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

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The function field is independent of the chosen nonempty principal affine open

Statement

Assume the Axiom of Choice, inherited from the Nullstellensatz route. For every nonempty affine open $U$ in an affine variety $X$ , restriction gives the canonical identification $Frac (O_{X} (U)) = k (X)$ . In particular, for $f \neq 0$ , $k [D_{X} (f)] = k [X]_{f}$ and $Frac (k [D_{X} (f)]) = k (X)$ . The identifications commute with further nonempty affine-open restriction.

Facts & Assumptions

Given: AC, an affine variety $X$ over algebraically closed $k$ , and a nonempty affine open $U \subseteq X$ . For the principal case, $f \in k [X]$ is nonzero.

[F1]

Nonempty open section algebras embed in k(X), compatibly with restriction (Regular functions on a nonempty open embed in the affine function field).

[F2]

A embeds in its fraction field (The function field of an irreducible classical affine variety).

[F3]

An injective map from a domain into a field extends uniquely to its fraction field (Every injective ring map from a domain into a field factors uniquely through its field of fractions).

[F4]

Principal-open sections are A_f (Regular functions on a principal open are the principal localization).

[F5]

Nonempty principal opens are affine (Every nonempty principal open is a classical affine variety).

Proof

technique · direct

1.1

Put $A = k [X]$ , $B = O_{X} (U)$ and $K = k (X)$ . Restriction sends A into B, and the embedding $B ↪ K$ of F1 sends each restricted polynomial a to $a / 1$ . Thus $A \subseteq B \subseteq K$ with these specified maps, and B is a domain as a subring of a field.

F1F2given

2.1

F3 extends $B ↪ K$ to an embedding $Frac (B) ↪ K$ . Its image contains every $a / b$ for $a, b \in A$ , $b \neq 0$ , since $A \subseteq B$ . Such fractions exhaust K by F2, so this extension is surjective and is the claimed isomorphism.

F2F3step 1.1

3.1

For $U = D (f)$ , F4 and F5 identify B with the affine coordinate ring $A_{f}$ , so step 2.1 is the asserted principal-open identification. For a further nonempty affine open V, both restriction embeddings into K agree on sections by F1; their extensions agree on every ratio by the uniqueness in F3.

F1F3F4F5step 2.1∎

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, §3k p. 74 and Proposition 3.32 p. 71. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Used by

Dependency tree · two levels

31 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, §3k p. 74 and Proposition 3.32 p. 71 (standard reference, not scraped)