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 vanishing ideal of a subset of affine space
Definition
Let be an algebraically closed field and let . The vanishing ideal of is
Thus records the polynomial equations satisfied by every point of . At the two boundary cases,
Depends on
Used by
- The coordinate ring of an affine algebraic set Definition
- The empty affine algebraic set corresponds to the unit ideal and the zero coordinate ring Example
- Vanishing ideals and zero loci form a Galois connection Lemma
- Affine algebraic sets correspond to radical ideals, and irreducible ones to prime ideals Theorem
- Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms Theorem
- Dominant rational maps to an affine variety correspond to injective homomorphisms of function fields Theorem
- Polynomial functions on an affine algebraic set are exactly its coordinate ring Theorem
- Regular functions on a principal open are the principal localization of the coordinate ring Theorem
Dependency tree · one level
1 result within one dependency step 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, Chapter 2d-e (standard reference, not scraped)
- Donu Arapura, Notes on Basic Algebraic Geometry, §1.3 and §1.6 (standard reference, not scraped)