LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-09-01

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Height in a quotient measures chains between two primes

Statement

Let $R$ be a commutative ring and let $p \subseteq q$ be prime ideals. Then the height of $q / p$ in $R / p$ is the supremum of the lengths of strict prime chains

p = q_{0} ⊊ q_{1} ⊊ \dots ⊊ q_{n} = q

in $R$ .

Facts & Assumptions

Given: A commutative ring $R$ and prime ideals $p \subseteq q$ .

[L1]

Prime ideals of $R / p$ correspond exactly to the prime ideals of $R$ containing $p$ , with strict inclusions preserved (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal).

[L2]

Height is the supremum of the lengths of strict prime chains ending at the chosen prime (Height equals local dimension).

Proof

technique · direct

1.1

By [L1], strict prime chains in $R / p$ ending at $q / p$ are in bijection with strict prime chains in $R$ beginning at $p$ and ending at $q$ . Corresponding chains have the same length.

L1given

2.1

Applying [L2] to the prime $q / p$ of the quotient ring $R / p$ , the height of $q / p$ is exactly the supremum of the lengths of those quotient chains. Step 1.1 translates that supremum into the displayed chains in $R$ .

L2step 1.1

3.1

Therefore height in the quotient is the relative chain length from $p$ to $q$ .

step 2.1∎

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, §21 (standard reference, not scraped)
The Stacks Project, Section 10.60: Dimension (standard reference, not scraped)