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

How statement and proof provenance work

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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

Used by

Twisting the exact sequence of an effective Cartier divisor Corollary
Global sections do not determine a sheaf on P1 Counterexample
Symmetric algebra of a quasi-coherent module Definition
Two-affine projective line and its twists Definition
An upper jump of h0 in a flat projective family Example
Twists on the two-affine projective line Example
Units satisfying the cocycle law glue local rank-one free modules into a line bundle Example
Pullback of a Cartier divisor computes the pullback of its line bundle Lemma
Symmetric algebras are quasi-coherent and commute with pullback Lemma
Compatible local sheaves glue uniquely up to unique isomorphism Theorem
Gluing affine schemes along compatible open isomorphisms Theorem

Dependency tree · two levels

2 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 26.14 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Section 2.5.D (standard reference, not scraped)