CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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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.
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A rational function with no codimension-one poles is regular

Statement

Assume the Axiom of Choice. Let $X$ be an irreducible normal classical variety and let $φ \in k (X)$ . If, on every affine chart $U$ , $φ$ belongs to $k [U]_{p}$ for every height-one prime $p$ of $k [U]$ , then $φ$ is globally regular. This is the meaning of having no codimension-one poles in the classical register.

Facts & Assumptions

Given: AC, $X$ and $φ$ satisfying the stated chartwise prime-local membership condition.

[F1]

On an irreducible normal classical variety, global regularity is equivalent to membership in every height-one prime localization on every affine chart (Regular functions on a normal variety are cut out in codimension one). Classical closed-point regularity means membership in the germ ring (A rational function is regular at a point exactly when it lies in the local ring); the prime-local criterion in this item uses the common chart function field of Integral classical varieties in the compatible affine-atlas register.

Proof

1.1F1given

The hypothesis is precisely the prime-local membership condition in [F1]. The implication from this condition to global regularity gives that $φ$ is regular on $X$ .

2.1F1step 1.1∎

Thus a rational function with no codimension-one poles, in the explicit chartwise sense of the Statement, is globally regular.

Depends on

Used by

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Dependency tree · two levels

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Sources

J. S. Milne, Algebraic Geometry (2025 version), Ch. 8 §a: Corollary 8.15 (a rational function regular outside codimension two is regular) (standard reference, not scraped)