Discrete Valuation Ring

Definition

A discrete valuation ring (DVR) is a principal ideal domain that is also a local ring with a unique non-zero maximal ideal. This structure allows for a well-defined valuation, which assigns to each non-zero element of the ring a unique integer that describes its divisibility properties. The DVR is significant because it provides a framework for understanding how prime ideals behave within local contexts, particularly in algebraic number theory.

5 Must Know Facts For Your Next Test

1. Every discrete valuation ring has a unique maximal ideal generated by a uniformizer, which is an element that generates the maximal ideal and represents the 'smallest' non-unit in terms of divisibility.
2. In a DVR, any non-zero element can be uniquely expressed as a product of a unit and a power of the uniformizer, highlighting its structure in terms of divisibility.
3. The residue field of a discrete valuation ring is finite if and only if the DVR is complete with respect to its valuation, which ties the concept closely to local fields.
4. Discrete valuation rings play a crucial role in algebraic geometry and number theory, particularly in the study of local properties of schemes and algebraic varieties.
5. The completion of a discrete valuation ring with respect to its valuation gives rise to a field that is often used to study extensions and local properties of algebraic structures.

Review Questions

• How does the unique maximal ideal of a discrete valuation ring impact its structure and properties?
  • The unique maximal ideal of a discrete valuation ring simplifies its structure significantly. This uniqueness means that any non-zero element can be classified in terms of how many times it can be divided by the uniformizer associated with the maximal ideal. This property allows for a clear understanding of divisibility within the ring, making it easier to analyze prime ideals and their behavior. As such, DVRs serve as important building blocks in understanding local properties in algebraic number theory.
• Discuss the significance of uniformizers in the context of discrete valuation rings and their applications.
  • Uniformizers are essential in discrete valuation rings because they serve as generators for the maximal ideal. Each non-zero element in the DVR can be expressed as a product involving a unit and some power of the uniformizer. This relationship helps establish valuations that give insight into how elements behave with respect to divisibility. The concept of uniformizers also finds application in various areas such as algebraic geometry, where they assist in studying local properties of schemes and their interactions with global structures.
• Evaluate how discrete valuation rings relate to other algebraic structures such as fields and schemes, particularly regarding their role in local fields.
  • Discrete valuation rings are intimately connected to local fields and schemes through their inherent structural properties. By completing a DVR with respect to its valuation, we obtain a local field, which is crucial for studying extensions and understanding local-global principles. Additionally, when considering schemes, DVRs provide local models that allow mathematicians to analyze how global properties manifest in localized contexts. This interplay between DVRs, fields, and schemes underscores their importance in modern algebraic geometry and number theory.

Related terms

Local Ring: A local ring is a ring with a unique maximal ideal, allowing for the study of properties related to convergence and completion in algebraic geometry.

Valuation: A valuation is a function that measures the 'size' or 'multiplicity' of elements in a field or ring, often associated with the divisibility of elements by prime ideals.

Principal Ideal Domain: A principal ideal domain is an integral domain in which every ideal can be generated by a single element, making the structure simpler and more manageable.