Gorenstein ring

Definition

A Gorenstein ring is a commutative ring that is Cohen-Macaulay and has finite injective dimension. These rings are significant because they possess duality properties that simplify their homological structure, especially in the context of projective modules and resolutions. Gorenstein rings are a special subclass of Cohen-Macaulay rings, and they have nice symmetry in their minimal free resolutions, which relates to their dualizing complex.

5 Must Know Facts For Your Next Test

1. Gorenstein rings have a very symmetrical structure in terms of their projective resolutions, which leads to simpler calculations in commutative algebra.
2. If a local ring is Gorenstein, then it has a unique maximal ideal and its residue field is Cohen-Macaulay.
3. All regular rings are Gorenstein rings, making Gorenstein rings generalizations of regular local rings.
4. The definition of a Gorenstein ring can be extended to graded rings, where the graded components satisfy the necessary duality conditions.
5. Gorenstein rings often arise in algebraic geometry, particularly in the study of singularities and resolutions of varieties.

Review Questions

• How does the property of being Cohen-Macaulay relate to Gorenstein rings?
  • Gorenstein rings are always Cohen-Macaulay, meaning they maintain the property where depth equals Krull dimension. This relationship indicates that while all Gorenstein rings have this stronger structure, not all Cohen-Macaulay rings will necessarily have the additional symmetry and homological properties that characterize Gorenstein rings. This highlights how Gorenstein rings sit within the broader category of Cohen-Macaulay rings.
• What implications does finite injective dimension have for the structure of Gorenstein rings?
  • The finite injective dimension of a Gorenstein ring implies that its modules behave well with respect to homological algebra. Specifically, this condition facilitates the existence of well-defined projective resolutions for its modules, allowing for simpler computation of derived functors. Additionally, this property indicates that certain homological conjectures can hold true within the context of Gorenstein rings, enhancing their significance in commutative algebra.
• Evaluate the role of dualizing complexes in understanding the properties and applications of Gorenstein rings.
  • Dualizing complexes play a critical role in providing insights into the homological behavior of Gorenstein rings. They help establish relationships between different cohomological dimensions and facilitate computations involving derived categories. By studying these complexes within Gorenstein contexts, mathematicians can derive duality statements that reveal more about module categories and geometric structures arising from these rings. This understanding is essential for both theoretical advancements and practical applications in algebraic geometry.

Related terms

Cohen-Macaulay ring: A Cohen-Macaulay ring is a type of ring where the depth equals the Krull dimension, indicating a well-behaved structure with respect to its prime ideals.

Injective dimension: The injective dimension of a module measures the minimum length of an injective resolution of that module, reflecting its homological properties.

Dualizing complex: A dualizing complex is a bounded complex of finitely generated modules that provides a notion of duality for homological dimensions, essential in the study of Gorenstein rings.