Regular local ring

Definition

A regular local ring is a type of local ring where the dimension of the ring equals the minimal number of generators of its maximal ideal. This means that in a regular local ring, the depth and dimension coincide, indicating that it has 'nice' properties like being a Cohen-Macaulay ring. Regular local rings serve as a foundational concept connecting to various aspects such as Krull dimension, Gorenstein rings, and regular sequences, showcasing their significance in algebraic geometry and commutative algebra.

5 Must Know Facts For Your Next Test

1. In a regular local ring, every sequence of elements that generate the maximal ideal can be chosen to be a regular sequence.
2. The concept of regular local rings generalizes to schemes where regularity relates to smoothness.
3. A regular local ring is automatically Noetherian, meaning it satisfies the ascending chain condition on ideals.
4. Regular local rings are integral domains, which implies they have no zero divisors.
5. The completion of a regular local ring is also regular, preserving its nice structural properties.

Review Questions

• How do regular local rings relate to the concept of Krull dimension and what implications does this have for their structure?
  • Regular local rings have a Krull dimension equal to the number of generators of their maximal ideal. This relationship implies that they exhibit desirable structural features, such as being Cohen-Macaulay. The equality between depth and dimension indicates that these rings behave well under various algebraic operations, which is essential when working with them in both algebra and geometry.
• Discuss the significance of regular sequences in the context of regular local rings and how they contribute to the overall properties of such rings.
  • Regular sequences are crucial in regular local rings because they provide insight into the depth of the ring. In these rings, any generating set of the maximal ideal can be arranged into a regular sequence. This property means that the structure of the ring can be fully understood by examining these sequences, leading to deeper insights into homological dimensions and associated graded rings.
• Evaluate how the characteristics of regular local rings influence their relationship with Cohen-Macaulay and Gorenstein rings, particularly in terms of homological dimensions.
  • Regular local rings form an important subset of both Cohen-Macaulay and Gorenstein rings. Since all regular local rings are Cohen-Macaulay due to their equal depth and dimension, they share many favorable homological properties, such as finite projective dimensions. Additionally, Gorenstein rings often arise as duals to Cohen-Macaulay structures in specific contexts. The interplay between these types highlights significant aspects of algebraic geometry where singularities are examined through their regularity conditions.

Related terms

Cohen-Macaulay ring: A Cohen-Macaulay ring is a ring where the depth equals the Krull dimension, reflecting good homological properties.

Depth: Depth is a measure of how many elements can be found in a maximal ideal that form a regular sequence in a ring.

Maximal ideal: The maximal ideal of a local ring is the unique ideal that contains all non-units, and its generators relate closely to the structure of the ring.