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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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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

1.1

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.

given

2.1

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.

F1F2step 1.1∎

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

A rational map to an affine target has a unique maximal open domain Theorem

Dependency tree · two levels

5 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 and §5d pp. 103–104 (standard reference, not scraped)