DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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A sheaf on a topological space

Definition

Let $F$ be a presheaf on a topological space $X$ . We say that $F$ is a sheaf if for every open set $U \subseteq X$ and every open cover $U = ⋃_{i \in I} U_{i}$ , including the empty cover of $\emptyset$ , the following two conditions hold.

Locality. If $s, t \in F (U)$ satisfy $s ∣_{U_{i}} = t ∣_{U_{i}} for all i \in I,$ then $s = t$ .

Gluing. If sections $s_{i} \in F (U_{i})$ satisfy $s_{i} ∣_{U_{i} \cap U_{j}} = s_{j} ∣_{U_{i} \cap U_{j}} for all i, j \in I,$ then there exists $s \in F (U)$ such that $s ∣_{U_{i}} = s_{i} for all i \in I .$

By locality, such a section $s$ is automatically unique. Thus a sheaf is exactly a presheaf whose compatible local sections glue uniquely.

Depends on

A presheaf on a topological space

Used by

Twisting the exact sequence of an effective Cartier divisor Corollary
A global section of the quotient that does not lift, and its nonzero connecting class Counterexample
A non-quasi-coherent module with H1 on an affine scheme Counterexample
Bounded continuous functions need not form a sheaf Counterexample
Global generation does not imply very ampleness Counterexample
Proper cohomology need not be finite for noncoherent sheaves Counterexample
The constant sheaf of integers on the line is not flasque Counterexample
The one-member cover of the circle has no Čech H1, but the sheaf H1 is nonzero Counterexample
Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
A ringed space Definition
A skyscraper sheaf of abelian groups at a point Definition
Algebraically constructible complexes of analytic field sheaves Definition
c-soft sheaves Definition
Canonical bundle and canonical divisors Definition
Cartier divisor Definition
Cech cohomology of holomorphic sections of a line bundle on finite good covers Definition
Cohomology and base-change map Definition
Compactly supported global sections and their right derived functors Definition
Derived operations on analytic field sheaves Definition
Equivariant constructible derived category of complex varieties Definition
Flasque sheaf Definition
Godement resolution of an abelian sheaf Definition
Invertible sheaf of cartier divisor Definition
Locally constant sheaves of finite rank Definition
Ordered Čech cochain complex of a cover Definition
Picard group of a scheme Definition
Principal parts of an invertible sheaf on a curve Definition
Refinement map of ordered open covers Definition
Sheaf of relative Kähler differentials Definition
Sheaf total quotient rings Definition
Subsheaves Definition
The etale space of a sheaf of sets Definition
A coherent closed-point skyscraper Example
A hyperplane in projective space is effective Cartier with O(H) = O(1) Example
A set-valued skyscraper sheaf and its stalks Example
A skyscraper sheaf is flasque and acyclic Example
Čech cohomology of the two-arc cover of the circle Example
Compact support and softness on a finite discrete space Example
Continuous real-valued functions form a sheaf Example
Dualizing and support shifts on the complex line Example

…and 99 more results.

Dependency tree · two levels

4 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, Sheaves on Spaces, Definition 7.1 (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, Class 3 (standard reference, not scraped)