LemmaStatement: Literature-sourcedProof: Literature-sourced

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.

Pointwise regularity and smoothness over a perfect field

PDF p. 10: Dense smooth locus in Lemma 5.1; PDF p. 15: Codimension of the singular locus

Statement

Let $k$ be a perfect field and $X$ a reduced scheme locally of finite type over $k$ . A point $x \in X$ is smooth over $k$ if and only if $O_{X, x}$ is a regular local ring.

A perfect field is a field whose algebraic extensions are separable. Smooth at $x$ means that the structure morphism is smooth at that point. A regular local ring is a noetherian local ring whose Krull dimension equals the dimension of its maximal ideal modulo its square over its residue field.

External input

The equivalence is imported from The Stacks Project, Lemma 33.25.8, under the displayed hypotheses. The cited result also states density and openness, but only its pointwise equivalence is used here. Its underlying proof is not reconstructed locally.

Facts & Assumptions

Given: A perfect field $k$ , a reduced locally finite-type $k$ -scheme $X$ , and $x \in X$ .

Goal: Establish the stated pointwise equivalence as an imported result.

[F1]

The external input in Stacks Project Lemma 33.25.8 states precisely this equivalence for every point of such an $X$ ; its cited proof uses Algebra Lemma 10.140.5.

Proof

1.1F1given

The hypotheses in Given match those of the external input [F1], and $x$ is a point of the specified scheme.

2.1F1step 1.1∎

Applying [F1] at $x$ gives both directions of the equivalence. This is an application of the imported theorem, not a local reconstruction of its proof.

Depends on

The Stacks Project, Schemes smooth over fields · Lemma 33.25.8, tag 0B8X

Used by · 1

Smooth density and normal codimension over a perfect field

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

The Stacks Project, Schemes smooth over fields · Lemma 33.25.8, tag 0B8X

For a reduced scheme locally of finite type over a perfect field, its smooth locus is exactly the set of points with regular local ring; this locus is dense and open.

Hypotheses: The base is a perfect field and the scheme is reduced and locally of finite type over it.

Reading limits: The statement and its brief proof in Section 33.25 were read; the proof of the cited Algebra Lemma 10.140.5 was not reconstructed or independently verified.

Sources

The Stacks Project, Schemes smooth over fields · Lemma 33.25.8, tag 0B8X