Discrete valuation ring

(Redirected from Uniformizer)

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:

  1. is a local ring, a principal ideal ___domain, and not a field.
  2. is a valuation ring with a value group isomorphic to the integers under addition.
  3. is a local ring, a Dedekind ___domain, and not a field.
  4. is Noetherian and a local ___domain whose unique maximal ideal is principal, and not a field.[1]
  5. is integrally closed, Noetherian, and a local ring with Krull dimension one.
  6. is a principal ideal ___domain with a unique non-zero prime ideal.
  7. is a principal ideal ___domain with a unique irreducible element (up to multiplication by units).
  8. is a unique factorization ___domain with a unique irreducible element (up to multiplication by units).
  9. 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 on the field of fractions of such that .

Examples

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Algebraic

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Localization of Dedekind rings

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Let   be the localization of   at the ideal generated by 2. Formally,

 .

The field of fractions of   is  . For any nonzero element   of  , we can apply unique factorization to the numerator and denominator of   to write   as   where  ,  , and   are integers with   and   odd. In this case, we define  .

Then   is the discrete valuation ring corresponding to  . The maximal ideal of   is the principal ideal generated by 2; i.e.,  , 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

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The ring   of p-adic integers is a DVR, for any prime  . Here   is an irreducible element; the valuation assigns to each  -adic integer   the largest integer   such that   divides  .

Formal power series

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Another important example of a DVR is the ring of formal power series   in one variable   over some field  . The "unique" irreducible element is  , the maximal ideal of   is the principal ideal generated by  , and the valuation   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

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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 of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is   and the valuation assigns to each function   the order (possibly 0) of the zero of   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

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Henselian trait

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For a DVR   it is common to write the fraction field as   and   the residue field. These correspond to the generic and closed points of   For example, the closed point of   is   and the generic point is  . Sometimes this is denoted as

 

where   is the generic point and   is the closed point .

Localization of a point on a curve

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Given an algebraic curve  , the local ring   at a smooth point   is a discrete valuation ring, because it is a principal valuation ring. Note because the point   is smooth, the completion of the local ring is isomorphic to the completion of the localization of   at some point  .

Uniformizing parameter

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

If we fix a uniformizing parameter  , then   is the unique maximal ideal of  , and every other non-zero ideal is a power of  ; i.e. has the form   for some  . All the powers of   are distinct, and so are the powers of  . Every non-zero element   of   can be written in the form   with   a unit in   and  , both uniquely determined by  . The valuation is given by  . Thus, to understand the ring completely, one needs to know the group of units of   and how the units interact additively with the powers of  .

The function   also makes any discrete valuation ring into a Euclidean ___domain.[citation needed]

Topology

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

 ,

where  . Intuitively, an element   is "small" and "close to 0" iff its valuation   is large. The above metric, along with the condition  , is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring.

A DVR   with maximal ideal   is compact if and only if it is complete and its residue field   is a finite field.

Examples of complete DVRs include the ring of  -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.

The ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined 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   (which can be seen as the set of all rational numbers that are  -adic integers) is the ring of all  -adic integers  .

See also

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References

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