TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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

Assume the Axiom of Choice.

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: Under the Axiom of Choice, if $p$ is minimal over an ideal generated by $n \geq 1$ elements in a Noetherian ring, then $ht (p) \leq n$ .

Proof

1.1F1

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.

2.1F2step 1.1algebra∎

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 $\dim R = ht m \leq e$ .

Depends on

Dependency tree · 2 levels

Sources

Remark 12.4, p.115