Regular sequences

Definition

Regular sequences are sequences of elements in a ring that satisfy certain criteria, mainly that each element is not a zero-divisor on the quotient of the ring by the ideal generated by the previous elements. This concept is vital as it ties into the depth of a module and the dimension theory, showing how regular sequences can lead to understanding properties like Cohen-Macaulay rings. Furthermore, these sequences play a significant role in flatness, where they help determine when certain algebraic properties can be preserved under base change.

5 Must Know Facts For Your Next Test

1. A regular sequence in a Noetherian ring can be used to construct a resolution for modules, indicating the relationship between the elements in the sequence and the module's structure.
2. If an ideal is generated by a regular sequence, then the quotient by this ideal maintains many desirable properties, such as being Cohen-Macaulay if the ring itself is Cohen-Macaulay.
3. In the context of flatness, regular sequences are crucial because if you have a flat module over a Noetherian ring, then every sequence of elements that is regular in one module remains regular in all tensor products with that module.
4. The number of elements in a maximal regular sequence corresponds to the depth of the module, which provides valuable information about its geometric or algebraic properties.
5. Regular sequences can be utilized to study local cohomology, revealing how they influence the vanishing and non-vanishing properties of certain modules.

Review Questions

• How do regular sequences relate to depth and provide insight into the structure of modules within Noetherian rings?
  • Regular sequences directly relate to the concept of depth as they define the maximum length of sequences that can be formed without leading to zero-divisors. This indicates structural qualities of modules, where a longer regular sequence typically implies a richer structure. The depth serves as a measure of complexity for modules over Noetherian rings, making regular sequences essential for understanding how these structures behave.
• Discuss how regular sequences impact the properties of Cohen-Macaulay rings and why this matters in algebraic geometry.
  • Regular sequences are instrumental in defining Cohen-Macaulay rings because if an ideal is generated by such sequences, it ensures that the ring retains critical properties like having its depth equal to its dimension. This balance is essential in algebraic geometry because Cohen-Macaulay schemes represent nice geometric objects with well-behaved singularities and provide a framework for studying intersections and dimension theory. Thus, recognizing and working with regular sequences aids in understanding these geometric structures.
• Evaluate how the concept of regular sequences can influence flatness and the preservation of exactness in algebraic operations.
  • Regular sequences significantly influence flatness by ensuring that when tensoring with flat modules, the properties of regularity are preserved. This means that if a sequence is regular in one context, it remains so even after performing operations like tensor products. This preservation is crucial for maintaining exactness in sequences across various algebraic manipulations, which ultimately supports deeper results in homological algebra and allows mathematicians to draw connections between different algebraic entities effectively.

Related terms

Cohen-Macaulay ring: A type of ring where the depth equals its Krull dimension, signifying a balanced structure between ideals and modules.

Depth: The length of the longest regular sequence in a module, giving insight into its structure and dimensionality.

Flatness: A property of a module that ensures it behaves well under tensor products, important for preserving exact sequences.