Valuation (algebra)

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. They generalize 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

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 $(Γ, +, \geq)$ (which could also be given in multiplicative notation as $(Γ, \cdot, \geq)$ ).

The ordering and group law on $Γ$ are extended to the set $Γ \cup {\infty}$ ^[1] by the rules

$\infty \geq α$ for all $α$ in $Γ$ ,
$\infty + α = α + \infty = \infty$ for all α in $Γ$ .

Then a valuation of $K$ is any map

v : K \to Γ \cup {\infty}

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

$v (a) = \infty$ if, and only if, $a = 0$ ,
$v (ab) = v (a) + v (b)$ ,
$v (a + b) \geq min(v (a), v (b))$ , with equality if 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^[2] is given and the ordering and group law on Γ are extended by the rules

$O \leq α$ for all $α$ in $Γ$ ,
$O \cdot α = α \cdot O = O$ for all α in $Γ$ ,

then a valuation of K is any map

v : K \to Γ \cup {O}

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

$v (a) = O$ if, and only if, $a = 0$ ,
$v (ab) = v (a) \cdot v (b)$ ,
$v (a + b) \leq max(v (a), v (b))$ , with equality if 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.

Associated objects

If $v : K \to Γ \cup {\infty}$ is a valuation, then there are several objects that can be defined from it:

the value group of v (or valuation group of v), denoted $Γ v$ , is v(K^×), it is a subgroup of $Γ$ ,
the valuation ring of v, denoted R_v is the set of elements a of $K$ such that v(a) ≥ 0, it is a valuation ring,
the prime ideal of v (or the maximal ideal of v), denoted m_v is the set of elements a of K such that v(a) > 0, it is a maximal ideal of R_v,
the residue field of v, denoted k_v is R_v/m_v, it is a field.
a place of $K$ into $k v \cup {\infty},$ whose restriction to R_v is the natural projection.

Basic properties

Equivalence of valuations

Two valuations v₁ and v₂ of $K$ with valuation group Γ₁ and Γ₂, respectively, are said to be equivalent if there is an order-preserving group isomorphism $φ : Γ 1 \to Γ 2$ 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 $Q$ : these are precisely the equivalence classes of valuations for the p-adic completions of $Q$ .

Extension of valuations

Let v be a valuation of $K$ and let L be a field extension of $K$ . An extension of v (to L) is a valuation w of L such that the restriction of w to $K$ 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 $Γ$ is the additive group of the integers, the associated valuation induces a metric on the field $K$ . If $K$ is complete with respect to this metric, then it is called a complete valued field. In general, a valuation induces a uniform structure on $K$ , and $K$ 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 $Γ = Z$ , but stronger in general.

Examples

$π$ -adic valuation

Let $R$ be a principal ideal domain, $K$ be its field of fractions, and $π$ be an irreducible element of $R$ . Since every principal ideal domain is a unique factorization domain, every non-zero element a of $R$ can be written (essentially) uniquely as

a=\pi ^{e_{a}}p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{n}^{e_{n}}

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

The π-adic valuation of K is then given by

$v_{\pi }(0)=\infty$
$v_{\pi }(a/b)=e_{a}-e_{b},{\text{ for }}a,b\in R,a,b\neq 0.$

If π' is another irreducible element of $R$ 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).

When $R = Z$ , then $K = Q$ , and $π$ is some prime number p (or its negative). The π-adic valuation obtained is the p-adic valuation on $Q$ .

P-adic valuation on a Dedekind domain

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

Geometric notion of contact

Let $C [x, y], C (x, y)$ be the ring of complex polynomials of two variables and the field of complex rational functions respectively. Consider the (convergent) power series

f(x,y)=y-\sum _{n=3}^{\infty }x^{n}\in \mathbf {C} \{x,y\}

whose zero set, the analytic variety $V f$ , can be parametrized by one coordinate $t$ as follows

V_{f}=\left\{(x,y)\in \mathbf {C} ^{2}\ :\ f(x,y)=0\right\}=\left\{(x,y)\in \mathbf {C} ^{2}\ :\ (x,y)=\left(t,\sum _{n=3}^{\infty }t^{n}\right)\right\}

It is possible to define a map $v : C [x, y] \to Z$ as the value of the order of the formal power series in the variable $t$ obtained by restriction of any polynomial $P$ in $C [x, y]$ to the points of the set $V f$

v(P)=\mathrm {ord} _{t}\left(P|_{V_{f}}\right)=\mathrm {ord} _{t}\left(P\left(t,\sum _{n=3}^{\infty }t^{n}\right)\right),\qquad \forall P\in \mathbf {C} [x,y]

It is also possible to extend the map $v$ from its original ring of definition to the whole field $C (x, y)$ as follows

v(P/Q)={\begin{cases}v(P)-v(Q)&P/Q\in {\mathbf {C} (x,y)}^{*}\\\infty &P\equiv 0\in \mathbf {C} (x,y)\end{cases}}

As the power series $f$ is not a polynomial, it is easy to prove that the extended map $v$ is a valuation: the value $v (P)$ is called intersection number between the curves (1-dimensional analytic varieties) $V P$ and $V f$ . As an example, the computation of some intersection numbers follows:

{\begin{aligned}v(x)&=\mathrm {ord} _{t}(t)=1\\v(x^{6}-y^{2})&=\mathrm {ord} _{t}\left(t^{6}-t^{6}-2t^{7}-3t^{8}-\cdots \right)=\mathrm {ord} _{t}\left(-2t^{7}-3t^{8}-\cdots \right)=7\\v\left({\tfrac {x^{6}-y^{2}}{x}}\right)&=\mathrm {ord} _{t}\left(-2t^{7}-3t^{8}-\cdots \right)-\mathrm {ord} _{t}(t)\\&=7-1=6\end{aligned}}

Vector spaces over valuation fields

Suppose that $Γ$ is the set of non-negative real numbers. Then we say that the valuation is non-discrete if its range is not finite.

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 in K such that λ in 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. And 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 $f^{-1}(B)$ . 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.

Notes

↑ The symbol ∞ denotes an element not in $Γ$ , and has not any other meaning. Its properties are simply defined by axioms, as in every formal presentation of a mathematical theory.
↑ As for the symbol ∞, O denotes an element not in Γ and has not any other meaning, its properties being again defined by axioms.

References

Efrat, Ido (2006), Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs 124, Providence, RI: American Mathematical Society, ISBN 0-8218-4041-X, Zbl 1103.12002
Jacobson, Nathan (1989) [1980], "Valuations: paragraph 6 of chapter 9", Basic algebra II (2nd ed.), New York: W. H. Freeman and Company, ISBN 0-7167-1933-9, Zbl 0694.16001 . A masterpiece on algebra written by one of the leading contributors.
Chapter VI of Zariski, Oscar; Samuel, Pierre (1976) [1960], Commutative algebra, Volume II, Graduate Texts in Mathematics 29, New York, Heidelberg: Springer-Verlag, ISBN 978-0-387-90171-8, Zbl 0322.13001
Schaefer, Helmuth H.; Wolff, M.P. (1999). Topological Vector Spaces. GTM 3. New York: Springer-Verlag. pp. 10–11. ISBN 9780387987262.

External links

Danilov, V.I. (2001), "Valuation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Discrete valuation at PlanetMath.org.
Valuation at PlanetMath.org.
Weisstein, Eric W., "Valuation", MathWorld.

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