Depth Lemma

Definition

The depth lemma states that for any finitely generated module over a Noetherian ring, the depth of the module is at least as large as the length of any regular sequence contained in it. This concept highlights the connection between regular sequences and the depth of modules, particularly in Cohen-Macaulay rings, which are characterized by having a well-defined depth that matches their Krull dimension.

5 Must Know Facts For Your Next Test

1. The depth lemma is crucial for understanding the relationship between depth and regular sequences in module theory.
2. In a Cohen-Macaulay ring, the depth and dimension are equal, making the depth lemma particularly powerful in this context.
3. The lemma implies that if a module has a regular sequence of length $n$, then its depth is at least $n$.
4. The depth of a module can provide insight into its homological properties and how it behaves under various operations.
5. Applications of the depth lemma extend to many areas of algebraic geometry and commutative algebra, influencing both theoretical and practical aspects.

Review Questions

• How does the depth lemma relate to the concept of regular sequences in Noetherian rings?
  • The depth lemma directly connects the notion of regular sequences to the depth of modules over Noetherian rings. It establishes that for any finitely generated module, if there exists a regular sequence of length $n$ within that module, then its depth cannot be less than $n$. This relationship emphasizes how understanding regular sequences can provide valuable insights into the structural properties of modules.
• In what ways does the depth lemma contribute to our understanding of Cohen-Macaulay rings?
  • The depth lemma plays a significant role in characterizing Cohen-Macaulay rings because these rings are defined by having equal depth and Krull dimension. Since the lemma guarantees that a module's depth is at least as large as any regular sequence it contains, it reinforces the property that all finitely generated modules over Cohen-Macaulay rings possess this crucial equality. Understanding this link helps deepen our insight into their structural characteristics.
• Evaluate how the application of the depth lemma influences broader topics within algebraic geometry and commutative algebra.
  • The application of the depth lemma significantly impacts both algebraic geometry and commutative algebra by providing tools to analyze module behaviors in various contexts. Its ability to connect the concepts of regular sequences, depth, and Cohen-Macaulay properties allows mathematicians to derive important results regarding singularities and projective varieties. Consequently, mastering this lemma equips students with foundational knowledge necessary for tackling more complex problems in these fields, ultimately enhancing their mathematical intuition and skills.

Related terms

Regular Sequence: A sequence of elements in a ring such that each element is a non-zero divisor on the quotient by the ideal generated by the previous elements.

Cohen-Macaulay Ring: A type of ring where the depth of every finitely generated module equals its Krull dimension, implying a strong relationship between algebraic and geometric properties.

Krull Dimension: The supremum of the lengths of chains of prime ideals in a ring, reflecting its structural complexity.