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.

Kernel sheaves are objectwise, while cokernels and images are sheafified

Let $φ : F \to G$ be a morphism of sheaves of abelian groups on a topological space $X$ .

The kernel sheaf is the subsheaf

\ker (φ) \subseteq F

defined objectwise by

\ker (φ) (U) : = \ker ⁣ (φ_{U} : F (U) \to G (U)) .

The cokernel presheaf is

U ⟼ coker ⁣ (φ_{U} : F (U) \to G (U)),

and the cokernel sheaf $coker (φ)$ is its sheafification.

The image presheaf is

U ⟼ im ⁣ (φ_{U} : F (U) \to G (U)),

and the image sheaf $im (φ)$ is its sheafification.

For sheaves of modules on a ringed space, the same formulas are taken in the corresponding module categories on each open set.

Depends on