CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-5.6-terra)audited 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.

The cusp parametrization t mapsto (t^2,t^3) is bijective but not an isomorphism

Statement refuted

A bijective morphism of affine varieties need not be an isomorphism.

Let $C : = V (v^{2} - u^{3}) \subseteq A_{k}^{2},$ and define $ν : A_{k}^{1} \to C, t \mapsto (t^{2}, t^{3}) .$ Every point of $C$ has the form $(a^{2}, a^{3})$ for a unique $a \in k$ , so $ν$ is bijection on points.

Its pullback on coordinate rings is $k [C] = k [u, v] / (v^{2} - u^{3}) ⟶ k [t], u \mapsto t^{2}, v \mapsto t^{3} .$ This homomorphism is injective but not surjective, because $t$ is not in its image. Hence Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms shows that $ν$ is not an isomorphism.

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Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Example 3.29(a) (standard reference, not scraped)
Michael Artin, Notes for a Course in Algebraic Geometry, Example 2.5.7 (standard reference, not scraped)