TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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 varieties are normal

Statement

Assume the Axiom of Choice. At every regular point of a classical variety over an algebraically closed field the local ring is an integrally closed domain; hence every regular variety is normal. No characteristic or perfectness hypothesis is needed.

Facts & Assumptions

Given: AC, the algebraically closed field $k$ , the classical variety $X$ , a regular point $x \in X$ .

[F1]

A point $x$ of a classical variety is regular exactly when its local ring $O_{X, x}$ is a regular local ring; on an affine chart this local ring is the localisation of the coordinate ring at the maximal ideal of $x$ (Regular points of locally Noetherian schemes, The local ring at a point of an affine variety is the localization at its maximal ideal).

[F2]

Every regular local ring is an integrally closed domain (regular local rings are normal). AC is used there.

[F3]

$x$ is a normal point exactly when $O_{X, x}$ is an integrally closed domain, and $X$ is normal exactly when all of its points are normal (Normal points and normal varieties).

Proof

1.1F1F2F3given

Let $x$ be a regular point of $X$ . By [F1] the local ring $O_{X, x}$ is a regular local ring, and by [F2] it is an integrally closed domain. By [F3] the point $x$ is therefore normal.

2.1F1F2F3step 1.1∎

Since $x$ was an arbitrary regular point and [F3] decides normality pointwise, every regular point of $X$ is normal; hence a variety all of whose points are regular is normal, that is, every regular variety is normal. Neither [F1] nor [F2] used any hypothesis on the characteristic of $k$ or its perfectness, so the conclusion carries no such hypothesis.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

33 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

J. S. Milne, Algebraic Geometry (2025 version), Ch. 8 §a: normal points (Definition 8.1) together with the commutative-algebra theorem that regular local rings are integrally closed (standard reference, not scraped)