ExampleConstruction: AI-adaptedVerification: 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 ideal of a polynomial graph

Example

Let $k$ be a field, $n, m \geq 0$ , and let $u : A_{k}^{n} \to A_{k}^{m}$ be given by polynomials $f_{1}, \dots, f_{m} \in k [x_{1}, \dots, x_{n}]$ . Its graph is the closed subscheme of $A_{k}^{n + m}$ with ideal $(y_{1} - f_{1} (x), \dots, y_{m} - f_{m} (x)) .$

Facts & Assumptions

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

[F1]

For an $S$ -morphism $u : X \to Y$ (as in def-scheme-over-base), the graph morphism is $Γ_{u} = ({id}_{X}, u) : X \to X \times_{S} Y$ , supplied by thm-fibre-products-of-schemes-exist. Its first projection is the identity and its second projection is $u$ . The definition alone does not assert that its image is closed. (The graph morphism over a base)

[F2]

For an $S$ -morphism $u : X \to Y$ , put $H = (u {pr}_{X}, {pr}_{Y}) : X \times_{S} Y \to Y \times_{S} Y$ . The square with top arrow $Γ_{u} : X \to X \times_{S} Y$ , bottom arrow $Δ_{Y / S} : Y \to Y \times_{S} Y$ , left arrow $u$ , and right arrow $H$ is Cartesian. (The graph is a pullback of the diagonal)

[F3]

For a ring $A$ , closed immersions $Z \to Spec A$ are, up to unique isomorphism over $Spec A$ , precisely the morphisms $Spec (A / I) \to Spec A$ for ideals $I \subseteq A$ . (Closed immersions into affine schemes are quotient spectra)

[F4]

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

1.1givenF1F3algebra

By F1 the graph has coordinate map sending $x_{i}$ to $x_{i}$ and $y_{j}$ to $f_{j} (x)$ . This is a surjection to $k [x_{1}, \dots, x_{n}]$ . In the quotient by the displayed ideal, successive substitution replaces every $y_{j}$ by $f_{j} (x)$ , giving inverse ring maps with $k [x]$ . Thus the displayed ideal is exactly the kernel; F3 makes the morphism a closed immersion.

2.1F2F4step 1.1∎

The diagonal in $A_{k}^{m} \times_{k} A_{k}^{m}$ has ideal $(z_{j} - y_{j})$ by the same substitution calculation. Pulling it back along $(u {pr}_{X}, {pr}_{Y})$ sends those generators to $f_{j} (x) - y_{j}$ . By F2 this pullback is the graph; F4 computes its quotient ideal and gives the identical presentation. If $m = 0$ the list is empty and the graph is the identity; if $n = 0$ it is the point given by the constants $f_{j}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

Affine specialization of Vakil 11.1.17–18 (standard reference, not scraped)