LemmaStatement: 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.

Sections of the associated sheaf on basic opens

Statement

Assume the Axiom of Choice, inherited from the existence theorem for the associated sheaf. Let $A$ be a commutative ring, $M$ an $A$ -module and $X = Spec A$ ; let $\tilde{M}$ be the $O_{X}$ -module associated to $M$ , so that $\tilde{M}$ is the sheaf whose distinguished-open sections are the localisations $M_{f}$ .

Then for every $f \in A$ there is a canonical identification $Γ (D (f), \tilde{M}) = \tilde{M} (D (f)) ≅ M_{f},$ natural in $f$ and in $M$ : for $D (g) \subseteq D (f)$ the restriction $Γ (D (f), \tilde{M}) \to Γ (D (g), \tilde{M})$ is the canonical localisation $M_{f} \to M_{g}$ , and an $A$ -linear map $u : M \to N$ induces the morphism of sheaves $\tilde{u} : \tilde{M} \to \tilde{N}$ whose component at $D (f)$ is the localisation $u_{f} : M_{f} \to N_{f}$ . In particular, for $f = 1$ , $Γ (X, \tilde{M}) ≅ M$ , and for $f = 0$ the open set is empty and the identification reads $Γ (\emptyset, \tilde{M}) = 0 = M_{0}$ .

Facts & Assumptions

Given: The Axiom of Choice; a commutative ring $A$ ; an $A$ -module $M$ ; the associated sheaf $\tilde{M}$ of $M$ .

[F1]

$\tilde{M}$ is a sheaf of $O_{X}$ -modules whose distinguished open sections are the data $M_{f}$ with restriction maps $ρ_{f g}$ , and it is unique up to unique isomorphism compatible with that data (The associated module sheaf exists).

[F2]

The maps $ρ_{f g}$ are functorial and are the canonical localisations compatible with the canonical maps $M \to M_{f}$ ; an $A$ -linear $u : M \to N$ induces $u_{f} : M_{f} \to N_{f}$ compatible with the $ρ$ 's (Module sheaf on an affine scheme, Localisation of a module at a multiplicative subset).

Proof

technique · direct, comparing the canonical identifications delivered by the existence theorem

1.1F1

By [F1] the extension $\tilde{M}$ is unique up to unique isomorphism over the distinguished-open data, and the identification $\tilde{M} (D (f)) ≅ M_{f}$ it provides is canonical: on the basis-family description of $\tilde{M}$ of [F1] the inverse is evaluation of a family at the member $D (f)$ , and the forward map sends $s \in M_{f}$ to the family $(ρ_{f g} (s))_{D (g) \subseteq D (f)}$ . Consequently a morphism of $O_{X}$ -modules out of $\tilde{M}$ is determined by its components on distinguished opens, and a family of compatible maps on distinguished opens extends uniquely to a morphism.

1.2F1F2given

The identifications are natural in $f$ : for $D (g) \subseteq D (f)$ the restriction $\tilde{M} (D (f)) = M_{f} \to M_{g} = \tilde{M} (D (g))$ is $ρ_{f g}$ , the canonical localisation, by [F1] and [F2].

2.1F2step 1.1

They are natural in $M$ : an $A$ -linear $u : M \to N$ gives localisations $u_{f} : M_{f} \to N_{f}$ compatible with the restriction maps $ρ$ by [F2], hence a compatible family of $O_{D (f)}$ -linear maps between the distinguished-open data, which extends uniquely to a morphism $\tilde{u} : \tilde{M} \to \tilde{N}$ with component $u_{f}$ on $D (f)$ ; this assignment preserves identities and compositions because each $u \mapsto u_{f}$ does.

3.1F1givenstep 1.2step 2.1∎

In particular $Γ (X, \tilde{M}) = \tilde{M} (D (1)) = M_{1} = M$ and more generally $Γ (D (f), \tilde{M}) = M_{f}$ for every $f$ , with the stated restriction maps; for $f = 0$ one has $D (0) = \emptyset$ , $M_{0} = 0$ and $Γ (\emptyset, \tilde{M}) = 0$ by the sheaf axiom for the empty cover, so the degenerate case is included; the Axiom of Choice is inherited from [F1] and no further choice is made.

Depends on

Used by

Dependency tree · two levels

14 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)