Regular Sequence

Definition

A regular sequence is a sequence of elements in a ring that behaves nicely in terms of generating ideals and behaving well with respect to modules. In the context of Koszul complexes, regular sequences are crucial because they help define certain properties related to the construction of the Koszul complex, such as depth and homological dimensions. A regular sequence can be used to form a free resolution, which is essential in understanding how certain modules can be decomposed or related to others.

5 Must Know Facts For Your Next Test

1. A sequence is called regular if each element is a non-zero divisor on the quotient by the ideal generated by the previous elements in the sequence.
2. Regular sequences provide important information about the depth of modules, which is key for understanding their structure and behavior.
3. In a Noetherian ring, every regular sequence can be extended to a maximal regular sequence, which relates to the concept of depth.
4. The Koszul complex built from a regular sequence gives rise to exact sequences that reveal properties about syzygies and resolutions.
5. Regular sequences are connected to concepts like Cohen-Macaulay rings and Gorenstein rings, which further explore the relationships between algebraic geometry and commutative algebra.

Review Questions

• How does the definition of a regular sequence influence the construction of Koszul complexes?
  • The definition of a regular sequence is central to constructing Koszul complexes because it determines how elements interact within the ideal they generate. When building a Koszul complex from a regular sequence, each element must be a non-zero divisor on the quotient formed by the ideal generated by its predecessors. This property ensures that the complex remains exact, leading to insights into the module's homological characteristics and depth.
• Discuss the relationship between regular sequences and the concept of depth in commutative algebra.
  • Regular sequences are directly linked to the concept of depth, which measures how many non-zero divisors can be found in an ideal. The depth reflects how robust an ideal is against non-regular elements. If a regular sequence exists in an ideal, it implies that the depth can be at least as long as the length of that sequence. This relationship highlights the importance of regular sequences in determining not just local properties of modules but also their global behavior in terms of dimensions and resolutions.
• Evaluate how regular sequences affect the properties of perfect modules and their projective resolutions.
  • Regular sequences play a significant role in the structure of perfect modules, which have finite projective resolutions. When examining perfect modules, regular sequences facilitate understanding their syzygies and the behavior of their resolutions. If a module has a regular sequence associated with it, this often implies better homological properties, such as having fewer generators in its projective resolution. Evaluating these connections reveals how algebraic structures interact through their sequences and resolutions.

Related terms

Koszul Complex: A chain complex constructed from a regular sequence that provides insights into the homological properties of modules over a ring.

Depth: The length of the longest regular sequence contained in an ideal, reflecting how 'far' one can go without encountering a non-regular element.

Perfect Module: A module that has a projective resolution of finite length, often related to regular sequences when considering their homological dimensions.