Cohen-Macaulay ring

Definition

A Cohen-Macaulay ring is a type of commutative ring that has well-behaved properties concerning its depth and dimension, specifically when the depth equals the Krull dimension. This characteristic implies that every ideal in the ring can be generated by a number of elements that corresponds to the height of that ideal, which reflects a certain regularity in its structure. Cohen-Macaulay rings play a crucial role in algebraic geometry and commutative algebra, linking various properties like duality and homological dimensions.

5 Must Know Facts For Your Next Test

1. In a Cohen-Macaulay ring, every finitely generated module over the ring has depth at least equal to its dimension.
2. The property of being Cohen-Macaulay is preserved under many operations, such as taking direct products or localizations.
3. Cohen-Macaulay rings can be seen as generalizations of regular local rings, which have an even nicer structure.
4. Examples of Cohen-Macaulay rings include polynomial rings over fields and coordinate rings of projective varieties.
5. The concept of Cohen-Macaulayness is closely tied to duality theories in algebraic geometry, such as Serre's duality.

Review Questions

• How does the concept of depth relate to the definition of a Cohen-Macaulay ring?
  • Depth is fundamental to understanding Cohen-Macaulay rings because a ring is defined as Cohen-Macaulay if its depth matches its Krull dimension. This means that for every ideal in the ring, its generators can account for the height of that ideal, indicating that there is a good balance between the structure and complexity of the ring. When depth equals dimension, it implies certain homological properties are satisfied, which is crucial for further applications in algebraic geometry.
• Discuss the implications of being a Cohen-Macaulay ring on module theory and how it affects finitely generated modules.
  • In Cohen-Macaulay rings, every finitely generated module behaves nicely because its depth is at least equal to its dimension. This relationship allows for a more structured understanding of module properties and decomposition. For instance, if you have a Cohen-Macaulay ring and work with its finitely generated modules, you can infer important facts about their resolutions and how they interact with other modules based on their depth and projective dimensions.
• Evaluate how Cohen-Macaulay rings interact with Gorenstein rings and what this means for algebraic geometry applications.
  • Cohen-Macaulay rings serve as a foundational concept for understanding Gorenstein rings, which are a subclass that not only maintains Cohen-Macaulay properties but also exhibits stronger duality behaviors due to their dualizing modules. In algebraic geometry, Gorenstein rings are particularly significant because they help establish better results regarding resolutions and singularities. The interaction between these two types leads to deeper insights into geometric structures, allowing mathematicians to use properties from one area to infer results in another, ultimately enhancing our understanding of algebraic varieties.

Related terms

Depth: The minimum number of elements needed to generate a maximal ideal in the ring, indicating how many 'dimensions' the ring has in a sense.

Krull Dimension: The maximum length of chains of prime ideals in the ring, serving as a measure of the complexity of the ring's structure.

Gorenstein Ring: A special case of Cohen-Macaulay rings that have dualizing modules, indicating more symmetry and allowing for stronger duality properties.