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, +, ≥).
The ordering and group law on Γ are extended to the set Γ ∪ {∞}[1] by the rules
- ∞ ≥ α for all α ∈ Γ,
- ∞ + α = α + ∞ = ∞ for all α ∈ 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)), 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, ·, ≥): instead of ∞, we adjoin a formal symbol O to Γ, with the ordering and group law extended by the rules
- O ≤ α for all α ∈ Γ,
- O · α = α · O = O for all α ∈ Γ.
Then a valuation of K is any map
- v : K → Γ ∪ {O}
satisfying the following properties for all a, b ∈ K:
- v(a) = O if and only if a = 0,
- v(ab) = v(a) · v(b),
- v(a + b) ≤ 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) ≤ 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 → Γ ∪ {∞} ;
- the value group or valuation group Γv = v(K×), a subgroup of Γ (though v is usually surjective so that Γv = Γ);
- the valuation ring Rv is the set of a ∈ K with v(a) ≥ 0,
- the prime ideal mv is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of Rv),
- the residue field kv = Rv/mv,
- 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 : Γ1 → Γ2 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 Γv in Γw, e(w/v) = [Γw : Γv], 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) = [Rw/mw : Rv/mv] (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 Rw/mw over Rv/mv.
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 vp 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 vp(a) measures the divisibility of a by powers of p:
and for a fraction, vp(a/b) = vp(a) − vp(b).
The completion of Q with respect to vp is the field Qp 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 with , define va(f) = k, the order of vanishing at x = a; and va(f /g) = va(f) − va(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
where the e's are non-negative integers and the pi are irreducible elements of R that are not associates of π. In particular, the integer ea is uniquely determined by a.
The π-adic valuation of K is then given by
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 RP, is a principal ideal domain whose field of fractions is K. The construction of the previous section applied to the prime ideal PRP of RP 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 va(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 vh on R = C[x,y] so that vh(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 , 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 . Then the above valuation vh is the π-adic valuation corresponding to the irreducible element π = h ∈ R.
Example: Consider the curve defined by , namely the graph near the origin . This curve can be parametrized by t ∈ C as:
with the special point (0,0) corresponding to t = 0. Now define 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 Vh:
This extends to the field of rational functions C(x, y) by , along with .
Some intersection numbers:
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 . 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
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.