Cohen-Macaulay

Definition

Cohen-Macaulay refers to a class of rings and their associated varieties that exhibit particularly nice properties in commutative algebra and algebraic geometry. Specifically, a ring is Cohen-Macaulay if the depth of the ring equals its Krull dimension, indicating that it has a well-behaved structure. This concept plays an important role in understanding the singularities of varieties, especially in the context of toric resolutions where the geometric properties are closely tied to algebraic characteristics.

5 Must Know Facts For Your Next Test

1. Cohen-Macaulay rings have desirable properties, such as having a well-defined dualizing complex, which simplifies many calculations in algebraic geometry.
2. In the case of varieties, being Cohen-Macaulay ensures that they have 'nice' singularities and can often be resolved via toric methods.
3. The concept extends to schemes, where a Cohen-Macaulay scheme has the property that all its local rings are Cohen-Macaulay.
4. When studying toric resolutions, Cohen-Macaulay varieties help in understanding how singular points can be smoothed out through combinatorial data.
5. A key result is that if a ring is Cohen-Macaulay, then its associated projective variety will also have good cohomological properties.

Review Questions

• How does the concept of Cohen-Macaulay relate to the depth and Krull dimension of a ring?
  • Cohen-Macaulay rings are defined by the equality of their depth and Krull dimension. This means that the longest sequence of non-zerodivisors (depth) matches the maximum length of chains of prime ideals (Krull dimension). This property indicates that such rings have a well-structured form, which is essential for understanding their algebraic properties and how they relate to geometrical concepts.
• Discuss why Cohen-Macaulay varieties are significant when examining singularities and toric resolutions.
  • Cohen-Macaulay varieties are important because they exhibit good behavior in terms of their singularities. When trying to resolve singularities through toric methods, having a variety that is Cohen-Macaulay ensures that the geometric structure is manageable and that resolutions can be effectively constructed using combinatorial data. This interplay is crucial for simplifying complex singularities into more tractable forms.
• Evaluate how the properties of Cohen-Macaulay rings influence broader results in algebraic geometry, particularly concerning dualizing complexes.
  • The properties of Cohen-Macaulay rings significantly impact results in algebraic geometry by providing a framework for understanding dualizing complexes. Since Cohen-Macaulay rings have well-defined dualizing complexes, they facilitate easier calculations and deeper insights into cohomological aspects. This allows mathematicians to derive important results regarding sheaves and their cohomology on varieties, leading to advancements in resolving singularities and classifying algebraic structures.

Related terms

Depth: The length of the longest sequence of non-zerodivisors in a ring, which gives insight into the 'size' or 'structure' of the ring.

Krull Dimension:

A measure of the 'size' of a ring, defined as the supremum of the lengths of chains of prime ideals.

Toric Variety:

A type of algebraic variety that can be described combinatorially via fans and polytopes, often arising in the study of toric resolutions.