DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

A rational map as an equivalence class of morphisms on nonempty opens

Definition

For affine varieties $X, Y$ , a representative of a rational map $X ⇢ Y$ is a pair $(U, ϕ)$ with $U \subseteq X$ nonempty open and $ϕ : U \to Y$ a morphism. Two pairs represent the same rational map when their maps agree on some nonempty open subset of their common domain. A rational map is an equivalence class for this relation; its equivalence-relation well-definedness is supplied by The rational-map relation is transitive ↗. The domain of a representative need not be all of $X$ .

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, §5l p. 117. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Used by

Birational maps and birational equivalence of classical varieties Definition
Dominant classical morphisms and rational maps Definition
The candidate domain of a rational map Definition
The rational-map relation is transitive Lemma
A rational map to an affine target has a unique maximal open domain Theorem

Dependency tree · two levels

5 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 v6.10, §5l p. 117 (standard reference, not scraped)