Krull Dimension

The Krull dimension, named after mathematician Wolfgang Krull, is a foundational concept in commutative algebra and algebraic geometry that provides a rigorous way to measure the "dimension" of algebraic structures.

Definition

For a commutative ring R, the Krull dimension is defined as the supremum of the lengths of all chains of prime ideal:

P₀ ⊂ P₁ ⊂ P₂ ⊂ ... ⊂ Pₙ

where each inclusion is strict.

Key Properties

1. For a field, the Krull dimension is 0
2. For a principal ideal domain, the Krull dimension is 1
3. For a polynomial ring k[x₁,...,xₙ] over a field k, the Krull dimension is n

Geometric Interpretation

The Krull dimension closely relates to our geometric intuition of dimension:

• The Krull dimension of k[x] corresponds to the dimension of a line (1)
• The Krull dimension of k[x,y] corresponds to the dimension of a plane (2)
• The Krull dimension of k[x₁,...,xₙ] corresponds to n-dimensional space

This connection makes it a crucial bridge between algebraic geometry and ring theory perspectives.

Applications

The Krull dimension finds applications in:

• Classification of Noetherian ring
• Study of algebraic variety
• dimension theory in topology
• local ring analysis

Related Concepts

• height of prime ideal
• depth of ring
• transcendence degree
• Hilbert's Nullstellensatz

Historical Context

The concept was developed by Wolfgang Krull in the 1920s as part of his work on dimension theory in abstract algebra. It represented a significant advancement in understanding the structure of commutative rings and their geometric counterparts.

Examples

1. Fields: k has Krull dimension 0 (no proper prime ideals)
2. Integers: Z has Krull dimension 1 (chains of prime ideals have maximum length 1)
3. Polynomial Rings: k[x,y] has Krull dimension 2 (corresponding to coordinate planes)

The study of Krull dimension continues to be active in modern research, particularly in its interactions with homological algebra and algebraic K-theory.