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

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Classical regular functions satisfy locality and unique gluing

Statement

Regular functions on opens in an affine algebraic set form unital $k$ -algebras, have restriction homomorphisms, and glue uniquely from every compatible open cover. Regularity can be checked on an open cover.

Facts & Assumptions

Given: An open $U$ in an affine algebraic set $X$ over algebraically closed $k$ . For gluing, an open cover $U = ⋃_{i} U_{i}$ and regular functions $s_{i}$ agreeing on overlaps.

[F1]

Regularity means having a quotient expression near each point (A regular function on an open subset of a classical affine variety).

Proof

technique · direct

1.1

Near a given point, intersect the neighbourhoods on which $s = g / h$ and $t = a / b$ with $h, b$ nowhere zero. Then $s + t = (g b + a h) / (h b)$ , $s t = g a / (h b)$ , and $- s = - g / h$ on this intersection; the denominators are nonzero there. Constants are $c / 1$ . Pointwise ring laws therefore give a $k$ -algebra, including the zero function algebra on the empty open.

F1givenalgebra

2.1

Restricting a quotient expression to its intersection with a smaller open preserves regularity. Pointwise sums, products and constants restrict to themselves, so restriction is a unital algebra homomorphism; iterated restrictions agree.

F1step 1.1

3.1

Given $U = ⋃_{i} U_{i}$ and regular $s_{i}$ with equal restrictions on every overlap, the union of their function graphs is a function $s : U \to k$ : for a fixed point all available values coincide. On $U_{i}$ it equals $s_{i}$ , hence near each point it has the quotient expression of that section. Thus it is regular. A function with these restrictions must have that same value at every point, proving uniqueness. For the empty cover of the empty open its graph is empty.

F1given∎

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, Proposition 3.9, p. 61. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

A regular function on an open subset of a classical affine variety

Used by

Dependency tree · two levels

4 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 3.9, p. 61 (standard reference, not scraped)