Cohen-Macaulay ring

Definition

A Cohen-Macaulay ring is a type of commutative ring that satisfies certain depth and dimension conditions, making it a key object of study in algebraic geometry and commutative algebra. It is characterized by having a well-behaved structure, where the depth of every ideal equals its height, ensuring that the ring has desirable properties such as finite generation of its modules. This notion plays an important role in understanding primary decomposition and associated primes.

5 Must Know Facts For Your Next Test

1. A Cohen-Macaulay ring has the property that for any prime ideal, the depth equals the height, which ensures a balance between algebraic and geometric properties.
2. Cohen-Macaulay rings are particularly significant because they ensure that certain modules over them have desirable homological properties, such as being finitely generated.
3. One of the most common examples of Cohen-Macaulay rings is polynomial rings over a field, which exhibit both good depth and dimension behavior.
4. The concept extends to graded rings and modules, allowing for applications in various contexts including projective geometry and intersection theory.
5. In primary decomposition, associated primes provide information about the structure of ideals in a Cohen-Macaulay ring, linking algebraic properties to geometric interpretation.

Review Questions

• How does the concept of depth relate to Cohen-Macaulay rings and their properties?
  • In Cohen-Macaulay rings, the relationship between depth and dimension is crucial because the depth of any ideal must equal its height. This balance indicates that every prime ideal contributes equally to the ring's structure. Understanding depth helps in analyzing how ideals behave within these rings, leading to insights into their module structures and further properties.
• Discuss the implications of being a Cohen-Macaulay ring in terms of module generation and associated primes.
  • Being a Cohen-Macaulay ring implies that all finitely generated modules over it exhibit well-behaved generation properties. This means there are systematic ways to understand the structure of these modules via their associated primes. Each associated prime corresponds to certain decompositions in primary ideals, providing deep insights into both algebraic relations and geometric representations in associated varieties.
• Evaluate the significance of Cohen-Macaulay rings in the context of algebraic geometry and how they influence geometric interpretations.
  • Cohen-Macaulay rings are fundamental in algebraic geometry as they establish a bridge between algebraic properties and geometric interpretations. Their well-defined depth conditions facilitate understanding singularities and regularity in algebraic varieties. By providing an efficient framework for working with primary decomposition and associated primes, Cohen-Macaulay rings enhance our ability to study the geometric aspects of solutions to polynomial equations and their intersections, thus influencing various aspects of modern algebraic geometry.

Related terms

Depth: The minimum number of elements needed to generate a regular sequence on a module, reflecting how 'far' a module is from being free.

Dimension: The Krull dimension of a ring, representing the maximum length of chains of prime ideals, which provides insight into the geometric structure of the associated varieties.

Regular local ring: A local ring where the dimension equals the Krull dimension of its residue field, serving as an important example of Cohen-Macaulay rings.