Discrete valuation ring explained

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.^[1]
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 =

\cup .

Examples

Algebraic

Localization of Dedekind rings

Let

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

Z _{(p)} : = \left.\left\{ (zn\right|)/(z,n\in Z,p\nmid n\right\}) for any prime p in complete analogy.

p-adic integers

Z _{p} of p-adic integers is a DVR, for any prime p . Here p is an irreducible element; the valuation assigns to each p -adic integer x the largest integer k such that p ^{k} divides x .

Formal power series

R = k[[T]] in one variable T over some field k . The "unique" irreducible element is T , the maximal ideal of R is the principal ideal generated by T , and the valuation \nu 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 =, 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

R it is common to write the fraction field as K = Frac (R) and \kappa = R/ ak{m} the residue field. These correspond to the generic and closed points of S = Spec (R). For example, the closed point of Spec ( Z _{p)} is F _{p} and the generic point is Q _{p} . Sometimes this is denoted asη \to S \leftarrow s

where

η is the generic point and s is the closed point .

Localization of a point on a curve

(X, l{O} _{X)} , the local ring l{O} _{X,ak{p} } at a smooth point ak{p} is a discrete valuation ring, because it is a principal valuation ring. Note because the point ak{p} is smooth, the completion of the local ring is isomorphic to the completion of the localization of A ^{1} at some point ak{q} .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:

|x - y| = 2 ^{-\nu(x-y)} (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

\Z _{(p)} = \Q \cap \Z _{p} (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

• • Local ring
• Ramification of local fields
• Cohen ring
• Valuation ring

References

• • • Discrete valuation ring, The Encyclopaedia of Mathematics.

Notes and References

1. Web site: ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian. MathOverflow.