TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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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.

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Global regular functions on a classical affine variety are its coordinate ring

Statement

Assume the Axiom of Choice, inherited from the Nullstellensatz route. For every affine algebraic set $X$ , the canonical map $k [X] \to O_{X} (X)$ is an isomorphism. This includes varieties and the empty set.

Facts & Assumptions

Given: AC, an algebraically closed field $k$ , and an affine algebraic set $X$ .

[F1]

For any $f$ , $A_{f} ≅ O_{X} (D (f))$ (Regular functions on a principal open are the principal localization).

Proof

technique · direct

1.1

Apply F1 to $f = 1$ . Then $D (1) = X$ , so the displayed isomorphism is $A_{1} ≅ O_{X} (X)$ . The maps $A \to A_{1}$ , $a \mapsto a / 1$ , and $A_{1} \to A$ , $a / 1^{r} \mapsto a$ , are mutually inverse algebra maps.

F1givenalgebra

2.1

The composite sends $a$ to its function on $X$ , which is exactly the canonical map in the statement. If $X$ is empty both algebras are the zero ring by the empty case of F1.

F1step 1.1∎

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, Proposition 3.11 final paragraph, p. 62. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Used by

Dependency tree · two levels

13 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, Proposition 3.11 final paragraph, p. 62 (standard reference, not scraped)