TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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

Statement

Assume the Axiom of Choice.

Let $X$ be a classical affine variety over an algebraically closed field $k$ . Then evaluation induces a ring isomorphism $k [X] \tilde{\to} O_{X} (X) .$

Facts & Assumptions

Given: The Axiom of Choice and a classical affine variety $X$ over an algebraically closed field $k$ .

[L1]

For every classical affine variety, the principal open determined by $1$ is the whole variety: $D_{X} (1) = X$ (A principal open subset of a classical affine variety).

[L2]

The principal localisation at $1$ is the original ring: $R_{1} ≅ R$ (Principal localisation $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$ ).

[L3]

Assuming the Axiom of Choice, for every $\overline{f} \in k [X]$ , evaluation induces a ring isomorphism $k [X]_{\overline{f}} \tilde{\to} O_{X} (D_{X} (\overline{f}))$ (Regular functions on a principal open are the principal localization of the coordinate ring).

Proof

technique · direct

1.1L1L2L3given

Apply [L3] to the element $1 \in k [X]$ . By [L1], the target is $O_{X} (X)$ , and by [L2] the source is canonically $k [X]$ . The resulting isomorphism is exactly the evaluation map from $k [X]$ to global regular functions on $X$ .

2.1step 1.1∎

Therefore evaluation induces a ring isomorphism $k [X] \tilde{\to} O_{X} (X)$ .

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources