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Discrete valuation ring
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{{More footnotes needed|date=July 2024}} {{Short description|Concept in abstract algebra}} 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 and all of the following equivalent conditions: # ''R'' is a [[local ring]], a [[principal ideal domain]], and not a [[field (mathematics)|field]]. # ''R'' is a [[valuation ring]] with a value group isomorphic to the integers under addition. # ''R'' is a local ring, a [[Dedekind domain]], and not a field. # ''R'' is [[Noetherian ring|Noetherian]] and a [[local domain]] whose unique maximal [[ideal (ring theory)|ideal]] is principal, and not a field.<ref>{{Cite web|url=https://mathoverflow.net/questions/155621/condition-for-a-local-ring-whose-maximal-ideal-is-principal-to-be-noetherian|title=ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian|website=MathOverflow}}</ref> # ''R'' is [[integrally closed domain|integrally closed]], Noetherian, and a 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 [[unit (ring theory)|unit]]s). # ''R'' is a [[unique factorization domain]] with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a [[field (mathematics)|field]], and every nonzero [[fractional ideal]] of ''R'' is [[irreducible ideal|irreducible]] in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it. # There is some [[discrete valuation#Discrete valuation rings and valuations on fields|discrete valuation]] ν on the [[field of fractions]] ''K'' of ''R'' such that ''R'' = {0} <math> \cup </math> {''x'' <math> \in </math> ''K'' : ν(''x'') ≥ 0}. ==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>. ==<span id="uniformizer"></span>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''<sup> ''k''</sup>) 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 α''t''<sup> ''k''</sup> 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|date=May 2015}} == Topology == 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 ''x'' and ''y'' can be measured as follows: :<math>|x-y| = 2^{-\nu(x-y)}</math> (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 (algebra)|valuation]] ν(''z'') is large. The function |x-y|, supplemented by |0|=0, is the restriction of an [[Absolute value (algebra)|absolute value]] defined on the [[field of fractions]] of the discrete valuation ring. A DVR is [[compact space|compact]] if and only if it is [[complete space|complete]] and its [[residue field]] ''R''/''M'' is a [[finite field]]. Examples of [[complete space|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 of a ring|completion]], a [[complete space|complete]] DVR containing the given ring that is often easier to study. This [[Completion of a ring|completion]] procedure can be thought of in a geometrical way as passing from [[rational function]]s to [[power series]], or from [[rational number]]s to the [[real number|reals]]. 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 <math>\Z_{(p)}=\Q \cap \Z_p</math> (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'''<sub>''p''</sub>. ==See also== * [[:Category:Localization (mathematics)]] * [[Local ring]] *[[Ramification of local fields]] * [[Cohen ring]] * [[Valuation ring]] ==References== {{Reflist}} * {{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}} * {{Citation | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract algebra | publisher=[[John Wiley & Sons]] | location=New York | edition=3rd | isbn=978-0-471-43334-7 | mr=2286236 | year=2004}} *[https://encyclopediaofmath.org/wiki/Discrete_valuation_ring Discrete valuation ring], The ''[[Encyclopaedia of Mathematics]]''. [[Category:Commutative algebra]] [[Category:Localization (mathematics)]]
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