nLab
valuation

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 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 $v:K\to G\cup \mathrm{\infty }$ such that $v(K^{\times })\subset G$ and

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

• $v(0)=\mathrm{\infty }$

• $v(x+y)\ge \min \{v(x),v(y)\}$

with usual conventions for $\mathrm{\infty }$. Field equipped with a valuation is a valued field.

If the abelian group is the group of integers $Z$ then we talk about discrete valuations.

Properties

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

Literature

• wikipedia valuation (algebra)
• A. Fröhlich, J. W. S. Cassels (editors), 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. (Especially chapters 1,2)
• 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 $\mathrm{SL}2$, 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