Basic Concepts of Valuation and Discrete Valuation Rings

Introduction

Valuation theory is a foundational pillar in algebraic geometry and number theory, providing a framework to study the behavior of rings under localizations and the structure of their residue fields. A valuation is a function that assigns a non-negative integer to elements of a ring, capturing the "size" or "valuation" of elements with respect to a specific ideal. Discrete valuation rings (DVRs) are a special class of rings that extend this concept by imposing a discrete valuation, making them particularly useful in analyzing the structure of rings of fractions and their associated fields. This article explores the core concepts of valuation theory and discrete valuation rings, emphasizing their properties, relationships, and applications in algebraic contexts.

Core Content

Valuation Theory

A valuation on a ring $ R $ is a function $ v: R \to \mathbb{Z}_{\geq 0} $ satisfying the following properties:

  1. $ v(0) = \infty $, ensuring that the zero element is not assigned a finite value.
  2. $ v(ab) = v(a) + v(b) $ for all $ a, b \in R $.
  3. $ v(a + b) \leq \max{v(a), v(b)} $, reflecting the subadditivity of valuations.

The valuation ring $ \mathcal{O}_v $ of a valuation $ v $ is the set of elements $ a \in R $ for which $ v(a) = 0 $. This ring is always a subring of $ R $ and is integrally closed. A discrete valuation ring (DVR) is a noetherian local ring with a discrete valuation, meaning it has a unique maximal ideal $ \mathfrak{m} $ such that $ \mathfrak{m}^2 = 0 $, and the quotient $ R/\mathfrak{m} $ is a field.

The valuation ideal $ \mathfrak{v} = { a \in R \mid v(a) > 0 } $ is a prime ideal, and $ \mathcal{O}v $ is the integral closure of $ R/\mathfrak{v} $ in $ R $. Valuations are closely tied to the concept of rings of fractions, where $ R{\mathfrak{v}} $, the localization of $ R $ at $ \mathfrak{v} $, is a discrete valuation ring.

Discrete Valuation Rings

A discrete valuation ring is a noetherian local ring $ R $ with a discrete valuation $ v $, and it satisfies the following properties:

  1. $ R $ is a Dedekind domain, meaning it is a noetherian integral domain with finite Krull dimension and a finite number of prime ideals.
  2. The residue field $ R/\mathfrak{m} $ is a finite field, and the quotient ring $ R/\mathfrak{m} $ is a field.
  3. $ R $ is integrally closed, ensuring that the quotient $ R/\mathfrak{m} $ is the integral closure of $ R/\mathfrak{v} $ in $ R $.

The valuation ideal $ \mathfrak{v} $ is a prime ideal, and the local ring $ R_{\mathfrak{v}} $ is a discrete valuation ring. In the case of a DVR, $ R_{\mathfrak{v}} $ is isomorphic to $ \mathbb{C}[x]/(x^n - a) $ for some $ a \in \mathbb{C} $, where $ x $ is a local variable.

Examples and Applications

A classic example of a discrete valuation ring is the ring of integers in a number field, such as $ \mathbb{Z}[\sqrt{2}] $, where the valuation is defined by the norm. Another example is the ring of formal power series $ \mathbb{C}[[x]] $, which is a DVR under the valuation associated with the power series.

Discrete valuation rings play a crucial role in the study of algebraic geometry, particularly in the analysis of schemes and their local rings. The valuation ring of a discrete valuation is used to construct local rings and study the behavior of functions near primes. Additionally, DVRs are essential in the construction of fields of fractions and in the study of algebraic curves and their singularities.

Conclusion

Valuation theory provides a powerful tool for understanding the structure of rings and their associated fields, while discrete valuation rings are central to the study of algebraic geometry and number theory. By examining the properties of valuations, the relationship between valuation rings and discrete valuation rings, and their applications in algebraic contexts, this article highlights the foundational role of these concepts in modern mathematics. The interplay between valuations, localizations, and integral closures ensures that discrete valuation rings remain a cornerstone of algebraic research, offering both theoretical depth and practical utility in the study of rings of fractions and their associated fields.