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.

A gluing datum for sheaves on an open cover

Definition

Let $X$ be a topological space with an open cover $X = ⋃_{i} U_{i}$ . A gluing datum for sheaves on this cover consists of

a sheaf $F_{i}$ on each open set $U_{i}$ , and
for each pair $(i, j)$ , an isomorphism of sheaves on $U_{i} \cap U_{j}$ , $φ_{i j} : F_{i} ∣_{U_{i} \cap U_{j}} \overset{≅}{\to} F_{j} ∣_{U_{i} \cap U_{j}},$

such that

$φ_{i i} = {id}_{F_{i}}$ , and
on every triple overlap $U_{i} \cap U_{j} \cap U_{k}$ one has $φ_{j k} \circ φ_{i j} = φ_{i k} .$

The same definition applies to sheaves of abelian groups, rings, or modules.

Depends on

Restriction of a sheaf to an open subspace

Dependency tree · 2 levels

Sources