Regular Sequence

Definition

A regular sequence is a sequence of elements in a ring that is both a part of an ideal and behaves nicely with respect to the ring's structure, meaning that each element is not a zero divisor on the quotient of the ring by the ideal generated by the preceding elements. This property ensures that regular sequences play a crucial role in defining the depth of modules over rings and are integral in characterizing Cohen-Macaulay rings, which exhibit desirable geometric properties.

5 Must Know Facts For Your Next Test

1. For a sequence to be regular, it must consist of elements that are non-zero divisors on the module formed by taking the quotient of the ring and the ideal generated by the previous elements in the sequence.
2. Regular sequences can have various lengths, and their length is directly related to the depth of the module they are associated with.
3. In Cohen-Macaulay rings, every maximal ideal has a regular sequence whose length matches the dimension of the ring.
4. If you have a regular sequence in a local ring, it can be used to construct a resolution for modules, which helps in understanding their structure and properties.
5. Regular sequences can also be linked to algebraic geometry, as they often arise when studying varieties and their defining equations.

Review Questions

• What role do regular sequences play in determining the depth of modules over rings?
  • Regular sequences are essential for establishing the depth of modules because they provide a measure of how many times you can remove elements from an ideal without hitting zero divisors. The depth of a module is defined as the length of the longest regular sequence contained within its associated ideal. Therefore, understanding regular sequences allows for a deeper insight into the structural characteristics and dimensions of modules.
• How does the concept of a Cohen-Macaulay ring relate to regular sequences and their lengths?
  • In Cohen-Macaulay rings, every finitely generated module achieves its maximum depth, which matches its Krull dimension. This relationship means that within Cohen-Macaulay rings, there exists a regular sequence whose length equals the dimension of the ring. This property makes Cohen-Macaulay rings particularly important in algebraic geometry since they correspond to well-behaved varieties.
• Evaluate how avoiding zero divisors in regular sequences impacts their significance in algebraic structures.
  • Avoiding zero divisors in regular sequences is crucial because it ensures that each element interacts properly with subsequent elements in terms of multiplication. This characteristic allows for stable construction of ideals and modules, leading to well-defined properties such as depth and dimension. By ensuring that elements are non-zero divisors, we maintain coherence in algebraic structures, facilitating deeper understanding and applications in both algebra and geometry.

Related terms

Depth: The depth of a module is the length of the longest regular sequence contained in the ideal associated with that module.

Cohen-Macaulay Ring: A Cohen-Macaulay ring is a type of ring where every finitely generated module has depth equal to its Krull dimension, reflecting a balanced geometric structure.

Zero Divisor: A zero divisor in a ring is an element that can multiply with some non-zero element to give zero, which is something regular sequences avoid.