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. 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 $K$ and its multiplicative group K^×,
an abelian totally ordered group $(Γ, +, \geq)$ .

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

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

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).

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. 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.

Multiplicative notation and absolute values

We could define^[2] the same concept writing the group in multiplicative notation as $(Γ, \cdot, \geq)$ : instead of ∞, we adjoin a formal symbol O to Γ, with the ordering and group law extended by the rules

$O \leq α$ for all $α$ ∈ $Γ$ ,
$O \cdot α = α \cdot O = O$ for all $α$ ∈ $Γ$ .

Then a valuation of K is any map

v : K \to Γ \cup {O}

satisfying the following properties for all a, b ∈ 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 the directions of the inequalities are reversed from those in the additive notation.)

If Γ is a subgroup of the positive real numbers under multiplication, the last condition is the ultrametric inequality, a stronger form of the triangle inequality $v (a + b) \leq v (a) + v (b)$ , and v is an absolute value. In this case, we may pass to the additive notation with value group Γ₊ ⊂ (R, +) by taking v₊(a) = −log v(a).

Each valuation on K defines a corresponding linear preorder: a ≼ b ⇔ v(a) ≤ v(b). Conversely, given a '≼' satisfying the required properties, we can define valuation v(a) = {b: b ≼ a ∧ a ≼ b}, with multiplication and ordering based on K and ≼.

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 $v : K \to Γ \cup {\infty}$ ;

the value group or valuation group $Γ v$ = v(K^×), a subgroup of $Γ$ (though v is usually surjective so that $Γ v$ = $Γ$ );
the valuation ring R_v is the set of a ∈ $K$ 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 $K$ associated to v, the class of v under the equivalence defined below.

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 is equivalent to an absolute value, and hence induces a metric on the field $K$ . If $K$ 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 $K$ , 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

p-adic valuation

The most basic example is the p-adic valuation v_p associated to a prime integer p, on the rational numbers K = Q, with valuation ring R = Z. The valuation group is the additive integers Γ = Z. For an integer a ∈ R = Z, the valuation v_p(a) measures the divisibility of a by powers of p:

v_{p}(a)=\max\{e\in \mathbb {Z} \mid p^{e}{\text{ divides }}a\};

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

The completion of Q with respect to v_p is the field Q_p of p-adic numbers.

Order of vanishing

Let K = C(x), the rational functions on the complex line X = C, and take a point a ∈ X. For a polynomial $f(x)=a_{k}(x{-}a)^{k}+a_{k+1}(x{-}a)^{k+1}+\cdots +a_{n}(x{-}a)^{n}$ with $a_{k}\neq 0$ , 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 C((x−a)).

$π$ -adic valuation

Generalizing the previous examples, 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).

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

Valuations can be defined for a field of functions on a space of dimension greater than one. Recall that the order-of-vanishing valuation v_a(f) on R = C[x] measures the multiplicity of the point x = a in the zero set of f; one may consider this as the order of contact (or local intersection number) of the graph y = f(x) with the x-axis y = 0 near the point (a,0). If, instead of the x-axis, one fixes another irreducible plane curve h(x,y) = 0 and a point (a,b), one may similarly define a valuation v_h on R = C[x,y] so that v_h(f) is the order of contact (the intersection number) between the fixed curve and f(x,y) = 0 near (a,b). This valuation naturally extends to rational functions f /g ∈ K = C(x,y).

In fact, this construction is a special case of the π-adic valuation on a PID defined above. Namely, consider the local ring $R=\mathbb {C} [x,y]_{(h)}=\{{\tfrac {f}{g}}\in \mathbb {C} (x,y)\mid h{\text{ does not divide }}g\}$ , the ring of rational functions which are defined on some open subset of the curve h = 0. This is a PID; in fact a discrete valuation ring whose only ideals are the powers $(h)^{k}$ . Then the above valuation v_h is the π-adic valuation corresponding to the irreducible element π = h ∈ R.

Example: Consider the curve $V_h$ defined by $h(x,y)=x^{3}-xy-y=0$ , namely the graph $\textstyle y={\frac {x^{3}}{1-x}}=\sum _{n=3}^{\infty }x^{n},$ near the origin $(a,b)=(0,0)$ . This curve can be parametrized by t ∈ C as:

\textstyle (x,y)\!=\!(t,{\frac {t^{3}}{1-t}})=(t,\sum _{n=3}^{\infty }t^{n}),

with the special point (0,0) corresponding to t = 0. Now define $v_{h}:\mathbf {C} [x,y]\to \mathbf {Z}$ as the order of the formal power series in $t$ obtained by restriction of any non-zero polynomial $f$ in $C [x, y]$ to the curve $V h$ :

\textstyle v_{h}(f)=\mathrm {ord} _{t}(f|_{V_{h}})=\mathrm {ord} _{t}\!\left(f(t,\sum _{n=3}^{\infty }t^{n})\right),\quad {\text{for }}f\in \mathbf {C} [x,y].

This extends to the field of rational functions $C (x, y)$ by $v_{h}(f/g)=v_{h}(f)-v_{h}(g)$ , along with $v_{h}(0)=\infty$ .

Some intersection numbers:

{\begin{aligned}v_{h}(x)&=\mathrm {ord} _{t}(t)=1\\v_{h}(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_{h}\!\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 $Γ$ ∪ {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 $f^{{-1}}(B)$ . 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 $Γ$ , with no other meaning. Its properties are simply defined by the given axioms.
↑ Emil Artin (1957) Geometric Algebra, page 48

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) [1994], "Valuation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Discrete valuation at PlanetMath.org.
Valuation at PlanetMath.org.
Weisstein, Eric W. "Valuation". MathWorld.

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[1] The symbol ∞ denotes an element not in $Γ$ , with no other meaning. Its properties are simply defined by the given axioms.

[2] Emil Artin (1957) Geometric Algebra, page 48