Regular Sequence

Definition

A regular sequence is a sequence of elements in a ring such that each element is not a zero divisor in the quotient of the ring by the ideal generated by the preceding elements. This concept plays a crucial role in understanding the depth of modules over rings, especially in the context of Cohen-Macaulay rings, where regular sequences help define the structure and properties of these rings and their associated geometrical aspects.

5 Must Know Facts For Your Next Test

1. A regular sequence can be thought of as a generalization of an independent set of vectors in linear algebra, where independence translates into no zero divisors.
2. In Cohen-Macaulay rings, a maximal regular sequence can help determine the dimension and homological properties of modules over that ring.
3. Regular sequences are particularly important for studying the local cohomology and singularities of varieties, as they reveal deep structural information.
4. The existence of a regular sequence within a module implies certain favorable properties, such as being torsion-free or having finite projective dimension.
5. Regular sequences are closely linked to shellability, as shellable complexes often have an associated regular sequence that leads to combinatorial implications in algebraic topology.

Review Questions

• How does the concept of a regular sequence relate to the depth of a module in a Cohen-Macaulay ring?
  • In a Cohen-Macaulay ring, the depth of a module corresponds to the length of its longest regular sequence. This connection signifies that regular sequences are fundamental in establishing the structure and homological properties of modules. When we have a regular sequence, it shows that we can choose elements from our module that do not create zero divisors with respect to previously chosen elements, directly contributing to measuring how 'deep' our module is.
• Discuss the significance of regular sequences in determining properties of shellable complexes and their relation to Cohen-Macaulay rings.
  • Regular sequences play a crucial role in defining shellability for simplicial complexes. A shellable complex can be constructed from a sequence of vertices where each vertex added respects the order given by a regular sequence. This relationship highlights how combinatorial properties can influence algebraic structures, particularly in Cohen-Macaulay rings where both concepts help bridge geometry with algebra.
• Evaluate how understanding regular sequences contributes to broader applications in algebraic geometry and commutative algebra.
  • Understanding regular sequences allows mathematicians to uncover significant features of algebraic varieties and schemes. Regular sequences indicate how well certain elements can span or generate ideals without leading to complexities like singularities. In algebraic geometry, these insights assist in studying various geometric structures and their intersections with commutative algebra, ultimately enhancing our grasp on topics like local cohomology, resolutions, and the overarching shape of algebraic objects.

Related terms

Cohen-Macaulay Ring: A Cohen-Macaulay ring is a type of commutative ring where every ideal has a depth equal to its dimension, reflecting a balanced relationship between algebraic and geometric properties.

Depth: Depth is a measure of the 'size' or 'complexity' of a module, defined as the length of the longest regular sequence contained in the module, which indicates how many elements can be chosen without becoming zero divisors.

Shellability: Shellability is a combinatorial property of a simplicial complex that allows for a specific ordering of its faces, related to the existence of a regular sequence in the associated polynomial ring.