DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Exact sequences of sheaves

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