Discrete valuation ring

Definition

A discrete valuation ring (DVR) is a type of local principal ideal domain that has a unique non-zero maximal ideal, which allows for the valuation of its elements to be defined in a way that assigns a non-negative integer to each non-zero element. This unique property facilitates the study of algebraic structures and local behavior in various mathematical fields. The concept is crucial when discussing Dedekind domains and local class field theory, as DVRs help understand how these structures behave under various mathematical operations and extensions.

5 Must Know Facts For Your Next Test

1. A discrete valuation ring has a unique maximal ideal generated by a single element, which can often be thought of as a 'uniformizer' or 'generator'.
2. The residue field of a DVR is finite or infinite, depending on whether the DVR is non-discrete or discrete, respectively.
3. Every discrete valuation ring is also a Noetherian ring, meaning every ascending chain of ideals stabilizes.
4. DVRs can be used to study the local behavior of schemes and algebraic varieties, providing insights into their structure at specific points.
5. In the context of class field theory, discrete valuation rings play a significant role in describing local fields and their extensions.

Review Questions

• How does the structure of a discrete valuation ring facilitate the understanding of algebraic varieties?
  • The structure of a discrete valuation ring helps in understanding algebraic varieties by providing a framework to study their local properties. A DVR allows mathematicians to analyze behavior around specific points on a variety, using the unique maximal ideal to capture local conditions. By focusing on these localized aspects, researchers can gain insights into the overall structure and properties of the variety as they relate to ideals and valuations.
• Compare and contrast discrete valuation rings with other types of local rings, particularly focusing on their unique features.
  • Discrete valuation rings differ from other types of local rings primarily in their structure: DVRs have a unique non-zero maximal ideal generated by a single element, which is not always the case for other local rings. This property allows DVRs to define valuations effectively. In contrast, other local rings might have more complex structures with multiple generators for their maximal ideals. As such, while all DVRs are local rings, not all local rings can be classified as DVRs due to this essential difference in ideal structure.
• Evaluate the impact of discrete valuation rings on the development of local class field theory and its applications.
  • Discrete valuation rings significantly impact local class field theory by providing essential tools for studying the arithmetic of local fields. Their unique properties allow for clear definitions of extension fields and residue classes, which are fundamental concepts in class field theory. This clarity aids in understanding how various extensions behave and how they relate back to the original fields. Consequently, DVRs play a crucial role in modern number theory and algebraic geometry by facilitating deeper insights into local-global principles and connections between algebraic structures.

Related terms

Valuation:

A valuation is a function that assigns a value (often an integer) to elements of a field, capturing information about divisibility and size.

Local ring: A local ring is a ring with a unique maximal ideal, which makes it useful for localizing problems in algebraic geometry and number theory.

Complete local ring: A complete local ring is a local ring that is complete with respect to its maximal ideal, meaning that every Cauchy sequence converges within the ring.