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Discrete valuation ring

A discrete valuation ring (DVR) is an integral domain that serves as the valuation ring for a discrete valuation on its field of quotients, where the valuation maps the multiplicative group of the field to the integers \mathbb{Z}.^[1] This valuation v satisfies v(xy) = v(x) + v(y) and v(x + y) \geq \min(v(x), v(y)) for nonzero elements, with the ring consisting of elements where v(x) \geq 0 and the maximal ideal comprising those with v(x) > 0.^[2] DVRs are principal ideal domains (PIDs) that are local rings with a unique nonzero prime ideal, which is maximal and principal, generated by a uniformizer \pi such that v(\pi) = 1.^[3] Every nonzero ideal in a DVR is a power of this maximal ideal, \mathfrak{m}^n = \{\alpha \in K \mid v(\alpha) \geq n\} for n \geq 0, and every nonzero element can be uniquely factored as u \pi^k where u is a unit and k \geq 0.^[1] They are Noetherian of Krull dimension 1, integrally closed in their fraction field, and regular local rings, meaning their completion with respect to the maximal ideal yields a power series ring over the residue field.^[2] Classic examples include the localization of the integers at a prime p, \mathbb{Z}_{(p)} = \{a/b \in \mathbb{Q} \mid p \nmid b\}, with p-adic valuation and residue field \mathbb{F}_p; the localization of a polynomial ring k_{(x)} over a field k at the origin, with valuation given by the order of vanishing at x=0; and the ring of formal power series k[], whose fraction field is the Laurent series field k((t)).^[1]^[2] In commutative algebra and algebraic number theory, DVRs play a central role as localizations of Dedekind domains at maximal ideals, facilitating the study of ramification, completions, and étale cohomology; they also model one-dimensional local rings in algebraic geometry, such as points on curves.^[3]

Definition and characterizations

Definition via valuation

A discrete valuation on a field K is a surjective group homomorphism v: K^\times \to \mathbb{Z} satisfying the ultrametric inequality v(x + y) \geq \min(v(x), v(y)) for all x, y \in K^\times with x + y \neq 0, extended by setting v(0) = +\infty.^[1]^[2] This valuation measures the "order of vanishing" of elements in K, with the properties ensuring multiplicativity and a non-Archimedean triangle inequality.^[4] Given such a valuation v, the associated valuation ring is defined as O_v = \{ x \in K \mid v(x) \geq 0 \} \cup \{0\}, which consists of all elements with non-negative valuation (including zero).^[1]^[2] The maximal ideal of O_v is then \mathfrak{m}_v = \{ x \in K \mid v(x) > 0 \}, comprising elements with positive valuation.^[1]^[4] The ring O_v is an integral domain with fraction field K, and its units are precisely the elements of valuation zero: O_v^\times = \{ x \in K^\times \mid v(x) = 0 \}.^[2] The ring O_v is called a discrete valuation ring (DVR) precisely when the value group v(K^\times) is isomorphic to \mathbb{Z}, making the valuation discrete in the sense that the possible valuations are exactly the integers.^[1]^[4] In this case, the residue field is \kappa = O_v / \mathfrak{m}_v, which is a field obtained by modding out the maximal ideal.^[2] A uniformizer \pi \in O_v is any element with v(\pi) = 1, which generates the maximal ideal \mathfrak{m}_v = (\pi).^[1]^[2] A key structural feature of a DVR is that every nonzero ideal is of the form \mathfrak{m}_v^n = \{ x \in K \mid v(x) \geq n \} for some integer n \geq 0, yielding a chain of ideals totally ordered by inclusion and corresponding to the discrete value group.^[1]^[2] This principal ideal structure underscores the ring's simplicity while highlighting its local nature with a single nonzero prime ideal.^[4]

Equivalent ring-theoretic definitions

A discrete valuation ring O is equivalently defined as a local principal ideal domain that is not a field and has a unique nonzero prime ideal \mathfrak{m}, which is maximal.$$] This characterization emphasizes the ring's structure as a principal ideal domain (PID) localized at its sole nonzero prime, ensuring all ideals are powers of \mathfrak{m}. Further ring-theoretic equivalents include the following: O is a discrete valuation ring if and only if it is a local ring with principal maximal ideal \mathfrak{m} such that every nonzero ideal of O is a power of \mathfrak{m}.[

Equivalently, $O$ is a one-dimensional [regular local ring](/page/Regular_local_ring).

