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

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.

Sections, restrictions, and global sections of a presheaf

Definition

Let $F$ be a presheaf on a topological space $X$ .

An element of $F (U)$ is called a section of $F$ over $U$ . We also write $Γ (U, F) : = F (U) .$

If $V \subseteq U$ and $s \in F (U)$ , its restriction to $V$ is $s ∣_{V} : = ρ_{V}^{U} (s) \in F (V) .$

A global section of $F$ is a section over the whole space: $Γ (X, F) = F (X) .$

The presheaf identities imply $s ∣_{U} = s, (s ∣_{V}) ∣_{W} = s ∣_{W} for W \subseteq V \subseteq U .$

Depends on

A presheaf on a topological space

Used by

Refinement choices differ on cochains but not on cohomology Counterexample
Affine-overlap separation condition Definition
c-soft sheaves Definition
Cech cohomology of holomorphic sections of a line bundle on finite good covers Definition
Compactly supported global sections and their right derived functors Definition
Flasque sheaf Definition
Germs of sections Definition
Global sections of an abelian sheaf Definition
Linear systems, base loci, and general members Definition
Ordered Čech cochain complex of a cover Definition
Refinement map of ordered open covers Definition
A skyscraper sheaf is flasque and acyclic Example
Three-open Čech sign cancellation Example
A section of an invertible sheaf has a canonical zero subscheme Lemma
Čech H0 equals global sections Lemma
Constant sheaves on irreducible spaces are flasque and acyclic Lemma
Extensions of line bundles on the projective line split after ordering Lemma
Global sections are left exact but need not preserve epimorphisms Lemma
Morphisms from the constant sheaf are global sections Lemma
Ordered and alternating Čech complexes agree Lemma
The Čech differential squares to zero Lemma
Refinement choices induce the same Čech map Theorem
The sheaf axiom is the equalizer condition on a cover Theorem

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, Section 3 (standard reference, not scraped)