nLab
valuation ring

Contents

Definition-Proposition

A valuation ring is a commutative integral domain OO which satisfies any of the following equivalent conditions

  • for every nonzero element xx in its field of fractions KK, either xOx \in O or x 1Ox^{-1} \in O.
  • The ideals of OO are totally ordered by inclusion
  • The principal ideals of OO are totally ordered by inclusion
Proposition

A valuation ring OO is a local ring. Its maximal ideal is said to be the valuation ideal?.

Proof

Suppose xx, yy are nonzero non-invertible elements of OO. Either x/yx/y or y/xy/x belongs to OO, say x/yx/y. Then (x+y)/y(x+y)/y belongs to OO as well. If x+yx+y were a unit of OO, it would follow that 1/y1/y belongs to OO, i.e., both yy and 1/y1/y belong to OO, so that yy is a unit of OO, contradiction. It follows that non-unit elements of OO are closed under addition. It is also clear that if xx is a non-unit element of OO and yy is any element of OO, then xyx y is a non-unit element of OO. Therefore the non-units form an ideal of OO, clearly the unique maximal ideal of OO.

Proposition

A valuation ring OO is integrally closed in its field of fractions FF.

Proof

Suppose xFx \in F satisfies an equation x n+a n1x n1++a 0=0x^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0 where the a ia_i belong to OO. Either xx or 1/x1/x belongs to OO, and if 1/x1/x belongs to OO, then so does xx because

x=a n1x 1+a n1x 2++a 0x nx = a_{n-1}x^{-1} + a_{n-1}x^{-2} + \ldots + a_0 x^{-n}

and this completes the proof.

Example: germs of definable functions

Local rings often arise as stalks of sheaves of real or complex-valued functions, and similarly, some of the more interesting examples of valuation rings arise by considering germs of functions at an “ideal” point, for example at an ultrafilter or infinite point where the functions are not formally defined. Examples such as these are often rich sources of rings and fields with infinitesimal elements. The following example should give the flavor of this phenomenon.

Consider the class of all functions \mathbb{R} \to \mathbb{R} which can be defined by a first-order formula, starting with the basic operations +,,,exp+, -, \cdot, \exp, any constant aa \in \mathbb{R}, and the relations <\lt and ==. This is an enormous class of functions: it includes the logarithm function and any function which can be built from polynomials, exponentials, and logarithms using the four basic arithmetic operations and composition, and also implicitly defined functions such as:

f(x)=max{y:y 5(e e x(logx) 2+1/x)y 2+log(1+x 2))y17=0}f(x) = max \{y: y^5 - (e^{e^x - (\log x)^2 + 1/x})y^2 + \log(1 + x^{\sqrt{2}}))y - 17 = 0\}

and many, many more. Of interest are rates of growth (cf. O notation?) of these functions, or, at an even more refined level, the precise ordering of these functions f(x)g(x)f(x) \leq g(x) when xx is large. A quite remarkable and deep fact is the following:

Theorem

Any such definable function f(x)f(x) is either positive for all sufficiently large xx, 00 for all sufficiently large xx, or negative for all sufficiently large xx.

This theorem implies that the germs at infinity of such functions (\sim-equivalence classes of functions where fgf \sim g if f(x)=g(x)f(x) = g(x) for all sufficiently large xx) form a totally ordered field, in fact a real closed field. The real numbers are embedded in this field as germs of constant functions, but lying between ordinary real numbers are other “numbers” infinitesimally close to reals, such as 2[1/x]2 - [1/x], 1+[1/loglog(x)]1 + [1/\log \log(x)], as well as infinite numbers such as [e x][e^x].

Sitting inside this field Germ( exp)Germ(\mathbb{R}_{exp}) is the valuation ring of germs of bounded definable functions, in other words the ring of finite “numbers”, which contains infinitesimals of incredibly rich variety.

Such examples are close in spirit to hyperreal numbers, which form a considerably larger real closed field. In this case, the procedure is similar, except that one takes germs of all functions \mathbb{N} \to \mathbb{R} in the neighborhood of a non-principal ultrafilter on \mathbb{N}, which can be considered an ideal point at “infinity”. This is called an ultrapower of the standard real numbers. Again the finite hyperreals form a valuation ring sitting inside.

The valuation function

(also called multiplicative or exponential valuation)

The nonzero elements of KK may be partially ordered as follows: write xyx \leq y if x/yx/y belongs to OO. For any two nonzero elements xx, yy of KK, exactly one of the following conditions holds:

  • x/yx/y is a non-unit of OO;

  • x/yx/y is a unit of OO: x/yx/y and y/xy/x belong to OO;

  • y/xy/x is a non-unit of OO.

We call this the trichotomy law. Thus, if we write O *O^* for the units of RR, it follows from trichotomy that the partial order on K *K^* descends to a total order on the quotient group G=K */O *G = K^*/O^*. This totally ordered group is called the value group of the valuation ring OO. When the value group is isomorphic to \mathbb{Z}, the ring OO is called a discrete valuation ring. Many local rings which arise in practice, for example localizations of rings of algebraic integers OO with respect to a prime ideal 𝓅\mathcal{p}, or their completions as rings of 𝓅\mathcal{p}-adic integers O (𝓅)O_{(\mathcal{p})}, are discrete valuation rings.

