libterm.com A discrete valuation ring (DVR) is a type of integral domain characterized by a unique non-zero maximal ideal generated by a single element called a uniformizer. This structure provides a powerful tool in number theory and algebraic geometry for understanding local properties of rings and fields. Explore the rest of the article to discover how DVRs apply to your mathematical studies and research.
Table of Comparison
| Feature | Discrete Valuation Ring (DVR) | Valuation Ring |
|---|---|---|
| Definition | Integral domain with a valuation that maps elements to the integers \(\mathbb{Z}\cup \{\infty\}\) | Integral domain associated with a valuation on a field, not necessarily discrete |
| Value Group | Isomorphic to \(\mathbb{Z}\) (discrete, rank 1) | Totally ordered abelian group, can be dense or non-discrete |
| Maximal Ideal | Principal ideal, generated by uniformizer | Usually prime, not always principal |
| Dimension | One-dimensional local ring | One-dimensional local ring (in many cases), but can be more general |
| Complete Description | Can be described by a uniformizer and discrete valuation function | Defined by valuation extending from the field, may be non-discrete |
| Examples | Ring of p-adic integers \(\mathbb{Z}_p\), formal power series ring \(k[[t]]\) | Rings of algebraic integers localized at a valuation, non-discrete valuation rings in algebraic geometry |
Introduction to Valuation Rings
Valuation rings are integral domains where, for every element in their field of fractions, either the element or its inverse belongs to the ring, providing a framework to study values assigned to elements in a field. Discrete valuation rings (DVRs) form a special class of valuation rings characterized by a unique non-zero maximal ideal generated by a single element, known as a uniformizer, reflecting a discrete valuation. These structures play a crucial role in algebraic geometry and number theory by enabling local analysis of fields through their valuation properties and ideal structure.
Defining Discrete Valuation Rings
A discrete valuation ring (DVR) is a specific type of valuation ring characterized by a valuation taking values in the integers, typically associated with a single discrete valuation defining its unique maximal ideal. Unlike general valuation rings, which correspond to valuations possibly taking values in arbitrary ordered groups, DVRs exhibit a well-ordered valuation structure that simplifies their ideal theory and valuation topology. This discrete valuation property ensures that every nonzero element's valuation is a nonnegative integer, providing a clear and algebraically tractable framework for studying local fields and algebraic curves.
Key Properties of Valuation Rings
Valuation rings are integral domains characterized by the property that for every element in their field of fractions, either the element or its inverse lies in the ring, ensuring a total ordering on principal ideals. Discrete valuation rings (DVRs) are a special class of valuation rings equipped with a valuation taking values in the integers, leading to a principal maximal ideal generated by a uniformizing parameter. Key properties of valuation rings include integrally closedness, local nature, and the existence of a valuation defining their structure, while DVRs stand out for their discrete valuation and principal ideal domain structure.
Distinguishing Discrete Valuation Rings
Discrete valuation rings (DVRs) represent a specific class of valuation rings characterized by a valuation group isomorphic to the integers, enabling a well-defined discrete valuation function. Unlike general valuation rings, which may have more complex or continuous valuation groups, DVRs possess a principal maximal ideal generated by a uniformizer, ensuring a simpler ideal structure. This distinction underpins their critical role in algebraic number theory and algebraic geometry, particularly in the study of local fields and regular local rings.
Structure Theorem: Valuation vs Discrete Valuation
A discrete valuation ring (DVR) is a special type of valuation ring characterized by a valuation with a discrete value group isomorphic to the integers, enabling a well-defined uniformizer and a principal ideal structure. Valuation rings in general may have value groups that are not discrete, leading to more complex structural behavior without a uniformizing element. The Structure Theorem for DVRs states that every nonzero ideal is a power of a single maximal ideal, reflecting their simplicity compared to arbitrary valuation rings with potentially non-finitely generated ideals.
Examples of Valuation Rings
Valuation rings include discrete valuation rings (DVRs) as key examples characterized by a discrete valuation mapping onto the integers, such as the ring of p-adic integers \(\mathbb{Z}_p\). Other examples of valuation rings encompass non-discrete valuation rings like the ring of all power series with real coefficients convergent on the unit disk, which have valuations taking values in more general ordered groups. Understanding these examples highlights the distinction between valuation rings with discrete value groups and those with continuous or more complex value structures.
Examples of Discrete Valuation Rings
Examples of discrete valuation rings include the ring of p-adic integers \(\mathbb{Z}_p\) and the ring of formal power series \(\mathbf{k}[[t]]\) over a field \(\mathbf{k}\). A discrete valuation ring (DVR) is a special case of a valuation ring where the value group is isomorphic to \(\mathbb{Z}\), giving a discrete valuation as opposed to a general valuation ring with possibly more complex value groups. These DVR examples serve as local models in algebraic geometry and number theory, highlighting their structure as principal ideal domains with unique maximal ideals generated by a uniformizer.
Applications in Algebraic Number Theory
Discrete valuation rings (DVRs) serve as foundational tools in algebraic number theory by providing local descriptions of Dedekind domains at prime ideals, enabling precise analysis of unique factorization of ideals and ramification in extensions. Valuation rings extend this framework to more general valuation theory, offering a broader classification of local fields and completions that facilitate the study of local-global principles and arithmetic of algebraic numbers. Applications of DVRs and valuation rings include the classification of places of number fields, explicit computations of integrality and divisibility properties, and understanding local behavior in arithmetic geometry.
Relationship to Local Fields
Discrete valuation rings (DVRs) serve as the local rings of non-archimedean local fields, providing a complete and discrete valuation that characterizes the field's arithmetic structure. Valuation rings generalize DVRs by allowing valuations with possibly non-discrete value groups, offering a broader framework for studying local properties within algebraic geometry and number theory. The tight connection between DVRs and local fields underpins much of local class field theory, while valuation rings extend these concepts to more general valuation spectra.
Summary: Comparing Discrete and General Valuation Rings
Discrete valuation rings (DVRs) are valuation rings characterized by a valuation with a value group isomorphic to the integers, providing a discrete and well-ordered structure. General valuation rings may have value groups that are more complex or dense, lacking the discrete stepwise ordering found in DVRs. DVRs are local, integrally closed domains with a principal maximal ideal, while valuation rings may not share these restrictive properties, making DVRs a specialized subset within the broader category of valuation rings.
Discrete valuation ring Infographic
