CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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.

Affine quasi-coherent sheaf determined by sections

Statement

Assume the Axiom of Choice, inherited from the affine equivalence (The Axiom of Choice). Let $A$ be a commutative ring with $1$ , put $X = Spec A$ , and let $F, G$ be quasi-coherent $O_{X}$ -modules (Quasi-coherent module on a scheme).

Then:

The canonical comparison morphism $κ_{F} : \tilde{Γ (X, F)} \to F$ , induced on $D (f)$ by the restriction map $Γ (X, F) \to Γ (D (f), F)$ followed by localisation, is an isomorphism (Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme).
A morphism $ψ : F \to G$ of quasi-coherent $O_{X}$ -modules is determined uniquely by its global section map $ψ_{X} : Γ (X, F) \to Γ (X, G)$ : if $ψ_{X} = φ_{X}$ for a second morphism $φ$ , then $ψ = φ$ , and the assignment ${Hom}_{O_{X}} (F, G) \to {Hom}_{A} (Γ (X, F), Γ (X, G))$ , $ψ \mapsto ψ_{X}$ , is a bijection.

Facts & Assumptions

Given: The Axiom of Choice; a commutative ring $A$ ; the scheme $X = Spec A$ ; quasi-coherent $O_{X}$ -modules $F, G$ .

[F1]

Unit, counit and full faithfulness of the affine equivalence: the functor $M \mapsto \tilde{M}$ from $A$ -modules to quasi-coherent $O_{X}$ -modules and $Γ (X, -)$ are quasi-inverse equivalences; the canonical map $M \to Γ (X, \tilde{M})$ is an isomorphism; the canonical comparison $κ_{F} : \tilde{Γ (X, F)} \to F$ , whose component on $D (f)$ is induced by the restriction $Γ (X, F) \to Γ (D (f), F)$ followed by the canonical localisation $M \to M_{f}$ , is an isomorphism and is natural in $F$ ; and for all $A$ -modules $M, N$ the map ${Hom}_{A} (M, N) \to {Hom}_{O_{X}} (\tilde{M}, \tilde{N})$ , $u \mapsto \tilde{u}$ , is a bijection with inverse $ψ \mapsto ψ_{X}$ (Affine quasi-coherent sheaves are modules, Module sheaf on an affine scheme).

[F2]

For a quasi-coherent $F$ and $f \in A$ the module $Γ (X, F)$ is an $A$ -module and the restriction $Γ (X, F) \to Γ (D (f), F)$ is $A$ -linear, so it factors through the canonical localisation $Γ (X, F) \to Γ (X, F)_{f}$ (Module sheaf on an affine scheme).

Proof technique: direct; apply the counit and the full faithfulness of the affine equivalence and use naturality to identify global sections.

Proof

1.1F1F2

Claim 1 is exactly the counit statement of [F1]: for quasi-coherent $F$ on $X = Spec A$ the canonical comparison $κ_{F} : \tilde{Γ (X, F)} \to F$ is an isomorphism, and by [F2] its component on $D (f)$ is indeed induced by the restriction map followed by localisation.

2.1F1step 1.1

Claim 2, determination: let $ψ, φ : F \to G$ be morphisms of quasi-coherent $O_{X}$ -modules with $ψ_{X} = φ_{X}$ . Using the isomorphisms $κ_{F}$ and $κ_{G}$ of step 1.1, form the morphisms of associated sheaves $\tilde{Γ (X, ψ)} : = κ_{G}^{- 1} \circ ψ \circ κ_{F}$ and $\tilde{Γ (X, φ)} : = κ_{G}^{- 1} \circ φ \circ κ_{F}$ from $\tilde{Γ (X, F)}$ to $\tilde{Γ (X, G)}$ . By naturality of the counit in [F1] these are the morphisms induced by the $A$ -linear maps $Γ (X, ψ)$ and $Γ (X, φ)$ , which are equal by hypothesis; hence the two morphisms of associated sheaves coincide and therefore $ψ = κ_{G} \circ \tilde{Γ (X, ψ)} \circ κ_{F}^{- 1} = κ_{G} \circ \tilde{Γ (X, φ)} \circ κ_{F}^{- 1} = φ$ .

3.1F1step 2.1

Claim 2, bijectivity: given any $A$ -linear map $u : Γ (X, F) \to Γ (X, G)$ , the composite $κ_{G} \circ \tilde{u} \circ κ_{F}^{- 1}$ is a morphism $F \to G$ whose induced map on global sections is $u$ , because the global component of $κ_{F}$ is the canonical identification $Γ (X, F) \to Γ (X, F)$ of [F1]; by step 2.1 this construction is inverse to $ψ \mapsto ψ_{X}$ , so ${Hom}_{O_{X}} (F, G) ≅ {Hom}_{A} (Γ (X, F), Γ (X, G))$ .

4.1F1step 3.1∎

Choice accounting: no choice is made beyond the one inherited from the affine equivalence [F1], which is the Axiom of Choice recorded in the Statement; the morphisms $κ_{F}$ , their inverses and the induced maps are canonical.

Depends on

Used by

Global sections do not determine a sheaf on P1 Counterexample
Finite locally free sheaves and geometric vector bundles Theorem

Dependency tree · two levels

22 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

The Stacks Project, Schemes, §§26.5, 26.7, 26.24 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Chapters 6, 14, 17 (standard reference, not scraped)