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.

polynomial local regularity fibre step

Statement

For a prime $q \subset R [t]$ with contraction $p \subset R$ , the closed fibre of $R_{p} \to R [t]_{q}$ is $k (p) [t]$ localized at a prime. That prime is either zero, giving a field, or generated by an irreducible polynomial, giving a DVR. In both cases the fibre is regular.

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]

one dimensional regular local rings are dvrs: A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring. Fields are excluded from the term DVR.

[F2]

Every Euclidean domain is a principal ideal domain: Every Euclidean domain is a principal ideal domain.

[F3]

For every field $F$ , $F [x]$ is a Euclidean domain with degree as Euclidean function: For every field $F$ , the ring $F [x]$ is a Euclidean domain with Euclidean function $δ (f) = \deg f$ on nonzero polynomials.

Proof

1.1

Localize first at $R ∖ p$ , quotient by $p R_{p}$ , and then localize at the image of $q$ . Fractions and the quotient relation identify the fibre with $k (p) [t]_{\overset{ˉ}{q}}$ . Over this field the polynomial ring is Euclidean and hence a PID.

F3F2algebra

2.1

In a PID every nonzero prime is generated by an irreducible $f$ and is maximal. Localization at it is a nonfield local PID; every nonzero element is a unit times $f^{n}$ , so the exponent gives a discrete valuation and the localization is a DVR. The zero-prime localization is the rational function field. DVRs are regular by the one-dimensional theorem, and a field has zero maximal ideal and dimension zero, hence is regular.

F1step 1.1algebra∎

Depends on

Used by

localisation and polynomial extension of regular rings Theorem

Dependency tree · two levels

15 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

Proposition 12.36 proof, p.124 (standard reference, not scraped)