Cohen-Macaulay Ring

Definition

A Cohen-Macaulay ring is a type of commutative ring that satisfies a specific depth condition, meaning the depth of any ideal is equal to its Krull dimension. This property implies that the ring has a nice geometric structure, especially in relation to its associated primes and the behavior of its modules. Cohen-Macaulay rings play a crucial role in understanding various algebraic and geometric properties, particularly when analyzing singularities and resolving dimension theory issues.

5 Must Know Facts For Your Next Test

1. Cohen-Macaulay rings have the property that for any ideal, the depth equals the Krull dimension, making them important in algebraic geometry.
2. They are characterized by having 'well-behaved' homological dimensions, which allows for good results regarding modules over such rings.
3. If a ring is Noetherian and Cohen-Macaulay, then all of its localizations at prime ideals are also Cohen-Macaulay.
4. Cohen-Macaulay rings arise naturally in many contexts, including projective varieties and intersection theory, emphasizing their importance in algebraic geometry.
5. A Gorenstein ring is a special case of a Cohen-Macaulay ring where the dualizing module is finitely generated, leading to additional properties related to symmetry.

Review Questions

• What does it mean for a ring to be Cohen-Macaulay in terms of depth and Krull dimension, and why is this significant?
  • For a ring to be Cohen-Macaulay means that for any ideal, the depth of that ideal matches its Krull dimension. This significance lies in how it relates to the overall structure and behavior of the ring, particularly in algebraic geometry where this relationship can indicate smoothness or singularities. Such a condition ensures that various homological dimensions behave predictably, allowing for deeper insights into the geometry associated with these rings.
• Discuss how Cohen-Macaulay rings relate to regular local rings and what implications this has for their properties.
  • Cohen-Macaulay rings share a close relationship with regular local rings in that both types exhibit desirable geometric and algebraic properties. Regular local rings have dimensions equal to their Krull dimensions, indicating they are Cohen-Macaulay. The implication is that results concerning modules over regular local rings can often be extended to Cohen-Macaulay rings, which aids in understanding how certain algebraic structures behave under localization and extension.
• Evaluate the importance of Gorenstein rings within the framework of Cohen-Macaulay rings and their applications in geometry.
  • Gorenstein rings are an important subclass of Cohen-Macaulay rings where their dualizing module is finitely generated, creating a symmetrical structure that enhances their utility in both algebra and geometry. This property allows for better resolution techniques and understanding of singularities in algebraic varieties. As Gorenstein rings retain all properties of Cohen-Macaulay rings while providing extra benefits, they are pivotal in applications such as intersection theory and studying syzygies, demonstrating their significant role in contemporary algebraic geometry.

Related terms

Depth: The minimum number of elements needed to generate an ideal that does not contain a non-zero element of the ring.

Krull Dimension: The supremum of the lengths of chains of prime ideals in a ring, indicating the 'size' or 'height' of the ring in terms of its prime spectrum.

Regular Local Ring: A local ring where the dimension equals the Krull dimension, often indicating that the ring behaves well under localization and has a nice structure.