TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

A sequence of abelian sheaves is exact exactly when it is exact on every stalk

Statement

Let

\dots ⟶ F_{i - 1} \overset{d_{i - 1}}{\to} F_{i} \overset{d_{i}}{\to} F_{i + 1} ⟶ \dots

be a sequence of sheaves of abelian groups on a topological space $X$ . Then the sequence is exact if and only if, for every point $x \in X$ , the stalk sequence

\dots ⟶ (F_{i - 1})_{x} \overset{(d_{i - 1})_{x}}{\to} (F_{i})_{x} \overset{(d_{i})_{x}}{\to} (F_{i + 1})_{x} ⟶ \dots

is exact.

Facts & Assumptions

Given: A sequence of sheaves of abelian groups on $X$ .

[F1]

Exactness of a sequence of sheaves means exactness in the abelian category of sheaves, so at each middle term the image sheaf equals the kernel sheaf (Exact sequences of sheaves).

[F2]

Stalks are germs of neighbourhood sections (The stalk of a presheaf at a point).

[L1]

Kernels are objectwise, while images and cokernels are obtained by sheafifying the corresponding presheaves (Kernel sheaves are objectwise, while cokernels and images are sheafified).

[L2]

Sheafification preserves stalks (Sheafification preserves stalks).

[L3]

A morphism of sheaves is an isomorphism exactly when it is so on every stalk (A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk).

Proof

technique · direct

1.1

Assume the sheaf sequence is exact. At each middle term $F_{i}$ , [F1] says that the canonical map $im (d_{i - 1}) \to \ker (d_{i})$ is an isomorphism. Using [F2] for the stalk construction and then [L1] and [L2], its stalk at $x$ is the canonical map $im ((d_{i - 1})_{x}) ⟶ \ker ((d_{i})_{x}),$ so the stalk sequence is exact at $x$ . Since $x$ was arbitrary, the whole stalk sequence is exact.

F1F2L1L2given

1.2

Conversely, assume every stalk sequence is exact. For each middle term, consider the canonical morphism $im (d_{i - 1}) \to \ker (d_{i})$ . Using [F2] for the stalk construction and then [L1] and [L2], its stalk at $x$ is $im ((d_{i - 1})_{x}) ⟶ \ker ((d_{i})_{x}),$ which is an isomorphism by the stalkwise exactness hypothesis. Therefore [L3] implies that $im (d_{i - 1}) \to \ker (d_{i})$ is an isomorphism of sheaves.

F2L1L2L3given

2.1

By [F1], step 1.2 is exactly the assertion that the original sequence is exact. Together with step 1.1, this proves the equivalence.

F1step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

21 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, Lemma 17.3.2 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Exercise 2.6.D (standard reference, not scraped)