Alphabeta MathHow 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 maximal domain of definition of a rational map to an affine variety
Definition
Let be a rational map between classical affine varieties. Its domain of definition is the subset where the union runs over all affine opens for which has a representative morphism .
The previous items guarantee that this is well defined: transitivity of the equivalence relation is provided by The rational-map equivalence relation is transitive, and Morphisms from an irreducible affine variety to an affine variety are determined by a dense open subset gives uniqueness of representatives on overlaps.
Depends on
Used by
- The rational map (x,y) mapsto y / x on the affine plane is undefined along x = 0 Counterexample
- Dominant morphisms and dominant rational maps Definition
- Dominant maps pull back function fields functorially Lemma
- Dominant rational maps to an affine variety correspond to injective homomorphisms of function fields Theorem
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)
- J. S. Milne, Algebraic Geometry, Proposition 5.8 (standard reference, not scraped)