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

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Compatible classical morphisms to an affine target glue over an open cover

Statement

For an open $U$ of an affine variety X and an open cover $U = ⋃_{i} U_{i}$ , compatible morphisms $ϕ_{i} : U_{i} \to Y$ to a fixed affine target glue uniquely to a morphism $ϕ : U \to Y$ .

Facts & Assumptions

Given: An affine variety $X$ over algebraically closed $k$ , an open cover $U = ⋃_{i} U_{i}$ of an open $U \subseteq X$ , an affine target $Y$ , and morphisms $ϕ_{i} : U_{i} \to Y$ agreeing on every overlap.

[F1]

Compatible regular functions on a cover glue uniquely (Classical regular functions satisfy locality and unique gluing).

[F2]

A set map to an affine target is a morphism when global regular functions pull back regularly (A morphism from an open subset of a classical affine variety to an affine variety).

Proof

technique · direct

1.1given

Compatibility means $ϕ_{i} (x) = ϕ_{j} (x)$ for every point of each overlap. Thus the union of their graphs is a function $ϕ : U \to Y$ restricting to each $ϕ_{i}$ . Every value is in Y because it is the value of a local map into Y. For empty U and the empty cover this is the empty graph.

2.1F1F2step 1.1∎

If $s \in O_{Y} (Y)$ , the functions $s \circ ϕ_{i}$ are regular by F2 and agree on overlaps. F1 makes their glued function regular, and pointwise it is $s \circ ϕ$ . Hence F2 makes $ϕ$ a morphism. Any map with the required restrictions equals the same graph union, proving uniqueness.

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, Proposition 3.9 p. 61 and §5d pp. 103–104. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

Used by

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Sources

J. S. Milne, Algebraic Geometry v6.10, Proposition 3.9 p. 61 and §5d pp. 103–104 (standard reference, not scraped)