9.1 Height and depth of prime ideals

Prime ideals are crucial in commutative algebra, with height and depth revealing their structure. Height measures the longest chain of contained primes, while depth shows the shortest chain to a maximal ideal. These concepts help us understand ideal relationships and ring properties.

Calculating height and depth in various rings gives insight into their structure. The fundamental theorems connect these concepts to ring dimensions and residue fields. Understanding these relationships is key to grasping the intricate world of commutative algebra.

Prime Ideal Height and Depth

Height and depth of prime ideals

• Height of a prime ideal P measures maximal chain length of contained primes denoted ht(P) or height(P) indicating ideal's "size" within ring
  • Example: In $k[x,y,z]$, ht((x,y)) = 2 as (0) ⊂ (x) ⊂ (x,y)
• Depth of prime ideal P represents minimal length of maximal chain containing P denoted depth(P) showing "distance" from being maximal
  • Example: In $Z$, depth((5)) = 0 as (5) is maximal
• Commutative ring R with unity provides context where prime ideals are proper and closed under multiplication
  • Example: In $k[x,y]$, (x) is prime but (x,y^2) is not

Calculation of ideal properties

• Polynomial ring $k[x,y,z]$ with prime ideal P = (x,y) has ht(P) = 2 and depth(P) = 1
  • Chain: (0) ⊂ (x) ⊂ (x,y) ⊂ (x,y,z)
• Integer ring $Z$ with prime ideal P = (p) has ht(P) = 1 and depth(P) = 0
  • Chain: (0) ⊂ (p)
• Localization $k[x,y]_{(x,y)}$ with prime ideal P = (x) has ht(P) = 1 and depth(P) = 1
  • Chain: (0) ⊂ (x) ⊂ (x,y)

Fundamental height and depth theorems

• Height and Krull dimension relation states ht(P) ≤ dim(R) for any prime P in R
  • Proof uses prime ideal chain definition of Krull dimension
• Residue field connection shows dim(R/P) + ht(P) = dim(R) where R/P is P's residue field
  • Example: In $k[x,y,z]$, dim($k[x,y,z]/(x,y)$) + ht((x,y)) = 1 + 2 = 3
• Additivity of height in Noetherian rings: ht(Q) = ht(P) + ht(Q/P) for prime ideals P ⊂ Q
  • Example: In $k[x,y,z]$, ht((x,y,z)) = ht((x)) + ht((y,z)/(x)) = 1 + 2 = 3
• Principal ideal theorem states minimal primes over principal ideals in Noetherian rings have height at most 1
  • Example: In $k[x,y]$, ht((x)) = 1

Ideals and ring dimensions

• Ring dimension equals maximum prime ideal chain length or maximum height of any prime ideal
  • Example: dim($k[x,y,z]$) = 3 as (0) ⊂ (x) ⊂ (x,y) ⊂ (x,y,z) is longest chain
• Depth and codimension relation: depth(P) = dim(R) - dim(R/P) represents P's codimension
  • Example: In $k[x,y,z]$, depth((x,y)) = 3 - 1 = 2
• Krull's principal ideal theorem: ht(P) ≤ n for P minimal over n-generated ideal in Noetherian ring
  • Example: In $k[x,y,z]$, ht((x,y)) ≤ 2 as (x,y) is 2-generated
• Cohen-Macaulay rings have equal height and depth for all prime ideals forming important ring class
  • Example: $k[x,y]$ is Cohen-Macaulay as ht((x)) = depth((x)) = 1
• Grade and depth relation: grade(P) ≤ depth(P) for any prime P with equality characterizing Cohen-Macaulay local rings
  • Example: In $k[x,y]_{(x,y)}$, grade((x)) = depth((x)) = 1