Artin-Rees Lemma

Definition

The Artin-Rees Lemma is a fundamental result in commutative algebra that provides a condition under which a certain ideal in a Noetherian ring can control the behavior of powers of another ideal. This lemma is particularly significant in the study of Artinian rings and has implications for the relationship between Artinian and Noetherian properties, as well as in the context of Koszul complexes, where it aids in establishing homological dimensions and relations between modules.

5 Must Know Facts For Your Next Test

1. The Artin-Rees Lemma states that if R is a Noetherian ring and I, J are ideals of R, there exists an integer n such that for all k ≥ n, the inclusion J^k ⊆ I implies that J^{k-m} ⊆ I for some m.
2. This lemma helps establish that the powers of an ideal can be controlled by another ideal, providing insights into the structure of modules over Noetherian rings.
3. In the context of Artinian rings, the Artin-Rees Lemma illustrates how the stability of ideals leads to stronger structural results and relationships among various submodules.
4. The lemma is essential for proving properties related to the homological dimensions of modules and their resolutions using Koszul complexes.
5. Applications of the Artin-Rees Lemma extend to resolving questions regarding intersection theory, depth, and regular sequences within commutative algebra.

Review Questions

• How does the Artin-Rees Lemma illustrate the connection between Noetherian rings and Artinian properties?
  • The Artin-Rees Lemma serves as a bridge between Noetherian and Artinian properties by demonstrating how ideals interact within these types of rings. Specifically, it shows that while Noetherian rings stabilize under ascending chains, certain ideal behaviors can exhibit characteristics akin to those in Artinian rings, especially through controlling powers of ideals. This interplay provides insight into how structures in one type of ring can inform properties in another, revealing deep connections within commutative algebra.
• Discuss how the Artin-Rees Lemma is utilized in the study of Koszul complexes and their applications.
  • In studying Koszul complexes, the Artin-Rees Lemma is utilized to establish essential relationships between modules over Noetherian rings. By applying this lemma, one can deduce conditions under which certain properties hold true for resolutions and homological dimensions. This understanding aids in simplifying complex interactions between modules, ultimately facilitating deeper insights into their structure and applications in commutative algebra.
• Evaluate the significance of the Artin-Rees Lemma in resolving questions related to depth and regular sequences in commutative algebra.
  • The Artin-Rees Lemma plays a crucial role in addressing questions regarding depth and regular sequences by providing tools to control the behavior of ideals across different powers. Its significance lies in its ability to facilitate understanding of how these sequences interact within Noetherian rings. By allowing mathematicians to manipulate ideal inclusions effectively, this lemma aids in developing strategies for proving results about depth, helping to clarify relationships between various submodules and ultimately enriching our comprehension of module theory.

Related terms

Noetherian Ring:

A Noetherian ring is a ring in which every ascending chain of ideals stabilizes, meaning it satisfies the ascending chain condition.

Artinian Ring: An Artinian ring is a ring in which every descending chain of ideals stabilizes, satisfying the descending chain condition.

Koszul Complex:

A Koszul complex is a specific type of chain complex constructed from a sequence of elements in a ring, used to study properties like depth and homological dimensions.