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.

Cylinder and zero section of a smooth affine surface

Statement

Assume AC. If $X = Spec A$ is a nonempty smooth affine surface over a field $k$ , then $Y = Spec A [t] = X \times_{k} A_{k}^{1}$ is a smooth affine threefold. Projection $p : Y \to X$ and the zero section $s : X \to Y$ satisfy $p s = {id}_{X}$ , and therefore $s^{*} p^{*} E ≅ E$ for every vector bundle $E$ .

Facts & Assumptions

Given: The data and hypotheses in the statement.

Goal: Establish the stated conclusion.

[F1]

A noetherian finite-dimensional ring gains exactly one dimension on adjoining a variable. A Noetherian polynomial ring has dimension one larger

[F2]

Smooth morphisms are stable under base change and composition. Smoothness survives base change and composition

[F3]

The finite presentation $k [t]$ has one variable and no equations; its empty Jacobian minor is $1$ . Under AC, the relative Jacobian criterion applied with this presentation proves that $A_{k}^{1} \to Spec k$ is smooth. Relative Jacobian criterion with its presentation hypothesis The meaning of smooth is fixed in Smooth morphism of schemes.

Proof

1.1F1F2F3

The smooth finite-type coordinate ring $A$ is noetherian, and the surface has dimension two. By [F1], $A [t]$ has dimension three. Its finite type over $k$ is immediate from generators for $A$ and $t$ . The map $p$ is a base change of the smooth affine line by [F3]; composition with the smooth map $X \to Spec k$ makes $Y$ smooth by [F2].

2.1constructstep 1.1∎

The homomorphisms $A \to A [t]$ and $A [t] \to A$ , $t \mapsto 0$ , compose to the identity. Hence $p s = {id}_{X}$ . Pullback of sheaves is functorial up to its canonical isomorphism, so $s^{*} p^{*} E ≅ E$ .

Depends on

Used by · 3

Dependency tree · 3 levels

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