ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05‡ 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.

Units satisfying the cocycle law glue local rank-one free modules into a line bundle

Example

Let $(X, O_{X})$ be a ringed space with an open cover $X = ⋃_{i} U_{i}$ . Suppose that for each pair $(i, j)$ there is a unit

g_{i j} \in O_{X} (U_{i} \cap U_{j})^{\times}

satisfying $g_{i i} = 1, g_{i j} = g_{j i}^{- 1}, g_{j k} g_{i j} = g_{i k}$ on triple overlaps. Then the free rank-one modules $O_{X} ∣_{U_{i}}$ glue to a line bundle on $X$ .

Facts & Assumptions

Given: A ringed space $(X, O_{X})$ , an open cover $X = ⋃_{i} U_{i}$ , and units $g_{i j}$ satisfying the displayed cocycle rules.

[F1]

An $O_{X}$ -module is a sheaf with compatible module structures on every open set (Modules on a ringed space).

[F2]

A gluing datum is given by overlap isomorphisms satisfying identity and cocycle conditions (A gluing datum for sheaves on an open cover).

[L1]

Compatible local sheaves glue uniquely up to unique isomorphism (Compatible local sheaves glue uniquely up to unique isomorphism).

Verification

technique · direct

1.1

On $U_{i} \cap U_{j}$ , define $φ_{i j} : O_{X} ∣_{U_{i} \cap U_{j}} ⟶ O_{X} ∣_{U_{i} \cap U_{j}}, s ⟼ g_{i j} s .$ Because $g_{i j}$ is a unit, $φ_{i j}$ is an isomorphism of $O_{X}$ -modules, and the rules for the $g_{i j}$ are exactly the identity and cocycle conditions of [F2].

F1F2given

2.1

By [L1], the local free rank-one modules $O_{X} ∣_{U_{i}}$ glue to an $O_{X}$ -module $L$ on $X$ with $L ∣_{U_{i}} ≅ O_{X} ∣_{U_{i}}$ for every $i$ . Therefore $L$ is locally free of rank one, i.e. a line bundle.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Ravi Vakil, The Rising Sea, Section 2.5.D (standard reference, not scraped)
The Stacks Project, Section 26.14 (standard reference, not scraped)