Wikipedia / Discrete valuation ring
Wikipedia / 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 that satisfies any and all of the following equivalent conditions:
is a local ring, a principal ideal domain, and not a field.
is a valuation ring with a value group isomorphic to the integers under addition.
is a local ring, a Dedekind domain, and not a field.
is Noetherian and a local domain whose unique maximal ideal is principal, and not a field.
is integrally closed, Noetherian, and a local ring with Krull dimension one.
is a principal ideal domain with a unique non-zero prime ideal.
is a principal ideal domain with a unique irreducible element (up to multiplication by unitss).
is a unique factorization domain with a unique irreducible element (up to multiplication by units).
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 on the field of fractions.
is a Noetherian local ring whose unique maximal ideal is generated by a single non-nilpotent element.
Examples
Algebraic
Localization of Dedekind rings
be the localization at the ideal generated by 2. Formally,
, we can apply unique factorization.
, and the "unique" irreducible element (up to units) is 2 (also known as a uniformizing parameter).
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
for any prime in complete analogy.
p-adic integers
The ring of p-adic integers is a DVR, for any prime is an irreducible element; the valuation the largest integer.
Formal power series
Another important example of a DVR is the ring of formal power series 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
considered as a subring of the field of rational functions can be identified with the ring of all real-valued rational functions defined (i.e., finite) on a neighborhood 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
the residue field. These correspond to the generic and closed. Sometimes this is denoted as
is the closed point.
Localization of a point
Given an algebraic curve, the local ring is smooth, the completion of the local ring is isomorphic to the completion of the localization.
For any nonsingular variety, one can similarly consider the local ring at a point of codimension one. More generally, on a (locally) Noetherian normal scheme, the local ring at the generic point of a prime Weil divisor is a discrete valuation ring, with the discrete valuation giving the order of vanishing of a regular function along the divisor.
Uniformizing parameter
and vice versa. Such an element is also called a uniformizing parameter (or a uniformizing element, a uniformizer, or a prime element).
.
also makes any discrete valuation ring into a Euclidean domain.
Topology
Every discrete valuation ring, being a local ring, carries a natural (adic) topology and is a topological ring. It also admits a metric space can be measured as follows:
is "small" and "close to 0" iff its valuation, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring.
is compact if and only if it is complete and its residue field is a finite field.
Examples of complete-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.
.
See also
References