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:

R is a local principal ideal domain, and not a field.
R is a valuation ring with a value group isomorphic to the integers under addition.
R is a local Dedekind domain and not a field.
R is a Noetherian local domain whose maximal ideal is principal, and not a field.^[1]
R is an integrally closed Noetherian local ring with Krull dimension one.
R is a principal ideal domain with a unique non-zero prime ideal.
R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
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.
There is some discrete valuation ν on the field of fractions K of R such that R = {0} $\cup$ {x $\in$ K : ν(x) ≥ 0}.

Examples

Algebraic

Localization of Dedekind rings

Let $\mathbb {Z} _{(2)}:=\{z/n\mid z,n\in \mathbb {Z} ,\,\,n{\text{ is odd}}\}$ . Then, the field of fractions of $\mathbb {Z} _{(2)}$ is $\mathbb {Q}$ . For any nonzero element $r$ of $\mathbb {Q}$ , 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 $\mathbb {Z} _{(2)}$ is the discrete valuation ring corresponding to ν. The maximal ideal of $\mathbb {Z} _{(2)}$ is the principal ideal generated by 2, i.e. $2\mathbb {Z} _{(2)}$ , and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that $\mathbb {Z} _{(2)}$ is the localization of the Dedekind domain $\mathbb {Z}$ 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

\mathbb {Z} _{(p)}:=\left.\left\{{\frac {z}{n}}\,\right|z,n\in \mathbb {Z} ,p\nmid n\right\}

for any prime p in complete analogy.

p-adic integers

The ring $\mathbb {Z} _{p}$ 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

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

\eta \to S\leftarrow s

where $\eta$ is the generic point and $s$ is the closed point .

Localization of a point on a curve

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

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.^{[citation needed]}

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 $\mathbb {Z} _{(p)}=\mathbb {Q} \cap \mathbb {Z} _{p}$ (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.

References

^ "ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian". MathOverflow.

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, MR 2286236
Discrete valuation ring, The Encyclopaedia of Mathematics.

[1] "ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian". MathOverflow.

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