ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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 affine plane and the pair of generic points

Example

For any field $k$ , $A_{k}^{1} \times_{k} A_{k}^{1} ≅ A_{k}^{2}$ functorially. Nevertheless the product has distinct points over the pair of generic points: in $k [x, y]$ , both $(0)$ and $(y - x)$ contract to $(0)$ in each of $k [x]$ and $k [y]$ .

Facts & Assumptions

Given: The objects, hypotheses and conventions in the statement above.

[F1]

Let $A \to B$ and $A \to C$ be maps of commutative unital rings, allowing the zero ring. In the category of all schemes, $Spec B \times_{Spec A} Spec C ≅ Spec (B \otimes_{A} C) .$ The projections correspond to $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$ . (Affine fibre products are spectra of tensor products)

[F2]

For scheme morphisms $f : X \to S$ and $g : Y \to S$ , points of $P = X \times_{S} Y$ are in bijection with quadruples $(x, y, s, r)$ where $f (x) = g (y) = s$ and $r \in Spec (κ (x) \otimes_{κ (s)} κ (y)) .$ The residue field at the corresponding point of $P$ is canonically $κ (r)$ . (Points of a fibre product via residue-field tensors)

Verification

1.1givenF1algebra

F1 identifies the product ring with $k [x] \otimes_{k} k [y] ≅ k [x, y]$ : the maps send $x \otimes 1$ to $x$ and $1 \otimes y$ to $y$ , with inverse on every polynomial specified by these images. Consequently morphisms from every $k$ -scheme $T$ into the plane are compatible pairs of morphisms into the two lines. Empty test schemes are allowed.

2.1F2step 1.1algebra∎

The ring $k [x, y]$ is a domain, so $(0)$ is prime. The quotient by $(y - x)$ is $k [x]$ , also a domain, so this is another prime, distinct because $y - x \neq 0$ . Substitution $y = x$ is injective on either single-variable subring, giving the asserted contractions. F2 explains these as distinct residue-tensor primes over the same point pair. This works over every field, including characteristic two.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Vakil 10.1.2–3 (standard reference, not scraped)