DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-09-12

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.

Rational maps between irreducible classical affine varieties

Definition

Let $X$ and $Y$ be classical affine varieties. A rational map $X ⇢ Y$ is an equivalence class of morphisms $φ : U \to Y$ whose source $U \subseteq X$ is a nonempty affine open subset.

Two representatives $φ : U \to Y$ and $ψ : V \to Y$ are equivalent when there exists a nonempty affine open subset $W \subseteq U \cap V$ such that $φ ∣_{W} = ψ ∣_{W} .$

The transitivity needed for this equivalence relation is discharged by The rational-map equivalence relation is transitive ↗.

Depends on

Used by

Dependency tree · two levels

9 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, paragraph before Proposition 5.38 (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, paragraph before Example 3.2.3 (standard reference, not scraped)