Discrete valuation ring

Definition

A discrete valuation ring (DVR) is a type of local integral domain that has a unique non-zero maximal ideal and allows for a valuation function that assigns to each non-zero element of the ring a non-negative integer, representing its 'order' in a sense. DVRs are important in algebraic geometry and number theory, particularly in studying local properties of schemes and in the context of completions of local rings. They provide a structured way to analyze the elements of the ring and their divisibility, making them crucial in many applications.

5 Must Know Facts For Your Next Test

1. Every discrete valuation ring is a principal ideal domain (PID), meaning every ideal in the DVR can be generated by a single element.
2. The maximal ideal of a DVR is generated by a uniformizer, which is an element whose valuation is 1, and it allows you to create all other non-zero ideals of the DVR.
3. DVRs are used to study algebraic curves and surfaces through their local properties, especially when examining singularities and intersection theory.
4. In a DVR, the residue field (the quotient of the DVR by its maximal ideal) is always a finite field extension over the prime field.
5. The completion of a discrete valuation ring with respect to its maximal ideal results in a complete discrete valuation field, which is useful for understanding local behavior in algebraic geometry.

Review Questions

• How does the structure of a discrete valuation ring support the concept of unique factorization within its ideals?
  • A discrete valuation ring supports unique factorization because it is a principal ideal domain. This means that every ideal can be generated by a single element, which simplifies the analysis of divisibility within the ring. The unique factorization property ensures that any non-zero element can be expressed uniquely in terms of its uniformizer and powers thereof, facilitating clear insights into the structure of ideals and how they interact within the DVR.
• Discuss the role of uniformizers in discrete valuation rings and how they influence the behavior of ideals.
  • Uniformizers play a crucial role in discrete valuation rings as they generate the maximal ideal and help define other ideals within the DVR. Each uniformizer corresponds to a unique valuation that categorizes elements based on their divisibility by this generator. Consequently, this means any non-zero ideal can be expressed as powers of this uniformizer, leading to an efficient way to understand how ideals are structured and how they can be manipulated through operations like addition or multiplication.
• Evaluate the implications of the completion of a discrete valuation ring and its application in algebraic geometry.
  • The completion of a discrete valuation ring leads to the formation of a complete discrete valuation field, which has significant implications in algebraic geometry. This completion provides a more refined framework for examining local properties of algebraic varieties, especially when analyzing singularities or intersections. Furthermore, working within this complete field allows for powerful techniques such as resolving singularities and applying tools from analytic geometry, making it essential for studying complex behaviors in algebraic structures.

Related terms

Valuation: A function that assigns to each element of a field a measure of its 'size' or 'divisibility,' often used in the context of discrete valuation rings.

Local ring: A ring that has a unique maximal ideal, often used to study algebraic structures in a localized setting.

Fractional ideal: An extension of the concept of an ideal in a ring, allowing for elements that are not necessarily whole but can be expressed as fractions of elements from a specific ring.