TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

dimension at most embedding dimension

Statement

Every nonzero commutative Noetherian local ring $R$ satisfies $\dim R \leq edim R < \infty$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

embedding dimension is minimal maximal ideal generator number: For a nonzero Noetherian local ring $(R, m, k)$ , $edim R$ is the least number of generators of $m$ .

[F2]

Krull's height theorem: Let $R$ be a Noetherian commutative ring, let $I = (x_{1}, \dots, x_{n})$ be an ideal generated by $n \geq 1$ elements, and let $p$ be a prime ideal minimal over $I$ . Then $ht (p) \leq n$ .

Proof

1.1

Let $e = edim R$ . The maximal ideal has a generating tuple of length $e$ . If $e = 0$ , it is zero; then every nonzero element is a unit and $R$ is a field of dimension zero.

2.1

If $e \geq 1$ , the maximal ideal is minimal over itself, so its height is at most $e$ by the height theorem. Every prime chain in a local ring can be extended to end at its maximal ideal; hence $\dim R = ht m \leq e$ .

F2step 1.1algebra∎

Depends on

Used by

betti numbers residue field regular ring Example
completion regularity invariance Example
cusp local ring not regular Example
embedding dimension versus dimension node Example
formal power series ring regular Example
localised polynomial ring regular Example
regular local quotient by parameter is regular Lemma
quotient and lifting regularity across a regular element Theorem
serre normality criterion Theorem

Dependency tree · two levels

8 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

Remark 12.4, p.115 (standard reference, not scraped)