ExampleConstruction: AI-generatedVerification: AI-generatedaudited 2026-09-12

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.

A polynomial map and its pullback on coordinate rings

Example

Let $X = A_{k}^{2}$ with coordinates $s, t$ , and let $Y = V (y - x^{2}) \subseteq A_{k}^{3} .$ Define $φ : X \to Y, (s, t) ⟼ (s, s^{2}, t) .$

The point lies in $Y$ because its coordinates satisfy $y - x^{2} = s^{2} - s^{2} = 0$ . Therefore $φ$ is a morphism. Since $k [Y] = k [x, y, z] / (y - x^{2}),$ the induced pullback sends $x \mapsto s, y \mapsto s^{2}, z \mapsto t .$ The defining relation pulls back to $s^{2} - s^{2} = 0$ , so the homomorphism respects the quotient relation exactly as the affine-morphism theorem predicts.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, Examples 3.29-3.30 (standard reference, not scraped)