LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

The rational-map relation is transitive

Statement

The relation used to define rational maps between affine varieties is reflexive, symmetric and transitive, hence is an equivalence relation.

Facts & Assumptions

Given: Affine varieties $X, Y$ over algebraically closed $k$ , and representatives $(U_{i}, ϕ_{i})$ with nonempty open domains in $X$ .

[F1]

Representatives have nonempty open domains, and equivalence is agreement on some nonempty open (A rational map as an equivalence class of morphisms on nonempty opens).

[F2]

Finite intersections of nonempty opens of X are nonempty (Every nonempty open of a classical affine variety is dense).

Proof

technique · direct

1.1

A representative $(U, ϕ)$ agrees with itself on the nonempty open U, proving reflexivity. If two maps agree on a nonempty open W, reversing the equality on that same W proves symmetry.

F1given

2.1

If $ϕ_{1} = ϕ_{2}$ on a nonempty open W and $ϕ_{2} = ϕ_{3}$ on a nonempty open V, F2 makes $W \cap V$ nonempty open in X. All three maps are defined there, and equality of their values gives $ϕ_{1} = ϕ_{3}$ there. This is the required transitivity witness by F1.

F1F2given∎

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

Dominant classical morphisms and rational maps Definition
The candidate domain of a rational map Definition
Dominant rational maps compose on nonempty open domains Lemma
A rational map to an affine target has a unique maximal open domain Theorem

Cited to discharge well-definedness by A rational map as an equivalence class of morphisms on nonempty opens.

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)