] These conditions capture the discrete nature without reference to an underlying valuation on the fraction field. A key theorem states that a Noetherian local domain of Krull dimension 1 is a discrete valuation ring if and only if its maximal ideal is principal.[$$ This result bridges dimension theory and ideal structure, showing that principality of the maximal ideal suffices for the discrete valuation property in this setting. Discrete valuation rings are integrally closed in their fraction fields, meaning the integral closure of O in \mathrm{Frac}(O) is O itself.$$] This integrality follows from the valuation-theoretic origins but holds as a purely ring-theoretic fact. In contrast to general valuation rings, where the associated value group may be any ordered abelian group, a discrete valuation ring requires the value group to be isomorphic to \mathbb{Z}, ensuring the "discreteness" in the ordering of ideal powers.[$$

Structural properties

Ideal and module structure

A discrete valuation ring (DVR) O is a principal ideal domain with a unique nonzero prime ideal, which serves as its maximal ideal \mathfrak{m}. All ideals of O are principal and form a total chain: (0) \subset \mathfrak{m}^n \subset \cdots \subset \mathfrak{m}^2 \subset \mathfrak{m} \subset O for n \geq 1, where \mathfrak{m}^k = (\pi^k) for a uniformizer \pi generating \mathfrak{m}. This structure arises because the residue field O/\mathfrak{m} is a field and the valuation ensures that every nonzero ideal is generated by a power of \pi.^[5]^[6] Finitely generated modules over a DVR classify neatly into torsion-free and torsion types. Torsion-free modules, those with no nonzero elements annihilated by nonzero elements of O, are free of finite rank; specifically, any such module of rank n is isomorphic to O^n. This follows from the local nature of the DVR and the fact that projective modules over PIDs are free. In contrast, torsion modules—those where every element is annihilated by some nonzero element of O—decompose as finite direct sums of cyclic modules of the form O/(\pi^k) for k \geq 1. This decomposition is unique up to isomorphism, reflecting the principal ideal chain.^[5]^[7]^[6] A key consequence for modules over DVRs is a specialization of Nakayama's lemma: for a finitely generated module M, if \mathfrak{m} M = M, then M = 0. This holds because \mathfrak{m} is principal and the DVR is local, ensuring that no nonzero module can be fully annihilated by the maximal ideal without being trivial. This lemma simplifies many arguments about module generation and minimality in the DVR setting.^[5]^[7] DVRs are Euclidean domains, admitting a division algorithm via the discrete valuation v: K^\times \to \mathbb{Z} on the fraction field K, where the Euclidean function is \nu(x) = v(x) for x \in O \setminus \{0\}. For any a, b \in O with b \neq 0, there exist q, r \in O such that a = q b + r with either r = 0 or v(r) > v(b), enabling the standard Euclidean algorithm for gcd computations.^[7]^[6]

Noetherian and dimension properties

Discrete valuation rings (DVRs) are Noetherian rings, satisfying the ascending chain condition on ideals. This follows from the fact that every nonzero proper ideal is a power of the unique maximal ideal \mathfrak{m}, and the discrete nature of the valuation ensures that any ascending chain of ideals must stabilize, as the exponents in the valuation cannot increase indefinitely without repetition.^[4] The Krull dimension of a DVR is 1. The spectrum consists solely of the zero ideal and the maximal ideal \mathfrak{m}, forming a unique chain of prime ideals of length 1, with height \mathrm{ht}(\mathfrak{m}) = 1.^[8] DVRs are regular local rings. The embedding dimension equals the Krull dimension, as the cotangent space \mathfrak{m}/\mathfrak{m}^2 is one-dimensional over the residue field, generated by the image of a uniformizing parameter \pi.^[9] By the Cohen structure theorem, in the equal characteristic case, a complete DVR with residue field k is isomorphic to the power series ring k[[ \pi ]], where \pi is a uniformizer.^[10] DVRs are excellent rings, being catenary and universally catenary, which ensures well-behaved dimension theory in extensions and localizations. Although DVRs are principal ideal domains and thus unique factorization domains, this property distinguishes them from more general valuation rings that may lack unique factorization.^[11]

