Editing Discrete valuation ring (section)

==Examples==

=== Algebraic ===

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

More generally, any [[Localization (commutative algebra)|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 [[ring (mathematics)|rings]]
:<math>\mathbb Z_{(p)}:=\left.\left\{\frac zn\,\right| z,n\in\mathbb Z,p\nmid n\right\}</math>
for any [[prime number|prime]] ''p'' in complete analogy.

==== ''p''-adic integers ====
The [[ring (mathematics)|ring]] <math>\mathbb{Z}_p</math> of [[p-adic integer|''p''-adic integers]] is a DVR, for any [[prime number|prime]] <math>p</math>. Here <math>p</math> is an [[irreducible element]]; the [[valuation (algebra)|valuation]] assigns to each <math>p</math>-adic integer <math>x</math> the largest [[integer]] <math>k</math> such that <math>p^k</math> divides <math>x</math>.

==== Formal power series ====
Another important example of a DVR is the [[formal power series|ring of formal power series]] <math>R = k[[T]]</math> in one variable <math>T</math> over some field <math>k</math>. The "unique" irreducible element is <math>T</math>, the maximal ideal of <math>R</math> is the principal ideal generated by <math>T</math>, and the valuation <math>\nu</math> assigns to each power series the index (i.e. degree) of the first non-zero coefficient.

If we restrict ourselves to [[real number|real]] or [[complex number|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'' [[polynomial]]s in '''R'''[''X'']  and ''g''(0) ≠ 0}, considered as a [[subring]] of the field of [[rational function]]s '''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 (topology)|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 curve]]s near non-[[Singular point of a curve|singular points]], the algebraic curve in this case being the real line.

=== Scheme-theoretic ===

==== Henselian trait ====
For a DVR <math>R</math> it is common to write the fraction field as <math>K = \text{Frac}(R)</math> and <math>\kappa = R/\mathfrak{m}</math> the [[residue field]]. These correspond to the [[Generic point|generic]] and closed points of <math>S=\text{Spec}(R).</math> For example, the closed point of <math>\text{Spec}(\mathbb{Z}_p)</math> is <math>\mathbb{F}_p</math> and the generic point is <math>\mathbb{Q}_p</math>. Sometimes this is denoted as

:<math>
\eta \to S \leftarrow s
</math>

where <math>\eta</math> is the generic point and <math>s</math> is the closed point <!-- https://math.stackexchange.com/questions/2321214/grothendiecks-vanishing-cycles (as in SGA) -->.

==== Localization of a point on a curve ====
Given an [[algebraic curve]] <math>(X,\mathcal{O}_X)</math>, the [[local ring]] <math>\mathcal{O}_{X,\mathfrak{p}}</math> at a smooth point <math>\mathfrak{p}</math> is a discrete valuation ring, because it is a principal valuation ring. Note because the point <math>\mathfrak{p}</math> is smooth, the [[Completion of a ring|completion]] of the [[local ring]] is [[isomorphism|isomorphic]] to the completion of the [[Localization (commutative algebra)|localization]] of <math>\mathbb{A}^1</math> at some point <math>\mathfrak{q}</math>.