Depth

Definition

Depth is a fundamental concept in commutative algebra that measures the 'size' of a certain set of generators for an ideal in a ring. It relates closely to the notion of regular sequences, and it helps characterize the structure of modules over a ring, particularly within the context of Cohen-Macaulay and Gorenstein rings. Understanding depth is crucial as it can inform properties like dimension, regularity, and the nature of singularities.

5 Must Know Facts For Your Next Test

1. The depth of a module over a Noetherian ring provides insight into its structure and how it behaves under various operations, such as localization.
2. For a local ring, if the depth is equal to the dimension, then the ring is Cohen-Macaulay, which means it has desirable properties related to its ideals and modules.
3. Regular sequences can be used to compute depth; specifically, the length of a maximal regular sequence in an ideal gives the depth of that ideal.
4. In a Gorenstein ring, the depth is particularly significant because it coincides with other important invariants, such as the global dimension.
5. The concept of depth helps in determining whether certain homological properties hold for modules over specific types of rings, linking algebraic geometry and commutative algebra.

Review Questions

• How does depth relate to regular sequences in commutative algebra?
  • Depth is directly connected to regular sequences as it quantifies how many elements can be included in a maximal regular sequence within an ideal. Each element in this sequence must not be a zero divisor on the quotient by the ideal generated by the previous elements. Thus, understanding regular sequences allows one to compute the depth of an ideal or module, which reveals essential information about its structure and properties.
• Discuss the implications of a local ring being Cohen-Macaulay in terms of its depth and dimension.
  • In a local ring, if the depth equals the Krull dimension, then it qualifies as Cohen-Macaulay. This condition implies that every maximal ideal corresponds to a well-behaved structure concerning its associated prime ideals. This relationship leads to more straightforward behavior in terms of syzygies and resolutions, making Cohen-Macaulay rings particularly important in both algebraic geometry and commutative algebra.
• Evaluate how understanding depth contributes to characterizing Gorenstein rings and their homological properties.
  • Understanding depth in Gorenstein rings allows us to analyze their homological characteristics more effectively. In Gorenstein rings, depth coincides with other invariants like global dimension, meaning that one can deduce significant information about module resolutions and duality. This coherence between depth and homological dimensions simplifies many aspects of studying these rings, allowing mathematicians to leverage their unique properties in various applications within algebraic geometry and beyond.

Related terms

Regular Sequence:

A sequence of elements in a ring is called a regular sequence if each element is not a zero divisor on the quotient by the ideal generated by the preceding elements.

Cohen-Macaulay Ring:

A ring is Cohen-Macaulay if its depth equals its Krull dimension, indicating a well-behaved structure regarding its ideals and modules.

Gorenstein Ring: A Gorenstein ring is a special type of Cohen-Macaulay ring that has finite injective dimension, offering duality properties that can simplify homological studies.