Examples

Valuation rings in number theory

In number theory, discrete valuation rings (DVRs) play a central role in the local study of number fields, particularly through the localization of Dedekind domains at prime ideals. A Dedekind domain A, such as the ring of integers \mathcal{O}_K in a number field K, is an integrally closed Noetherian domain of dimension 1 where every nonzero prime ideal is maximal. The localization A_{\mathfrak{p}} of A at a nonzero prime ideal \mathfrak{p} is a DVR, with unique maximal ideal \mathfrak{m} = \mathfrak{p} A_{\mathfrak{p}}, which is principal and generated by a uniformizer.^[12] This structure arises because A_{\mathfrak{p}} is a local principal ideal domain with exactly one nonzero prime ideal, and the associated valuation on the fraction field of A is discrete.^[12] A fundamental example is the localization of the rational integers \mathbb{Z} at the prime ideal (p) for a prime p, yielding the ring \mathbb{Z}_{(p)} = \{ a/b \in \mathbb{Q} \mid a, b \in \mathbb{Z}, p \nmid b \}. This is a DVR whose maximal ideal is (p) \mathbb{Z}_{(p)}, and it admits the p-adic valuation v_p(a/b) = v_p(a) - v_p(b), where v_p(n) for n \in \mathbb{Z} is the highest power of p dividing n.^[1] The residue field of \mathbb{Z}_{(p)} is the finite field \mathbb{Z}/p\mathbb{Z}, and every nonzero ideal is a power of the maximal ideal, reflecting its principal ideal domain property.^[1] The p-adic integers \mathbb{Z}_p provide another key DVR, constructed as the completion of \mathbb{Z}_{(p)} (or equivalently \mathbb{Z}) with respect to the p-adic metric induced by v_p. This completion is a compact DVR with maximal ideal (p), where elements are formal series \sum_{i=0}^\infty a_i p^i with a_i \in \{0, 1, \dots, p-1\}, and the group of units is \mathbb{Z}_p^\times = \{ x \in \mathbb{Z}_p \mid v_p(x) = 0 \}.^[13] The residue field remains \mathbb{Z}/p\mathbb{Z}, and \mathbb{Z}_p serves as the valuation ring for the p-adic numbers \mathbb{Q}_p.^[13] The discrete valuations on the field of rational numbers \mathbb{Q} are exactly the non-archimedean p-adic valuations for each prime p, defined by v_p\left( \pm \prod_q q^{e_q} \right) = e_p on nonzero rationals, extended by v_p(0) = \infty.^[14] These extend naturally to finite extensions of \mathbb{Q}, such as number fields, via places associated to prime ideals of the ring of integers. In global fields (finite extensions of \mathbb{Q} or of \mathbb{F}_p(t)), the DVRs correspond precisely to the valuation rings at finite places, which are the non-archimedean places induced by prime ideals, with completions yielding local fields.^[15]

Rings in commutative algebra

In commutative algebra, discrete valuation rings (DVRs) arise prominently as localizations of polynomial rings and as formal power series rings, providing essential models for studying local ring properties and valuations in abstract settings. A fundamental construction begins with any field K equipped with a discrete valuation v: K^\times \to \mathbb{Z}; the associated valuation ring is O_v = \{ x \in K \mid v(x) \geq 0 \} \cup \{0\}, which forms a DVR with maximal ideal \mathfrak{m}_v = \{ x \in K \mid v(x) > 0 \} \cup \{0\}.^[10] This structure captures the "integral" elements under the valuation and serves as a building block for more specific algebraic examples.^[16] A canonical example is the formal power series ring k[] over a field k, consisting of infinite series \sum_{i=0}^\infty a_i t^i with a_i \in k. This ring is a DVR, where the maximal ideal is \mathfrak{m} = (t), generated by the uniformizer t, and the valuation \mathrm{ord}_t(f) assigns to each nonzero f \in k[] the lowest power with a nonzero coefficient.^[10] The residue field is k[] / (t) \cong k, and the fraction field is the Laurent series field k((t)).^[16] The ring k[] is complete with respect to the \mathfrak{m}-adic topology, making it a key tool for analyzing completions in commutative algebra.^[10] Another core example is the localization of the polynomial ring k at the prime ideal (x), denoted k_{(x)} = \{ f/g \mid f, g \in k, g(0) \neq 0 \}. This is a DVR with maximal ideal \mathfrak{m} = (x) k_{(x)} and residue field k_{(x)} / \mathfrak{m} \cong k.^[17] The valuation corresponds to the order of vanishing at x=0, aligning with the general construction above. In the context of function fields, for K = k(x), the ring \{ f/g \mid f, g \in k, g(0) \neq 0 \} precisely realizes the valuation ring associated to the discrete valuation at x=0, measuring the order of zeros or poles at that point.^[10] These constructions yield complete DVRs when the base field k is complete (e.g., the complexes), and they model the local behavior of algebraic varieties at smooth points, facilitating the study of ideals, modules, and dimension in commutative algebra and beyond.^[16] For instance, k[] serves as the completion of k_{(x)} at \mathfrak{m}, bridging polynomial and power series perspectives.^[10]

