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A set-valued sheaf has a unique section over the empty open set
Statement
Let be a sheaf of sets on a topological space . Then is a singleton.
Facts & Assumptions
Given: A sheaf on .
A sheaf satisfies locality and gluing for every open cover, including the empty cover of (A sheaf on a topological space).
Proof
Apply [L1] to the empty cover of . The empty family of local sections is vacuously compatible, so gluing produces at least one section .
Let . Then also restricts to the same empty family on the empty cover. By the uniqueness part of [L1], one has . Therefore .
Depends on
Used by
- Acyclic open cover for a sheaf Definition
- Cohomological dimension relative to a sheaf class Definition
- Flasque sheaf Definition
- Ordered Čech cochain complex of a cover Definition
- Cohomology of the empty space and the empty cover Example
- The empty space has a unique sheaf section over the empty open set Example
- Three-open Čech sign cancellation Example
- A polyhedral dimension bound for every sheaf Lemma
- A sheaf with no stalks off a closed subset is a pushforward Lemma
- Čech H0 equals global sections Lemma
- Compact-support calculus along open and closed embeddings Lemma
- Constant sheaves on irreducible spaces are flasque and acyclic Lemma
- Extension by zero and the closed complement: a short exact sequence Lemma
- Flasque kernel lifts quotient sections Lemma
- Pushforward along a closed immersion preserves sheaf cohomology Lemma
- The Čech differential squares to zero Lemma
- The eventual Hilbert function of a zero-dimensional projective quotient equals its total length Lemma
- The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
- A point has no higher sheaf cohomology Theorem
- Flasque abelian sheaves are Γ-acyclic Theorem
- Grothendieck vanishing on a Noetherian space Theorem
- Sheaves of smooth-function modules are cohomologically acyclic 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, Sheaves on Spaces, Remark 7.2 (standard reference, not scraped)