nLab valuation

This page is about valuations on rings/fields. For valuation in measure theory see valuation (measure theory).

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Idea

Wikipedia says very succinctly

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.

Sometimes one also discusses exponential (or multiplicative) valuations (also called valuation functions, and viewed as generalized absolute values) which look more like norms, and their equivalence classes, places. See at absolute value for more on this common sense.

See also discrete valuation and valuation ring.

Definition

Given a totally ordered abelian group $G$, a $G$-valued valuation $v$ on a (commutative) field $K$ is a (typically required to be surjective) function $v:K\to G\cup \infty$ such that $v(K^\times)\subset G$ and

• $v$ defines a homomorphism of groups $v|_K : K^\times\to G$ where $K^\times$ is the multiplicative group of $K$

• $v(0) = \infty$

• $v(x+y) \geq min\{ v(x),v(y)\}$

with usual conventions for $\infty$. A field equipped with a valuation is a valued field.

If the abelian group is the group of integers $\mathbb{Z}$, then we talk about discrete valuations.

Properties

Valuative criteria

In algebraic geometry there are very important theorems, due to Chevalley, called the valuative criterion of properness and valuative criterion of separatedness.

Valuation rings and ideals

The valuation ring of a valued field is the subring of elements of valuation $\ge 0$. The valuation ideal is the ideal in the valuation field consisting of elements of valuation $\gt 0$.

Valuation uniformity

Any valued field admits a valuation uniformity, where each $g\in G$ generates an entourage $\{ (x,y) \mid v(x-y)\ge g \}$. For a discrete valuation, the restriction of this uniformity to the valuation ring coincides with the adic topology generated by the valuation ideal; but in general this need not be the case.

A complete valued field is a valued field whose valuation unifomity is complete. A non-complete valued field can be completed.

Examples

Rational functions

For any field $k$, the field $k(t)$ of formal rational functions in a variable $t$ has a discrete valuation given by the difference in the lowest nonzero degree of the numerator and denominator. Thus for instance

$$\frac{3t + 2t^3- 7t^4}{2t^2 - 8t^9}$$

would have valuation $1-2 = -1$. The valuation ring contains the polynomial ring $k[t]$, but is strictly larger, for instance it contains $\frac{x^2}{x+1}$. The restriction of the valuation ideal to $k[t]$ does coincide with $(t)$ however. The valued field $k(t)$ is not complete; its completion is the field $k[[t]]$ of formal power series.

Generalized rational functions

For any field $k$, a generalized polynomial? with exponents in $G$ is a finite formal sum of monomials $a q^\ell$, where $a\in k$ and $\ell\in G$; this defines a ring $k[q^G]$. The field $k(q^G)$ of generalized rational functions? is the field of fractions of $k[q^G]$. It is a valued field with value group $G$, with valuation defined analogously to the case of ordinary rational functions. As there, its valuation ring contains $k[q^G]$ but is larger.

It is not generally complete. At least in good cases (e.g. if $G$ is Archimedean in the sense that positive integers are cofinal therein, such as $\mathbb{R}$), its completion should be the subfield of the field $k((q^G))$ of Hahn series consisting of those Hahn series of order type $\omega$ that converge to themselves in the topology of $k((q^G))$.

$p$-adic valuations

The field $\mathbb{Q}$ admits a $p$-adic valuation, whose completion is the field $\mathbb{Q}_p$ of p-adic numbers.

Related concepts

algebraic structure	group	ring	field	vector space	algebra
(submultiplicative) norm	normed group	normed ring	normed field	normed vector space	normed algebra
multiplicative norm (absolute value/valuation)			valued field
completeness	complete normed group	Banach ring	complete field	Banach vector space	Banach algebra

Literature

• S. Bosch, U. Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

• Wikipedia, valuation (algebra)

• Albrecht Fröhlich, J. W. S. Cassels (eds.), Algebraic number theory, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, pdf, errata pdf by Kevin Buzzard)

• J. W. S. Cassels, Chapter 2 in: Local Fields, Cambridge University Press, 1986 (ISBN:9781139171885, doi:10.1017/CBO9781139171885)

• Serge Lang, Algebraic number theory, GTM 110, Springer 1970, 2000

• Ehud Hrushovski, David Kazhdan, The value ring of geometric motivic integration and the Iwahori Hecke algebra of $SL 2$, math.LO/0609115; Integration in valued fields, in Algebraic geometry and number theory, 261–405, Progress. Math. 253, Birkhäuser Boston, pdf

• A H Lightstone, Abraham Robinson, Nonarchimedean fields and asymptotic expansions, North-Holland Publ. 1976

check out (Scholze 11, def. 22, remark 2.3)