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. ^[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 = {0} ${\displaystyle \cup }$ {x${\displaystyle \in }$K : ν(x) ≥ 0}.

Examples

Algebraic

Localization of Dedekind rings

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

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

for any prime p in complete analogy.

p-adic integers

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

Formal power series

Another important example of a DVR is the ring of formal power series ${\displaystyle R=k[[T]]}$ in one variable ${\displaystyle T}$ over some field ${\displaystyle k}$. The "unique" irreducible element is ${\displaystyle T}$, the maximal ideal of ${\displaystyle R}$ is the principal ideal generated by ${\displaystyle T}$, and the valuation ${\displaystyle \nu }$ 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 ${\displaystyle R}$ it is common to write the fraction field as ${\displaystyle K={\text{Frac}}(R)}$ and ${\displaystyle \kappa =R/{\mathfrak {m}}}$ the residue field. These correspond to the generic and closed points of ${\displaystyle S={\text{Spec}}(R).}$ For example, the closed point of ${\displaystyle {\text{Spec}}(\mathbb {Z} _{p})}$ is ${\displaystyle \mathbb {F} _{p}}$ and the generic point is ${\displaystyle \mathbb {Q} _{p}}$. Sometimes this is denoted as

${\displaystyle \eta \to S\leftarrow s}$

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

Localization of a point on a curve

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

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:

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

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

See also

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

References

1. "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 .