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 self fibre product of the quadratic cover

Example

For $A_{k}^{1} \to A_{k}^{1}$ given by $t \mapsto x^{2}$ , the self fibre product is $Spec k [x, y] / (x^{2} - y^{2}) .$ If $char k \neq 2$ it is a reduced union of two distinct line components; if $char k = 2$ it is the doubled diagonal.

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]

Let $A \to C$ be a unital ring map. For any set of variables $(t_{i})$ and any ideal $I \subseteq A [t_{i}]$ , $(A [t_{i}] / I) \otimes_{A} C ≅ C [t_{i}] / I C [t_{i}] .$ Here the extended ideal is generated by the coefficient images of all elements of $I$ . For a multiplicative subset $M \subseteq A$ , $(M^{- 1} A) \otimes_{A} C ≅ {\overline{M}}^{- 1} C .$ These are ring isomorphisms; no flatness, finite-generation or nonzero-ring hypothesis is required. (Presentations and localization under base extension)

Verification

1.1

Use F1 with the two maps from $k [t]$ . F2 eliminates $t$ from the presentation $k [x, y, t] / (t - x^{2}, t - y^{2})$ and obtains $k [x, y] / (x^{2} - y^{2})$ , with the two required projections.

givenF1F2

2.1

If $2 \neq 0$ in $k$ , the invertible linear substitution $u = x - y, v = x + y$ turns the ring into $k [u, v] / (u v)$ . Every prime over $(u v)$ contains $u$ or $v$ . The ideals $(u), (v)$ are incomparable primes and their intersection is $(u v)$ by monomial comparison. Thus the ring is reduced with exactly two line components meeting at the origin. They are not comaximal: their sum is $(u, v)$ , so this is not a product-ring decomposition.

step 1.1algebra

3.1

If $char k = 2$ , then $x^{2} - y^{2} = (x - y)^{2}$ . Put $ϵ = x - y$ to obtain $k [y, ϵ] / (ϵ^{2})$ . Its class $ϵ$ is nonzero with square zero, and its reduced support is the diagonal. Both characteristic cases give nonzero rings and exhaust all fields.

step 1.1algebra∎

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

Vakil 10.3.E (standard reference, not scraped)