LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge 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.

Affineness from a finite principal cover

Statement

Let $X$ be a scheme, $R = Γ (X, O_{X})$ , and $f_{1}, \dots, f_{n} \in R$ . Write $X_{f}$ for the open locus where the germ of $f$ is a unit. If $(f_{1}, \dots, f_{n}) = R$ and each $U_{i} = X_{f_{i}}$ is affine, then $X$ is affine. In fact the canonical morphism $X \to Spec R$ is an isomorphism. Empty $X$ and $n = 0$ are allowed.

Facts & Assumptions

Given: The objects, hypotheses and conventions in the statement above.

[F1]

For a scheme $X$ and a ring $A$ , taking global sections induces a natural bijection $Hom (X, Spec A) ≅ {Hom}_{C R i n g} (A, Γ (X, O_{X})) .$ (Morphisms to an affine scheme and global sections)

[F2]

For $f \in A$ , $Γ (D (f), O) = A_{f}$ . If $D (g) \subseteq D (f)$ , the restriction is the canonical localization map $A_{f} \to A_{g}$ . (Sections and restrictions on distinguished opens of an affine scheme)

[F3]

If $0 ⟶ M^{'} \overset{f}{\to} M \overset{g}{\to} M^{''} ⟶ 0$ is a short exact sequence of $R$ -modules, then $0 ⟶ S^{- 1} M^{'} \overset{S^{- 1} f}{\to} S^{- 1} M \overset{S^{- 1} g}{\to} S^{- 1} M^{''} ⟶ 0$ is a short exact sequence of $S^{- 1} R$ -modules. (Localisation of modules is exact)

[F4]

Let $F$ be a presheaf of sets on a topological space $X$ . Then $F$ is a sheaf if and only if, for every open set $U \subseteq X$ and every open cover $U = ⋃_{i \in I} U_{i}$ , the restriction map $e : F (U) ⟶ \prod_{i \in I} F (U_{i}), e (s) = (s ∣_{U_{i}})_{i \in I},$ is an equalizer of the two maps $d_{0}, d_{1} : \prod_{i \in I} F (U_{i}) ⇉ \prod_{(i, j) \in I \times I} F (U_{i} \cap U_{j}),$ defined by $d_{0} ((s_{i})) = (s_{i} ∣_{U_{i} \cap U_{j}})_{i, j}, d_{1} ((s_{i})) = (s_{j} ∣_{U_{i} \cap U_{j}})_{i, j} .$ (The sheaf axiom is the equalizer condition on a cover)

Proof

1.1

Write $1 = \sum_{i} g_{i} f_{i}$ . At every point some $f_{i}$ is a unit, so the $U_{i}$ cover $X$ . Set $B_{i} = Γ (U_{i}, O_{X})$ . The intersection $U_{i} \cap U_{j}$ is the principal open defined by $f_{j} ∣_{U_{i}}$ in $U_{i}$ , hence affine, and F2 gives its ring by localization.

givenF2

2.1

F4 identifies $R$ as the kernel of the difference of restriction maps $\prod_{i} B_{i} \to \prod_{i, j} Γ (U_{i} \cap U_{j}, O_{X})$ , a homomorphism of $R$ -modules. Fix $k$ and localize this kernel at $f_{k}$ . Localization preserves kernels by F3: apply exactness to the kernel-image short exact sequence and to the inclusion of the image in the target. It commutes with these finite products, because a common denominator exists for every finite tuple.

F3F4step 1.1

3.1

By F2 each localized factor is the ring of sections on its intersection with $U_{k}$ . By F4 their kernel is $Γ (U_{k}, O_{X})$ , since the opens $U_{i} \cap U_{k}$ cover $U_{k}$ . Thus restriction induces $R_{f_{k}} ≅ B_{k}$ as rings; multiplicativity follows from restriction and fraction multiplication, not merely module exactness.

F2F4step 2.1

4.1

F1 supplies the canonical map $j : X \to Spec R$ induced by the identity of $R$ . Its inverse image of $D (f_{k})$ is $X_{f_{k}}$ , and on this open it is the isomorphism from step 3.1. The $D (f_{k})$ cover $Spec R$ because the $f_{k}$ generate 1; local inverse maps agree and glue to the inverse of $j$ . If $n = 0$ then $R = 0$ , which forces $X = \emptyset$ since each point has a nonzero local ring. For $n = 1$ , $f_{1}$ is a unit and $U_{1} = X$ . Zero sections merely contribute empty charts.

F1step 1.1step 3.1∎

Depends on

Used by

Affineness is local on the target Lemma

Dependency tree · two levels

18 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

Stacks 28.28.3, with elementary finite-equalizer proof (standard reference, not scraped)