Height

Definition

In algebraic geometry, height refers to the length of the longest chain of prime ideals contained within a given prime ideal in a ring. This concept is crucial for understanding the structure of rings, particularly in relation to regular sequences and regular local rings, as it provides insight into the dimensional properties and complexity of these algebraic structures.

5 Must Know Facts For Your Next Test

1. The height of a prime ideal can give insights into the algebraic structure of the ring and its geometric properties.
2. For a regular local ring, the height of its maximal ideal equals its Krull dimension, indicating that such rings have well-behaved dimensions.
3. A chain of prime ideals can be visualized as a sequence where each prime ideal is properly contained in the next one, helping to determine the height.
4. If a prime ideal has height 0, it means it is minimal and does not contain any other prime ideals besides itself.
5. The concept of height plays a significant role in determining regular sequences, which are sequences of elements that generate an ideal and maintain certain algebraic properties.

Review Questions

• How does the concept of height relate to the chain of prime ideals in a ring?
  • The concept of height is directly connected to the chains of prime ideals within a ring. Specifically, the height measures the length of the longest chain of prime ideals contained within a given prime ideal. Understanding this relationship helps reveal structural properties of rings and their prime ideals, enabling deeper insights into their algebraic and geometric characteristics.
• Discuss the implications of a regular local ring having a maximal ideal with height equal to its Krull dimension.
  • In a regular local ring, having a maximal ideal whose height matches its Krull dimension indicates that the ring has a well-defined structure. This alignment signifies that there is an exact correspondence between the dimension theory and generators for the maximal ideal. Thus, it highlights that regular local rings behave nicely with respect to dimensions, facilitating further analysis in algebraic geometry.
• Evaluate how understanding the height of prime ideals can affect our approach to constructing regular sequences in commutative algebra.
  • Understanding the height of prime ideals can greatly influence our strategy for constructing regular sequences. By knowing the heights, we can determine which elements will contribute to generating an ideal without causing issues like non-regularity or dependency. This insight allows us to carefully select elements that not only generate an ideal but also maintain desired algebraic properties, ultimately leading to more efficient proofs and applications in commutative algebra.

Related terms

Prime Ideal: A prime ideal is an ideal in a ring such that if the product of two elements is in the ideal, at least one of those elements must be in the ideal.

Regular Local Ring: A regular local ring is a local ring whose Krull dimension equals the minimal number of generators of its maximal ideal.

Krull Dimension: The Krull dimension of a ring is defined as the supremum of the lengths of chains of prime ideals in that ring.