LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Height equals local dimension

Statement

Let $R$ be a commutative ring and let $p \in Spec (R)$ . Then $ht (p) = \sup {n \geq 0 : p_{0} ⊊ \dots ⊊ p_{n} = p is a strict chain of prime ideals in R}$ . The supremum is allowed to be infinite.

Facts & Assumptions

Given: A commutative ring $R$ and a prime ideal $p \subset R$ .

[L1]

By definition, $ht (p) = \dim (R_{p})$ (The height of a prime ideal).

[L2]

Prime ideals of $R_{p}$ correspond exactly to prime ideals of $R$ contained in $p$ , with strict inclusions preserved (Primes of a localization at a prime).

[L3]

Krull dimension is the supremum of lengths of strict chains of prime ideals (Krull dimension of a nonzero ring).

Proof

technique · direct

1.1L2given

By [L2], strict chains of prime ideals in $R_{p}$ are in bijection with strict chains of prime ideals in $R$ that end at $p$ . Corresponding chains have the same length because strict inclusions are preserved in both directions.

2.1L1L3step 1.1

By [L1], the left-hand side is $\dim (R_{p})$ . By [L3], that dimension is the supremum of the lengths of the strict prime chains in $R_{p}$ , so step 1.1 identifies it with the displayed supremum over chains in $R$ ending at $p$ .

3.1step 2.1∎

Therefore height agrees with the chain-length description.

Depends on

Used by

Dependency tree · two levels

6 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, Definition 3.14 (standard reference, not scraped)
The Stacks Project, Section 10.60: Dimension of rings (standard reference, not scraped)