Cohen-Macaulay

Definition

A Cohen-Macaulay ring is a type of commutative ring that has desirable properties related to the dimensions of its modules and ideals. In particular, these rings have a well-behaved depth, which is equal to their Krull dimension, making them important in both algebra and geometry. This relationship helps in understanding regular sequences and the structure of the ring, which is key in studying properties like homological dimensions.

5 Must Know Facts For Your Next Test

1. Cohen-Macaulay rings are characterized by having their depth equal to their Krull dimension, which means they behave nicely in terms of homological algebra.
2. Every Cohen-Macaulay ring is Noetherian, meaning it satisfies the ascending chain condition on ideals, which ensures finiteness properties.
3. Cohen-Macaulay rings arise frequently in algebraic geometry, especially when studying varieties that are smooth or have nice geometric properties.
4. The property of being Cohen-Macaulay can be checked through specific sequences called regular sequences; if these exist, it implies Cohen-Macaulayness.
5. In local rings, being Cohen-Macaulay is equivalent to having a regular local ring at each point of the variety defined by the ring.

Review Questions

• How does the depth of a Cohen-Macaulay ring relate to its Krull dimension, and why is this significant?
  • In a Cohen-Macaulay ring, the depth is equal to the Krull dimension, which signifies that these rings have a uniform structure where all dimensions align well. This relationship is significant because it indicates that the ring behaves predictably under various algebraic operations and has desirable homological properties. When studying modules over such rings, this alignment helps in understanding how they can be decomposed or constructed using regular sequences.
• Discuss the importance of regular sequences in determining whether a ring is Cohen-Macaulay.
  • Regular sequences are crucial for identifying Cohen-Macaulay rings because their existence directly implies that the depth of the ring meets its Krull dimension. If one can find a sequence of elements that form a regular sequence, it demonstrates that there are no unexpected dependencies or zero-divisors disrupting the structure. This property ensures that not only does the ring have a well-defined depth, but it also provides insights into the ideal structure and how modules over this ring will behave.
• Evaluate how Cohen-Macaulayness impacts our understanding of algebraic varieties and their properties.
  • Cohen-Macaulayness greatly enhances our understanding of algebraic varieties by providing a framework where we can analyze their geometric and algebraic structures systematically. When varieties are associated with Cohen-Macaulay rings, it indicates they possess nice properties such as being smooth or having controlled singularities. This allows mathematicians to apply tools from homological algebra effectively, leading to deeper insights into intersection theory and deformation theory, ultimately enriching our comprehension of complex algebraic geometry.

Related terms

Depth: The length of the longest sequence of elements in an ideal that are regular on the quotient by that ideal, reflecting the 'height' of the ideal.

Regular Sequence: A sequence of elements in a ring such that each element is not a zero-divisor on the quotient by the ideal generated by the preceding elements.

Krull Dimension: The maximum length of chains of prime ideals in a ring, indicating its 'size' in terms of its structure.