Discrete valuation ring

Definition

A discrete valuation ring (DVR) is a special type of local ring that is a principal ideal domain (PID) and has a unique non-zero maximal ideal. This structure allows for a well-defined valuation, which measures the 'size' of elements in a way that is discrete, meaning it can only take integer values. DVRs play a critical role in algebraic geometry, particularly in studying local properties of schemes and their functions.

5 Must Know Facts For Your Next Test

1. Every discrete valuation ring has a unique non-zero prime ideal, which makes it very structured and easier to analyze.
2. The valuation associated with a DVR gives a way to classify elements into equivalence classes based on their 'size'.
3. Discrete valuation rings are often used to study the local behavior of schemes at a point, providing insights into their geometry.
4. The field of fractions of a DVR is always a field, indicating that every non-zero element can be inverted.
5. DVRs are crucial in number theory, as they can be used to describe local properties of algebraic integers.

Review Questions

• How does the unique maximal ideal of a discrete valuation ring influence its structure and properties?
  • The unique maximal ideal of a discrete valuation ring simplifies its structure by ensuring that any non-zero element not in this ideal can generate the entire ring through multiplication. This leads to the property that every non-zero element can be expressed as a unit multiplied by a power of the generator of the maximal ideal. This characteristic allows for straightforward analysis and manipulation within the ring, as it eliminates ambiguity regarding ideals and their generators.
• Discuss the importance of discrete valuation rings in understanding local properties of schemes.
  • Discrete valuation rings provide essential tools for studying local properties of schemes because they allow us to focus on behavior at specific points. The DVR's structure facilitates the examination of functions defined on schemes at these points, enabling us to assess how these functions behave locally. This local analysis is vital in algebraic geometry as it helps determine properties such as regularity, singularity, and intersections within the geometric framework.
• Evaluate the role of discrete valuation rings in number theory and how they relate to algebraic integers.
  • Discrete valuation rings are instrumental in number theory because they offer a framework for analyzing local properties of algebraic integers. They enable mathematicians to examine the behavior of integers at various primes, leading to insights about factorization and congruences. The relationship between DVRs and algebraic integers highlights how local properties can affect global structures, revealing deeper connections within the landscape of arithmetic and paving the way for further developments in algebraic geometry and number theory.

Related terms

Local ring: A local ring is a ring that has a unique maximal ideal, allowing for a focus on local properties of algebraic structures.

Principal ideal domain (PID): A principal ideal domain is an integral domain in which every ideal is generated by a single element, which simplifies many aspects of ring theory.

Valuation: A valuation is a function that assigns values to elements of a field, measuring their size or multiplicity in a structured way.