We may then define a valuation function

v:KK */O *{0}v: K \to K^*/O^* \cup \{0\}

where v(x)v(x) is the coset xO *x O^* if xK *x \in K^*, and v(0)=0v(0) = 0. The codomain becomes a totally ordered monoid if 00 is regarded as the bottom element and is absorbent (0x=00 x = 0 for all xx). This function satisfies the following conditions:

  1. v(x)=0v(x) = 0 if and only if x=0x = 0;

  2. v(xy)=v(x)v(y)v(x y) = v(x)v(y);

  3. v(x+y)max{v(x),v(y)}v(x + y) \leq \max \{v(x), v(y)\}

  4. vv is surjective.

Example: rates of growth

Let us return to our earlier example of a valuation ring OO, consisting of germs of bounded functions which are first-order definable in the theory of the reals as ordered field with exponentiation. Two “numbers” [f][f], [g][g] have the same coset, [f]O *=[g]O *[f]O^* = [g]O^*, if both [f/g][f/g] and [g/f][g/f] are bounded. But this is precisely to say ff is O(g)O(g) and gg is O(f)O(f).

Thus, the elements of the value group in this case can be described as the various “rates of growth” of definable functions. The order relation is that [f]O *[g]O *[f]O^* \leq [g]O^* if ff is O(g)O(g). Thus the classical analysis notion of ‘O notation’ fits within the theory of valuation rings.

Rates of growth, as elements of the value group K */O *K^*/O^*, can also be regarded as “numbers” containing infinitesimal and infinite quantities. Thus, ordinary real numbers aa would correspond to rates of growth of power functions x ax^a, whereas the rate of growth logx\log x is infinitesimal and the rate of growth of e xe^x is infinite. The fact that rates of growth of definable functions are totally ordered is essentially due to G.H. Hardy, and in his honor, fields of germs of definable functions are frequently called “Hardy fields”.

  • Hardy actually studied the class of functions definable from polynomials, exponential, and logarithmic functions using the four arithmetic operations and composition. This of course is a small subclass of the definable functions considered above, but contains all functions likely to be of interest in analytic number theory (for example).

Valuation rings from valuation functions

Quite generally, we may define a valuation on a field KK to be a function

v:KG{0}v: K \to G \cup \{0\}

(where GG is a totally ordered group, extended to a totally ordered monoid G{0}G \cup \{0\} as above), satisfying conditions 1 - 4 listed above. Two valuations vv, vv' are equivalent if there is an isomorphism

ϕ:G{0}G{0}\phi: G \cup \{0\} \to G' \cup \{0\}

of totally ordered monoids such that v=ϕvv' = \phi \circ v. In fact, valuations vv may be preordered: if we regard vv as a special sort of group homomorphism v:K *Gv: K^* \to G, then define vvv \leq v' if there is a surjective homomorphism of ordered groups ϕ:GG\phi: G \to G' such that v=ϕvv' = \phi \circ v.

From the data of a valuation on KK, we may construct a valuation ring O vO_v inside KK:

O v={xK:v(x)1}O_v = \{x \in K: v(x) \leq 1\}

where 11 is the identity element of GG. (The fact that O vO_v is a ring follows straightforwardly from the axioms 1 - 3, and that O vO_v is a valuation ring follows from the fact that GG is totally ordered.)

In summary, we have the following result whose proof is straightforward:

Proposition

There is a natural bijective correspondence between equivalence classes of valuations on KK and valuation rings in KK.

To express the naturality more precisely: there are two functors

V:Field opPos,V:Field opPosV: Field^{op} \to Pos, \qquad V': Field^{op} \to Pos

from fields to posets. Here VV assigns to a field KK the poset of equivalence classes of valuations on KK, where the partial order is inherited from the preorder described above; VV' assigns to a field KK the poset of valuation rings inside KK, ordered by inclusion. If i:KKi: K \to K' is a field homomorphism, then V(i):V(K)V(K)V'(i): V'(K') \to V'(K) takes a valuation ring OKO' \subseteq K to the valuation ring i 1(O)Ki^{-1}(O') \subseteq K, and V(i)V(i) is defined similarly. The proposition asserts that the functors VV, VV' are naturally isomorphic. We freely conflate them, denoting either functor as Val:Field opPosVal: Field^{op} \to Pos.

Examples

  1. The algebraic notion of Riemann surface from 19th century is constructed in a way in which valuations are used to construct points of Riemann surfaces over various fields. See Riemann surface via valuations.

  2. There is a very general construction which takes as input an arbitrary field kk and a totally ordered abelian group GG, and produces as output a valuation ring whose value group is naturally identified with GG. This is the ring of Hahn series, see there.

Literature

Related entries: valuation, discrete valuation, class field theory

  • 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.
  • Serge Lang, Algebraic number theory. GTM 110, Springer 1970, 2000
  • O. Zariski, Samuel, Commutative algebra

Revised on January 4, 2012 17:16:08 by Zoran Škoda (161.53.130.104)