentially) uniquely as

<math>a=\pi^{e_a}p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}</math>

where the e's are non-negative integers and the p_i are irreducible elements of Template:Mvar that are not associates of Template:Mvar. In particular, the integer e_a is uniquely determined by a.

The π-adic valuation of K is then given by

• <math>v_\pi(0)=\infty</math>
• <math>v_\pi(a/b)=e_a-e_b,\text{ for }a,b\in R, a, b\neq0.</math>

If π' is another irreducible element of Template:Mvar such that (π') = (π) (that is, they generate the same ideal in R), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called the P-adic valuation, where P = (π).

Template:Anchor P-adic valuation on a Dedekind domain

The previous example can be generalized to Dedekind domains. Let Template:Mvar be a Dedekind domain, Template:Mvar its field of fractions, and let P be a non-zero prime ideal of Template:Mvar. Then, the localization of Template:Mvar at P, denoted R_P, is a principal ideal domain whose field of fractions is Template:Mvar. The construction of the previous section applied to the prime ideal PR_P of R_P yields the Template:Mvar-adic valuation of Template:Mvar.

Vector spaces over valuation fields

Suppose that Template:Math ∪ {0} is the set of non-negative real numbers under multiplication. Then we say that the valuation is non-discrete if its range (the valuation group) is infinite (and hence has an accumulation point at 0).

Suppose that X is a vector space over K and that A and B are subsets of X. Then we say that A absorbs B if there exists a α ∈ K such that λ ∈ K and |λ| ≥ |α| implies that B ⊆ λ A. A is called radial or absorbing if A absorbs every finite subset of X. Radial subsets of X are invariant under finite intersection. Also, A is called circled if λ in K and |λ| ≥ |α| implies λ A ⊆ A. The set of circled subsets of L is invariant under arbitrary intersections. The circled hull of A is the intersection of all circled subsets of X containing A.

Suppose that X and Y are vector spaces over a non-discrete valuation field K, let A ⊆ X, B ⊆ Y, and let f : X → Y be a linear map. If B is circled or radial then so is <math>f^{-1}(B)</math>. If A is circled then so is f(A) but if A is radial then f(A) will be radial under the additional condition that f is surjective.

See also

• Discrete valuation
• Euclidean valuation
• Field norm
• Absolute value (algebra)

References

• Template:Citation
• Template:Citation. A masterpiece on algebra written by one of the leading contributors.
• Chapter VI of Template:Citation
• Template:Cite book

External links

• Template:Springer
• Template:PlanetMath
• Template:PlanetMath
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