Discrete Valuation Ring
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| Discrete Valuation Ring | |
|---|---|
| Name | Discrete Valuation Ring |
| Type | Commutative ring |
| Properties | Principal ideal domain (local), valuation ring |
| Typical examples | Localizations of Dedekind rings, power series rings |
Discrete Valuation Ring A discrete valuation ring is a principal local ring used in algebraic number theory and algebraic geometry, characterized by a single nonzero maximal ideal and a discrete valuation measuring divisibility. It figures in the study of local fields such as p-adic numbers and appears in the local analysis of schemes related to Dedekind domain, Noetherian ring, and regular local ring structures. Classical contributors include Krull, Weil, Ostrowski, and Hensel, whose work links discrete valuation rings with completions, valuations, and local class field theory.
Definition and basic properties
A discrete valuation ring is defined via a discrete valuation v on a field K that takes values in the integers, with the valuation ring O_v = {x in K : v(x) ≥ 0} a principal ideal domain having a unique maximal ideal generated by any uniformizer. Foundational results connect this definition to equivalences: a Noetherian local domain of Krull dimension one that is integrally closed is a discrete valuation ring, a statement appearing in theorems by Dedekind, Kronecker, and Noether. Key properties relate to the structure of ideals, the existence of a parameter element, and the behavior under localization at height-one primes as in Hilbert's Nullstellensatz contexts and Zariski's local uniformization techniques.
Examples and constructions
Standard examples include the localization of a Dedekind domain at a nonzero prime ideal such as localizations of the ring of integers O_K of a number field at primes above a rational prime studied by Dirichlet, Hilbert, and Artin. The ring of formal power series kt over a field k, and the valuation rings of discrete valuations arising from divisors on curves as in work by Riemann, Hurwitz, and Grothendieck provide geometric examples. Local rings at smooth points of algebraic curves studied by Weierstrass and Abel are discrete valuation rings. In arithmetic, the rings of integers inside local fields like the field of p-adic numbers introduced by Hensel and developed in Tate's harmonic analysis serve as archetypes. Constructions via completion link to studies by Ramanujan and Eisenstein when examining ramification and uniformizers in cyclotomic extensions related to Galois theory.
Valuation theory and equivalences
Valuation theory developed by Krull and Ostrowski establishes equivalences between discrete valuations, valuation rings, and rank-one discrete valuation groups such as Z. The Ostrowski theorem classifying absolute values on Q and the product formula in global fields studied by Weil and Artin highlight the role of discrete valuations at finite places. The equivalence between DVRs and principal ideal local domains of dimension one is central in the work of Zariski, Samuel, and Matsumura, and ties to ramification theory as in Abhyankar's valuation-theoretic approaches to resolution of singularities. Valuation spectra considered by Hochster and Fontaine further generalize discrete valuations in rigid-analytic and p-adic Hodge contexts formulated by Faltings.
Algebraic and geometric applications
Discrete valuation rings enter algebraic geometry through local analysis of curves, surfaces, and models over Dedekind bases as in the studies by Grothendieck, Serre, and Deligne. They underpin the definition of divisors and the Picard group on curves traced to Riemann and Roch, and they are essential in the formulation of intersection theory advanced by Fulton and Mumford. In arithmetic geometry, DVRs regulate integral models of elliptic curves investigated by Tate and Néron, and they control reduction types central to Modular forms and Shimura varieties studied by Langlands and Deligne. In deformation theory and moduli problems as in work by Artin and Schlessinger, DVRs frequently model discrete one-parameter degenerations and specializations.
Extensions, completions, and discrete valuation fields
Extensions of DVRs correspond to extensions of discrete valuation fields, a theme in local field theory developed by Hasse, Tate, and Iwasawa. Ramification theory, including upper and lower ramification groups studied by Herbrand and Sen, describes how maximal ideals and uniformizers transform under finite extensions. Completion with respect to the valuation produces complete DVRs such as the p-adic integers Z_p, central in local class field theory by Artin and Tate. The structure of finite extensions and integral closures of DVRs relates to the work of Eisenstein on irreducibility criteria and to Dedekind's discriminant theory in number fields such as cyclotomic fields studied by Kummer and Leopoldt.
Homological and category-theoretic aspects
From a homological perspective, DVRs are regular local rings of dimension one, so projective and flat module behavior simplifies following results by Serre and Auslander–Buchsbaum. The category of finite modules over a DVR is well-understood via classification into torsion and torsion-free parts, echoing the structure theorem for modules over PID by Frobenius and Smith. Derived and categorical methods applied by Verdier and Neeman utilize DVRs as test cases in triangulated categories and t-structures arising in coherent cohomology as developed by Hartshorne and Illusie. In topos-theoretic and motivic frameworks, DVRs appear in localization sequences and purity results pursued by Voevodsky and Beilinson.