LemmaStatement: AI-generatedProof: AI-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.

Fields are regular local rings

PDF p. 10: Dense smooth locus in Lemma 5.1

Statement

Every field is a regular local ring.

A regular local ring is a noetherian local ring whose Krull dimension equals the dimension of $m / m^{2}$ over its residue field, where $m$ is its maximal ideal. Krull dimension is the supremum of lengths of strict chains of prime ideals.

Facts & Assumptions

Given: A field $K$ .

Goal: Prove that $K$ is regular local.

[F1]

The displayed equality of Krull dimension and embedding dimension characterizes regular noetherian local rings (embedding dimension and regular local ring).

Proof

1.1given

The only proper ideal of $K$ is $(0)$ , which is both prime and maximal. Thus $K$ is noetherian and local, and its Krull dimension is zero.

2.1F1step 1.1∎

Its maximal ideal and its square are zero, so the dimension of $m / m^{2}$ over $K$ is zero. The criterion [F1] now makes $K$ regular local.

Depends on

embedding dimension and regular local ring

Used by · 1

Smooth density and normal codimension over a perfect field

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.