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 local parameter is nonzerodivisor

Statement

In a positive-dimensional regular local ring, every member of a regular system of parameters is a nonzerodivisor.

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 equivalent basis: 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.

[F2]

regular local domain induction: Every regular local ring is an integral domain.

Proof

1.1

The class of any member $x$ is a member of a cotangent basis and hence is nonzero. In particular $x \neq 0$ .

2.1

The ring is a domain, so multiplication by this nonzero $x$ is injective. This proves the assertion for every member of the supplied tuple.

F2step 1.1∎

Depends on

Used by

regular local rings are domains and cohen macaulay Theorem

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

10.106.2–10.106.3 (standard reference, not scraped)