LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-05‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Global sections are left exact but need not preserve epimorphisms

Statement

Let $X$ be a topological space. If

0 \to F^{'} \to F \to F^{''}

is exact in sheaves of abelian groups on $X$ , then

0 \to Γ (X, F^{'}) \to Γ (X, F) \to Γ (X, F^{''})

is exact. However, there exists an epimorphism of sheaves of abelian groups on some space $X$ whose induced map on global sections is not surjective.

Facts & Assumptions

Given: A topological space $X$ .

[F1]

Global sections are sections over the whole space: $Γ (X, F) = F (X)$ (Sections, restrictions, and global sections of a presheaf).

[L1]

Kernel sheaves are computed objectwise (Kernel sheaves are objectwise, while cokernels and images are sheafified).

[L2]

Exactness of sheaves means image equals kernel at each interior term (Exact sequences of sheaves).

Proof

technique · direct

1.1

Suppose $0 \to F^{'} \to F \to F^{''}$ is exact. By [L1], the kernel of the map $Γ (X, F) \to Γ (X, F^{''})$ is the section group of the kernel sheaf at $X$ . By [L2], that kernel sheaf is the image of $F^{'} \to F$ , so its sections over $X$ are exactly the image of $Γ (X, F^{'}) \to Γ (X, F)$ . Together with [F1], this is the displayed left exactness.

F1L1L2given

1.2

For the failure of surjectivity, let $X = S^{1}$ , let $C$ be the sheaf of continuous real-valued functions on $S^{1}$ , and let $\underline{Z}$ be the subsheaf of locally constant integer-valued functions. Let $Q$ be the quotient sheaf obtained by sheafifying the presheaf $V \mapsto C (V) / \underline{Z} (V)$ . Every germ of a section of $Q$ is represented locally by a continuous real-valued function, so the canonical map $C \to Q$ is locally surjective and hence an epimorphism of sheaves.

F1construct

1.3

Cover the circle by the arcs $U_{0} = S^{1} ∖ {(- 1, 0)}$ and $U_{1} = S^{1} ∖ {(1, 0)}$ . Choose continuous angle functions $θ_{0} : U_{0} \to (- 1 / 2, 1 / 2)$ and $θ_{1} : U_{1} \to (0, 1)$ with $e^{2 π i θ_{i} (z)} = z$ . On one connected component of $U_{0} \cap U_{1}$ the difference $θ_{1} - θ_{0}$ is $0$ , and on the other it is $1$ , so the classes of $θ_{0}$ and $θ_{1}$ agree in the quotient presheaf and glue to a global section $q \in Γ (X, Q)$ . If $q$ had a lift $g \in Γ (X, C)$ , then $g - θ_{i}$ would be an integer-valued continuous function on the connected arc $U_{i}$ , hence a constant $n_{i}$ . On the two components of $U_{0} \cap U_{1}$ this would force $n_{0} - n_{1}$ to be both $0$ and $1$ , a contradiction. Thus $Γ (X, C) \to Γ (X, Q)$ is not surjective.

constructcontradiction

2.1

Step 1.1 gives left exactness, and step 1.3 gives an epimorphism whose global sections map is not surjective.

step 1.1step 1.3∎

Depends on

Used by

Global sections need not preserve surjections Counterexample

Dependency tree · two levels

9 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 17.3 and Section 6.9 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Exercise 2.6.F (standard reference, not scraped)