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 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
[edit]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
- ∞ ≥
α for allα ∈Γ , - ∞ +
α =α + ∞ = ∞ + ∞ = ∞ for allα ∈Γ .
Then a valuation of K is any map
- v : K →
Γ ∪ {∞}
that 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 on K×. The third property is a version of the triangle inequality on metric spaces adapted to an arbitrary
The valuation can be interpreted as the order of the leading-order term.[b] The third property then corresponds to the order of a sum being the order of the larger term,[c] unless the two terms have the same order, in which case they may cancel and the sum may have larger order.
For many applications,
Multiplicative notation and absolute values
[edit]The concept was developed by Emil Artin in his book Geometric Algebra writing the group in multiplicative notation as (
Instead of ∞, we adjoin a formal symbol O to
- O ≤
α for allα ∈Γ , - O ·
α =α · O = O for allα ∈Γ .
Then a valuation of K is any map
- | ⋅ |v : K →
Γ ∪ {O}
satisfying the following properties for all a, b ∈ K:
- |a|v = O if and only if a = 0,
- |ab|v = |a|v · |b|v,
- |a+b|v ≤ max(|a|v, |b|v), with equality if |a|v ≠ |b|v.
(Note that the directions of the inequalities are reversed from those in the additive notation.)
If
Each valuation on K defines a corresponding linear preorder: a ≼ b ⇔ |a|v ≤ |b|v. Conversely, given a "≼" satisfying the required properties, we can define valuation |a|v = {b: b ≼ a ∧ a ≼ b}, with multiplication and ordering based on K and ≼.
Terminology
[edit]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
[edit]There are several objects defined from a given valuation v : K →
- the value group or valuation group
Γ v = v(K×), a subgroup ofΓ (though v is usually surjective so thatΓ v =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
[edit]Equivalence of valuations
[edit]Two valuations v1 and v2 of K with valuation group
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 these are precisely the equivalence classes of valuations for the p-adic completions of
Extension of valuations
[edit]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
Complete valued fields
[edit]When the ordered abelian group
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 but stronger in general.
Examples
[edit]p-adic valuation
[edit]The most basic example is the p-adic valuation
and for a fraction,
Writing this multiplicatively yields the p-adic absolute value, which conventionally has as base , so .
The completion of with respect to
Order of vanishing
[edit]Let K = F(x), the rational functions on the affine line X = F1, 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 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.
π -adic valuation
[edit]Generalizing the previous examples, let R be a principal ideal domain, K be its field of fractions, and
where the e's are non-negative integers and the pi are irreducible elements of R that are not associates of
The
If
P-adic valuation on a Dedekind domain
[edit]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.
Vector spaces over valuation fields
[edit]Suppose that
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
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
[edit]Notes
[edit]- ^ The symbol ∞ denotes an element not in
Γ , with no other meaning. Its properties are simply defined by the given axioms. - ^ With the min convention here, the valuation is rather interpreted as the negative of the order of the leading order term, but with the max convention it can be interpreted as the order.
- ^ Again, swapped since using minimum convention.
- ^ Every Archimedean group is isomorphic to a subgroup of the real numbers under addition, but non-Archimedean ordered groups exist, such as the additive group of a non-Archimedean ordered field.
- ^ In the tropical semiring, minimum and addition of real numbers are considered tropical addition and tropical multiplication; these are the semiring operations.
References
[edit]- ^ Emil Artin Geometric Algebra, pages 47 to 49, via Internet Archive
- Efrat, Ido (2006), Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs, vol. 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, vol. 29, New York, Heidelberg: Springer-Verlag, ISBN 978-0-387-90171-8, Zbl 0322.13001
- Schaefer, Helmut H.; Wolff, M.P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York: Springer-Verlag. pp. 10–11. ISBN 9780387987262.
External links
[edit]- Danilov, V.I. (2001) [1994], "Valuation", Encyclopedia of Mathematics, EMS Press
- Discrete valuation at PlanetMath.
- Valuation at PlanetMath.
- Weisstein, Eric W. "Valuation". MathWorld.