LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04‡ 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.

A section of a sheaf of groups is zero exactly when all of its germs are zero

Statement

Let $F$ be a sheaf of groups on a topological space $X$ , let $U \subseteq X$ be open, and let $s \in F (U)$ . Then $s = 0 in F (U) ⟺ s_{x} = 0 in F_{x} for every x \in U .$

Facts & Assumptions

Given: A sheaf of groups $F$ , an open set $U$ , and a section $s \in F (U)$ .

[F1]

A sheaf of groups has an identity section on every open set; when the group law is written additively, this section is denoted by $0$ , and all restrictions preserve it (Presheaves and sheaves of groups, rings, and modules).

[L1]

A sheaf is determined by local agreement of sections on an open cover (A sheaf on a topological space).

[F2]

The germ $s_{x}$ is the class of $s$ in the stalk at $x$ (Germs of sections).

Proof

technique · direct

1.1

If $s = 0$ in $F (U)$ , then for every $x \in U$ the induced germ is $s_{x} = 0_{x}$ because the stalk map respects restriction and the zero section by [F1] and [F2].

F1F2

1.2

Assume $s_{x} = 0$ for every $x \in U$ . Fix $x \in U$ . By [F2], equality of germs means that there exists an open neighbourhood $V_{x} \subseteq U$ of $x$ such that $s ∣_{V_{x}} = 0 ∣_{V_{x}}$ . The sets $V_{x}$ cover $U$ .

F2given

2.1

On the open cover ${V_{x}}_{x \in U}$ , the sections $s$ and $0$ have the same restriction by step 1.2. Locality in [L1] therefore gives $s = 0$ in $F (U)$ . Together with step 1.1, this proves both directions.

L1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Ravi Vakil, Foundations of Algebraic Geometry, Class 4 (standard reference, not scraped)
The Stacks Project, Sheaves on Spaces, Section 15 (standard reference, not scraped)