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 coordinate ring of an affine algebraic set
Definition
Let be an algebraically closed field and let be an affine algebraic set. Its coordinate ring is
The image of in is the th coordinate function on . Thus elements of are polynomial expressions modulo the relations that vanish everywhere on .
Depends on
Used by
- The ideals (x) and (x²) have the same zero locus but different quotient rings Counterexample
- A principal open subset of a classical affine variety Definition
- Regular functions on open subsets of a classical affine variety Definition
- A polynomial map and its pullback on coordinate rings Example
- Affine space has zero vanishing ideal and polynomial coordinate ring Example
- The coordinate cross V(xy) is reducible and its coordinate ring has zero divisors Example
- The empty affine algebraic set corresponds to the unit ideal and the zero coordinate ring Example
- The parabola y=x² has coordinate ring k[t] and isomorphic intrinsic geometry to the affine line Example
- Points of an affine algebraic set correspond to maximal ideals of its coordinate ring Lemma
- A classical affine variety has a domain as its coordinate ring, and conversely Theorem
- Affine algebraic sets and reduced affine k-algebras at the object level 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
- The local ring at a point of an affine variety is the localization at its maximal ideal Theorem
Dependency tree · two levels
7 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, Chapter 2i (standard reference, not scraped)
- Donu Arapura, Notes on Basic Algebraic Geometry, §1.5 (standard reference, not scraped)