Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R that satisfies any one of the following equivalent conditions:

1. R is a local principal ideal domain, and not a field.
2. R is a valuation ring with a value group isomorphic to the integers under addition.
3. R is a local Dedekind domain and not a field.
4. R is a Noetherian local domain whose maximal ideal is principal, and not a field.
5. R is an integrally closed Noetherian local ring with Krull dimension one.
6. R is a principal ideal domain with a unique non-zero prime ideal.
7. R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
8. R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
9. R is Noetherian, not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it.
10. There is some discrete valuation ν on the field of fractions K of R such that R = {0} [math]\displaystyle{ \cup }[/math] {x [math]\displaystyle{ \in }[/math] K : ν(x) ≥ 0}.

Examples

Algebraic

Localization of Dedekind rings

Let [math]\displaystyle{ \mathbb{Z}_{(2)} := \{ z/n\mid z,n\in\mathbb{Z},\,\, n\text{ is odd}\} }[/math]. Then, the field of fractions of [math]\displaystyle{ \mathbb{Z}_{(2)} }[/math] is [math]\displaystyle{ \mathbb{Q} }[/math]. For any nonzero element [math]\displaystyle{ r }[/math] of [math]\displaystyle{ \mathbb{Q} }[/math], we can apply unique factorization to the numerator and denominator of r to write r as 2^k z/n where z, n, and k are integers with z and n odd. In this case, we define ν(r)=k. Then [math]\displaystyle{ \mathbb{Z}_{(2)} }[/math] is the discrete valuation ring corresponding to ν. The maximal ideal of [math]\displaystyle{ \mathbb{Z}_{(2)} }[/math] is the principal ideal generated by 2, i.e. [math]\displaystyle{ 2\mathbb{Z}_{(2)} }[/math], and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that [math]\displaystyle{ \mathbb{Z}_{(2)} }[/math] is the localization of the Dedekind domain [math]\displaystyle{ \mathbb{Z} }[/math] at the prime ideal generated by 2.

More generally, any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings

[math]\displaystyle{ \mathbb Z_{(p)}:=\left.\left\{\frac zn\,\right| z,n\in\mathbb Z,p\nmid n\right\} }[/math]

for any prime p in complete analogy.

p-adic integers

The ring [math]\displaystyle{ \mathbb{Z}_p }[/math] of p-adic integers is a DVR, for any prime [math]\displaystyle{ p }[/math]. Here [math]\displaystyle{ p }[/math] is an irreducible element; the valuation assigns to each [math]\displaystyle{ p }[/math]-adic integer [math]\displaystyle{ x }[/math] the largest integer [math]\displaystyle{ k }[/math] such that [math]\displaystyle{ p^k }[/math] divides [math]\displaystyle{ x }[/math].

Formal power series

Another important example of a DVR is the ring of formal power series [math]\displaystyle{ R = kT }[/math] in one variable [math]\displaystyle{ T }[/math] over some field [math]\displaystyle{ k }[/math]. The "unique" irreducible element is [math]\displaystyle{ T }[/math], the maximal ideal of [math]\displaystyle{ R }[/math] is the principal ideal generated by [math]\displaystyle{ T }[/math], and the valuation [math]\displaystyle{ \nu }[/math] assigns to each power series the index (i.e. degree) of the first non-zero coefficient.

If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that converge in a neighborhood of 0 (with the neighborhood depending on the power series). This is a discrete valuation ring. This is useful for building intuition with the Valuative criterion of properness.

Ring in function field

For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.

Scheme-theoretic

Henselian trait

For a DVR [math]\displaystyle{ R }[/math] it is common to write the fraction field as [math]\displaystyle{ K = \text{Frac}(R) }[/math] and [math]\displaystyle{ \kappa = R/\mathfrak{m} }[/math] the residue field. These correspond to the generic and closed points of [math]\displaystyle{ S=\text{Spec}(R). }[/math] For example, the closed point of [math]\displaystyle{ \text{Spec}(\mathbb{Z}_p) }[/math] is [math]\displaystyle{ \mathbb{F}_p }[/math] and the generic point is [math]\displaystyle{ \mathbb{Q}_p }[/math]. Sometimes this is denoted as

[math]\displaystyle{ \eta \to S \leftarrow s }[/math]

where [math]\displaystyle{ \eta }[/math] is the generic point and [math]\displaystyle{ s }[/math] is the closed point .

Localization of a point on a curve

Given an algebraic curve [math]\displaystyle{ (X,\mathcal{O}_X) }[/math], the local ring [math]\displaystyle{ \mathcal{O}_{X,\mathfrak{p}} }[/math] at a smooth point [math]\displaystyle{ \mathfrak{p} }[/math] is a discrete valuation ring, because it is a principal valuation ring. Note because the point [math]\displaystyle{ \mathfrak{p} }[/math] is smooth, the completion of the local ring is isomorphic to the completion of the localization of [math]\displaystyle{ \mathbb{A}^1 }[/math] at some point [math]\displaystyle{ \mathfrak{q} }[/math].

Uniformizing parameter

Given a DVR R, any irreducible element of R is a generator for the unique maximal ideal of R and vice versa. Such an element is also called a uniformizing parameter of R (or a uniformizing element, a uniformizer, or a prime element).

If we fix a uniformizing parameter t, then M=(t) is the unique maximal ideal of R, and every other non-zero ideal is a power of M, i.e. has the form (t ^k) for some k≥0. All the powers of t are distinct, and so are the powers of M. Every non-zero element x of R can be written in the form αt ^k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = kv(t). So to understand the ring completely, one needs to know the group of units of R and how the units interact additively with the powers of t.

The function v also makes any discrete valuation ring into a Euclidean domain.

Topology

Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. We can also give it a metric space structure where the distance between two elements x and y can be measured as follows:

[math]\displaystyle{ |x-y| = 2^{-\nu(x-y)} }[/math]

(or with any other fixed real number > 1 in place of 2). Intuitively: an element z is "small" and "close to 0" iff its valuation ν(z) is large. The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring.

A DVR is compact if and only if it is complete and its residue field R/M is a finite field.

Examples of complete DVRs include

• the ring of p-adic integers and
• the ring of formal power series over any field

For a given DVR, one often passes to its completion, a complete DVR containing the given ring that is often easier to study. This completion procedure can be thought of in a geometrical way as passing from rational functions to power series, or from rational numbers to the reals.

Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of [math]\displaystyle{ \Z_{(p)}=\Q \cap \Z_p }[/math] (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Z_p.

See also

• Category:Localization (mathematics)
• Local ring
• Ramification of local fields
• Cohen ring
• Valuation ring

References

• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 
• Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: John Wiley & Sons, ISBN 978-0-471-43334-7 
• Discrete valuation ring, The Encyclopaedia of Mathematics.