Regular Local Rings

Definition

Regular local rings are a special class of Noetherian local rings where the Krull dimension equals the minimal number of generators of their maximal ideal. This property implies that the ring has 'nice' geometric and algebraic features, making it an important structure in dimension theory and algebraic geometry. In these rings, the local behavior closely resembles that of regular local domains, which allows for a better understanding of their properties and applications in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. For a Noetherian local ring to be regular, it must satisfy the condition that its maximal ideal can be generated by a number of elements equal to its Krull dimension.
2. Regular local rings are integral domains, which means they do not have zero divisors, further enhancing their algebraic properties.
3. The concept of regular local rings plays a crucial role in algebraic geometry, particularly in studying smooth varieties and schemes.
4. If a local ring is regular, then its completion is also regular, which means that it behaves well under taking limits and completions.
5. Regular local rings are important in deformation theory, as they provide a setting where local properties can be studied through deformation spaces.

Review Questions

• How do regular local rings relate to Krull dimension and generators of the maximal ideal?
  • Regular local rings are defined by the relationship between their Krull dimension and the number of generators of their maximal ideal. Specifically, for a local ring to be considered regular, its Krull dimension must equal the minimum number of generators needed for its maximal ideal. This connection is crucial because it indicates that the ring has a certain 'regularity' in its structure, which can lead to favorable algebraic and geometric properties.
• Discuss the significance of regular local rings in the context of algebraic geometry and smooth varieties.
  • In algebraic geometry, regular local rings serve as a fundamental concept when studying smooth varieties. Smooth varieties can be locally described by regular local rings, indicating that their points behave nicely without singularities. The regularity condition ensures that these varieties have well-defined tangent spaces at each point, facilitating analysis through methods like intersection theory and deformation theory, which are essential for understanding complex geometrical structures.
• Evaluate the implications of having a regular local ring on its completion and deformation theory.
  • The property of being a regular local ring has significant implications for both its completion and its applications in deformation theory. Since the completion of a regular local ring remains regular, it allows mathematicians to study limits and continuity in a structured way. In deformation theory, this regularity implies that deformations will behave predictably, allowing for better control over how geometric objects evolve. This predictability is critical in applications where one needs to understand how small changes affect large structures.

Related terms

Krull Dimension: The Krull dimension of a ring is the supremum of the lengths of chains of prime ideals, providing a measure of the 'size' or 'complexity' of the ring.

Noetherian Rings: A Noetherian ring is a ring that satisfies the ascending chain condition on ideals, ensuring that every ideal is finitely generated.

Cohen-Macaulay Rings:

Cohen-Macaulay rings are rings where the depth equals the Krull dimension, allowing them to have desirable properties related to their structure and singularities.