Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)
How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Morphisms of schemes are local on compatible open covers

Statement

Compatible morphisms of schemes on an open cover of a scheme X glue uniquely to a morphism from X; two morphisms out of X are equal if their restrictions to an open cover are equal. Both assertions may be checked after affine-open refinement of source and target.

Facts & Assumptions

Given: An open cover of X and compatible local scheme morphisms.

Proof

technique · direct
1.1

The underlying continuous maps agree on overlaps and therefore glue to a unique continuous map from X.

givenconstruct
2.1

Compatibility of the maps of structure sheaves on overlaps glues their sheaf maps, and locality on stalks is local on X; thus the glued map is a scheme morphism.

step 1.1construct
3.1

Restricting a hypothetical equality to the cover proves necessity, while uniqueness in the gluing construction proves sufficiency; refining by affine opens preserves the same argument.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

9 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