Valuation Rings

Core Idea

A valuation ring is an integral domain V such that for every nonzero element x of its fraction field, either x or x^{-1} belongs to V. This total ordering on divisibility makes valuation rings the algebraic analogue of "measuring size" in a field. Discrete valuation rings (DVRs) -- the Noetherian valuation rings -- are local PIDs with a single uniformizer, and they are exactly the regular local rings of dimension 1. DVRs arise as the local rings of smooth curves at points and as completions of number rings at primes, making them foundational in both algebraic geometry and algebraic number theory.

Explainer

A valuation ring is an integral domain V with fraction field K such that for every nonzero x in K, either x belongs to V or x^{-1} belongs to V. Equivalently, the ideals of V are totally ordered by inclusion. This extreme structural simplicity makes valuation rings the building blocks for understanding local behavior in algebra and geometry. Every valuation ring is local: the set of non-units is the unique maximal ideal m, consisting of elements x with x^{-1} not in V.

A valuation on a field K is a function v: K* → G to a totally ordered abelian group G satisfying v(xy) = v(x) + v(y) and v(x + y) ≥ min(v(x), v(y)). The valuation ring of v is V = {x in K : v(x) ≥ 0} ∪ {0}, and every valuation ring arises this way. The group G is called the value group. When G = Z, the valuation is discrete and V is a discrete valuation ring (DVR). DVRs are characterized by multiple equivalent conditions: they are the Noetherian valuation rings, the regular local rings of dimension 1, the local PIDs that are not fields, and the integrally closed local Noetherian domains of dimension 1. In a DVR, every nonzero ideal is a power (π^n) of the maximal ideal m = (π), where π is the uniformizer.

DVRs are ubiquitous in algebraic geometry and number theory. In algebraic geometry, the local ring of a smooth algebraic curve at a closed point is a DVR. The uniformizer is a local coordinate, and the valuation measures the order of vanishing of a function at the point. In number theory, the localization Z_(p) is a DVR with uniformizer p, and its completion is the p-adic integers Z_p. The p-adic valuation v_p(n) = max{k : p^k | n} measures divisibility by p. More generally, the localization of a Dedekind domain at a nonzero prime is always a DVR -- this is one of the equivalent characterizations of Dedekind domains.

Non-discrete valuation rings, while non-Noetherian, play important theoretical roles. The extension theorem for valuations (a corollary of Zorn's lemma) states that every valuation on a subfield extends to a valuation on any field extension. This is used in the proof of the going down theorem for integrally closed domains and in constructing integral closures. Places (equivalence classes of valuations) provide the "points at infinity" needed to compactify algebraic curves: the abstract Riemann surface of a function field K/k is the set of all valuation rings of K containing k, and it is a complete nonsingular curve. This perspective -- geometric objects built from valuation rings -- is a precursor to Grothendieck's scheme theory and remains essential in birational geometry.