DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

Exact sequences of sheaves

Definition

Let $X$ be a topological space. A sequence of sheaves of abelian groups on $X$ is exact when it is exact in the abelian category of Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories, equivalently in the sense of Exact sequence and short exact sequence in an abelian category.

Thus a sequence

\dots \to F_{i - 1} \to F_{i} \to F_{i + 1} \to \dots

is exact at $F_{i}$ when the image sheaf of the first morphism equals the kernel sheaf of the second.

The same language applies to $O_{X}$ -modules on a ringed space.

Depends on

Used by

Dependency tree · two levels

10 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, Section 17.3 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Section 2.6 (standard reference, not scraped)