DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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 graph morphism over a base

Definition

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

Depends on

Used by

The ideal of a polynomial graph Example
The graph is a pullback of the diagonal Lemma

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

Vakil 11.1.17, p.232 (standard reference, not scraped)