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:
The ordering and group law on are extended to the set } by the rules
Then a valuation of is any map
which satisfies the following properties for all "a", "b" in "K":
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. 
The valuation can be interpreted as the order of the leading-order term. The third property then corresponds to the order of a sum being the order of the larger term, unless the two terms have the same order, in which case they may cancel, in which case the sum may have larger order.
For many applications, is an additive subgroup of the real numbers formula_1 in which case ∞ can be interpreted as +∞ in the extended real numbers; note that formula_2 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, 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.
We could define the same concept writing the group in multiplicative notation as : instead of ∞, we adjoin a formal symbol "O" to Γ, with the ordering and group law extended by the rules
Then a valuation of "K" is any map
satisfying the following properties for all "a", "b" ∈ "K":
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 , and "&vert; - &vert;v" is an absolute value. In this case, we may pass to the additive notation with value group formula_3 by taking "v"+("a") = −log "&vert;a&vert;v".
Each valuation on "K" defines a corresponding linear preorder: "a" ≼ "b" ⇔ "&vert;a&vert;v" ≤ "&vert;b&vert;v". Conversely, given a '≼' satisfying the required properties, we can define valuation "&vert;a&vert;v" = {"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: 
Associated objects.
There are several objects defined from a given valuation ;
Basic properties.
Equivalence of valuations.
Two valuations "v"1 and "v"2 of with valuation group Γ1 and Γ2, respectively, are said to be equivalent if there is an order-preserving group isomorphism such that "v"2("a") = φ("v"1("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 formula_4 these are precisely the equivalence classes of valuations for the "p"-adic completions of formula_5
Extension of valuations.
Let "v" be a valuation of and let "L" be a field extension of . An extension of "v" (to "L") is a valuation "w" of "L" such that the restriction of "w" to 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") = ["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 is the additive group of the integers, the associated valuation is equivalent to an absolute value, and hence induces a metric on the field . If 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 , and 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 formula_6 but stronger in general.
Examples.
p-adic valuation.
The most basic example is the -adic valuation "v""p" associated to a prime integer "p", on the rational numbers formula_7 with valuation ring formula_8 where formula_9 is the localization of formula_10 at the prime ideal formula_11. The valuation group is the additive integers formula_12 For an integer formula_13 the valuation "v""p"("a") measures the divisibility of "a" by powers of "p":
and for a fraction, "v""p"("a"/"b") = "v""p"("a") − "v""p"("b").
Writing this multiplicatively yields the -adic absolute value, which conventionally has as base formula_15, so formula_16.
The completion of formula_17 with respect to "v""p" is the field formula_18 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 formula_19 with formula_20, 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" (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.
-adic valuation.
Generalizing the previous examples, let be a principal ideal domain, be its field of fractions, and be an irreducible element of . Since every principal ideal domain is a unique factorization domain, every non-zero element "a" of can be written (essentially) uniquely as
where the "e"'s are non-negative integers and the "pi" are irreducible elements of that are not associates of . 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 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 be a Dedekind domain, its field of fractions, and let "P" be a non-zero prime ideal of . Then, the localization of at "P", denoted "RP", is a principal ideal domain whose field of fractions is . The construction of the previous section applied to the prime ideal "PRP" of "RP" yields the -adic valuation of .
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 formula_24 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 formula_25 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 formula_26
In fact, this construction is a special case of the π-adic valuation on a PID defined above. Namely, consider the local ring formula_27, 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 formula_28. Then the above valuation "v""h" is the π-adic valuation corresponding to the irreducible element π = "h" ∈ "R".
Example: Consider the curve formula_29 defined by formula_30, namely the graph formula_31 near the origin formula_32. This curve can be parametrized by formula_33 as:
with the special point (0,0) corresponding to "t" = 0. Now define formula_35 as the order of the formal power series in obtained by restriction of any non-zero polynomial formula_36 to the curve :
This extends to the field of rational functions formula_38 by formula_39, along with formula_40.
Some intersection numbers:
Vector spaces over valuation fields.
Suppose that ∪ {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 formula_42. 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.