LemmaStatement: 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.

regular system of parameters equivalent basis

Statement

Let $(R, m, k)$ be a nonzero Noetherian local ring of dimension $d$ , and let $x = (x_{1}, \dots, x_{d}) \in m^{d}$ . Then $x$ is a regular system of parameters if and only if its classes form a $k$ -basis of $m / m^{2}$ . In particular every lift of a cotangent basis in a regular local ring generates $m$ and is a system of parameters.

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]

regular system of parameters: In a regular local ring $(R, m, k)$ of dimension $d$ , an ordered minimal generating tuple $(x_{1}, \dots, x_{d})$ of $m$ is a regular system of parameters. The tuple is empty when $d = 0$ . This definition concerns generators of the maximal ideal; the regular-sequence property is a theorem, not part of the definition.

[F2]

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 $m$ .

[F3]

Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators: Assume the Axiom of Choice. Let $R$ be a commutative ring, let $I ⊴ R$ satisfy $I \subseteq J (R)$ , and let $M$ be a finitely generated left $R$ -module. If elements $x_{1}, \dots, x_{r} \in M$ generate $M / I M$ , then $x_{1}, \dots, x_{r}$ generate $M$ .

Proof

1.1

If $x$ is a regular system, it minimally generates $m$ in a regular ring, whose cotangent dimension is $d$ . Its $d$ spanning classes therefore form a basis.

F1F2

2.1

Conversely, a basis of length $d$ makes the embedding dimension $d$ , so $R$ is regular. Nakayama lifts the spanning classes to generators of $m$ , and no generator can be removed since its class is independent. Their ideal has radical $m$ and length $d$ , which is exactly the parameter condition. For $d = 0$ , Nakayama gives $m = 0$ and the empty tuple has the same property.

F3F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

9 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

12.3–12.5, p.115 (standard reference, not scraped)