Geometric and scheme-theoretic examples

In algebraic geometry, a fundamental example of a discrete valuation ring (DVR) arises as the local ring at a smooth point on an algebraic curve. Consider an algebraic curve C defined over a field k, and let p be a closed point on C. The local ring \mathcal{O}_{C,p} is then a DVR, with maximal ideal \mathfrak{m}_p consisting of the germs of regular functions on C that vanish at p.^[18]^[19] This structure reflects the one-dimensional nature of the curve at a smooth point, where the valuation corresponds to the order of vanishing at p. More generally, in the context of function fields, the valuation ring associated to a place of a function field K/k provides another geometric realization of a DVR. Here, K is the function field of an algebraic curve over k, and a place corresponds to a point on the curve (or more precisely, a prime ideal in the integral closure). The valuation ring \mathcal{O}_P at such a place P is a DVR whose fraction field is K, with the maximal ideal generated by elements of positive valuation at P. This construction links points on the curve to the DVRs dominating the coordinate ring.^[20]^[21] In scheme theory, the spectrum of a DVR, denoted \operatorname{Spec}(O), forms a Henselian trait, consisting of a generic point \eta = \operatorname{Spec}(\operatorname{Frac}(O)) and a closed point s = \operatorname{Spec}(\kappa), where \kappa is the residue field of O. This trait structure models the "local behavior" of schemes over a DVR base, facilitating constructions like models of curves.^[22] For instance, such traits appear in the study of arithmetic surfaces or relative curves over DVRs. A key structural fact is that DVRs precisely classify one-dimensional regular local rings: a Noetherian local domain of dimension one is regular if and only if it is a DVR. This characterization is pivotal in resolution of singularities for curves, where the normalization of a curve ring at a height-one prime yields a DVR when the normalization is regular, effectively resolving singularities by replacing singular points with smooth ones.^[18]^[23] In dimension one, this process ensures the integral closure is a product of DVRs, providing a regular model of the curve.^[24]

Uniformizing parameter

Definition and role in ideals

In a discrete valuation ring O, equipped with a discrete valuation v: K^\times \to \mathbb{Z} where K is the fraction field of O, a uniformizing parameter (or uniformizer) \pi \in O is an element satisfying v(\pi) = 1 and generating the unique maximal ideal \mathfrak{m} of O principally, i.e., \mathfrak{m} = (\pi).^[8] This property follows from the discreteness of the valuation, which ensures that the value group v(O \setminus \{0\}) = \mathbb{Z}_{\geq 0}, allowing the selection of such a generator for \mathfrak{m}, and the principal ideal domain structure of O.^[25] The existence of \pi underscores the ring's local principal nature, distinguishing discrete valuation rings from more general valuation rings.^[8] Every nonzero element x \in O can be uniquely expressed as x = u \pi^k, where u is a unit in the multiplicative group O^\times and k = v(x) \geq 0 is a nonnegative integer.^[25] This decomposition highlights the uniformizer's role in capturing the valuation directly and facilitates the description of the ring's elements in terms of their "order of vanishing" at \mathfrak{m}. The principal generation extends to higher powers of the maximal ideal: for each n \geq 1, \mathfrak{m}^n = (\pi^n), and the zero ideal corresponds to n = \infty in this chain.^[8] Consequently, all nonzero ideals of O are of the form (\pi^n) for some n \geq 0, forming a total chain under inclusion, which reflects the one-dimensional Krull dimension of the ring.^[25]

Units and factorization

In a discrete valuation ring O with maximal ideal \mathfrak{m} and valuation v, the group of units O^\times consists of all elements x \in O such that v(x) = 0, equivalently x \notin \mathfrak{m}. This forms a multiplicative group under the ring multiplication, as any such x has an inverse also in O with valuation 0.^[26] For a complete discrete valuation ring O with residue field \kappa = O / \mathfrak{m}, the unit group admits a canonical decomposition O^\times \cong \kappa^\times \times (1 + \mathfrak{m}), where \kappa^\times is the multiplicative group of the residue field and $1 + \mathfrak{m} = \{ 1 + y \mid y \in \mathfrak{m} \} is a profinite group. This isomorphism arises from the natural projection O^\times \to \kappa^\times with kernel $1 + \mathfrak{m}, reflecting the topological and algebraic structure induced by the valuation.^[27] Every nonzero nonunit element of O factors uniquely as u \pi^k with u \in O^\times, k \geq 1, and \pi a uniformizer generating \mathfrak{m}. Discrete valuation rings are unique factorization domains, as they are principal ideal domains of Krull dimension 1; moreover, all irreducible elements are associates of \pi, i.e., of the form u \pi for some unit u.^[26] For example, in the ring of p-adic integers \mathbb{Z}_p, the units \mathbb{Z}_p^\times comprise all p-adic integers not divisible by the prime p.^[28]

Topological aspects

Discrete metric and topology

A discrete valuation on a field K induces a metric on K known as the \pi-adic metric, where \pi is a uniformizing parameter. For x, y \in K, the distance is defined as d(x, y) = b^{-v(x-y)} for some base b > 1, or equivalently via the \pi-adic absolute value |x|_\pi = b^{-v(x)}, with d(x, y) = |x - y|_\pi. This metric satisfies the non-archimedean property, specifically the ultrametric inequality |x + y|_\pi \leq \max(|x|_\pi, |y|_\pi), which implies that the valuation v is totally ordered on the elements of K up to units, and induces a total order on the associated residue classes via the filtration by powers of the maximal ideal.^[25]^[29] The topology on K is the one generated by this metric, making K a metric space. The open balls centered at 0, B(0, b^{-n}) = \{x \in K \mid |x|_\pi < b^{-n}\} = \{x \in K \mid v(x) \geq n\} = \mathfrak{m}^n, where \mathfrak{m} is the maximal ideal of the valuation ring \mathcal{O}_K, form a basis of neighborhoods of 0 and hence generate the topology on K. This topology is Hausdorff, as distinct points x \neq y satisfy v(x - y) = k < \infty, allowing disjoint open balls around them.^[25]^[29] Furthermore, the topology is totally disconnected, with clopen balls and singletons as connected components, reflecting the discrete nature of the valuation. If \mathcal{O}_K is complete and its residue field is finite, then \mathcal{O}_K is homeomorphic to the Cantor set as a topological space.^[25]^[29] Regarding completion, a sequence (x_n) in K is Cauchy with respect to the \pi-adic metric if v(x_n - x_m) \to \infty as n, m \to \infty, and in the completion, such sequences converge.^[25]^[29] A discrete valuation ring O is complete if every Cauchy sequence in O with respect to the \mathfrak{m}-adic metric converges in O, where \mathfrak{m} is the maximal ideal.^[30] This property ensures that O is closed in its fraction field under the valuation topology. A canonical example of a complete discrete valuation ring is the formal power series ring k[], where k is a field; here, the maximal ideal is (t), and completeness follows from the ability to take limits of series term by term.^[30] Not all discrete valuation rings are complete. For instance, the localization \mathbb{Z}_{(p)} of the integers at the prime ideal (p) is a discrete valuation ring that is incomplete with respect to its p-adic topology. Its completion \hat{\mathbb{Z}}_{(p)} is the ring of p-adic integers \mathbb{Z}_p, which is a complete discrete valuation ring with the same residue field \mathbb{F}_p. In general, for any discrete valuation ring O with maximal ideal \mathfrak{m}, the completion \hat{O} with respect to the \mathfrak{m}-adic topology is again a complete discrete valuation ring, and the residue fields satisfy \hat{O}/\hat{\mathfrak{m}} \cong O/\mathfrak{m}.^[30] Moreover, if O is complete and its residue field is finite, then O is compact in the \mathfrak{m}-adic topology. The henselization O^h of a discrete valuation ring O is the minimal henselian extension of O, which for DVRs is again a DVR with the same value group and residue field; it can be realized (under suitable hypotheses, such as O being excellent) as the subring of the completion \hat{O} consisting of elements algebraic over the fraction field K = \operatorname{Frac}(O).^[31] This construction yields a henselian ring, meaning it satisfies a version of Hensel's lemma: for a polynomial f(x) \in O^h and \alpha \in O^h/\mathfrak{m} such that f(\alpha) \equiv 0 \pmod{\mathfrak{m}} and f'(\alpha) \not\equiv 0 \pmod{\mathfrak{m}}, there exists a unique lift \tilde{\alpha} \in O^h with f(\tilde{\alpha}) = 0 and \tilde{\alpha} \equiv \alpha \pmod{\mathfrak{m}}. Henselian rings facilitate lifting solutions from the residue field to the ring, bridging algebraic and topological aspects. In the context of rigid analytic geometry, extensions related to henselization of discrete valuation rings provide constructions that preserve valuation properties while adapting to analytic spaces.^[32]

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