LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

Vanishing compactly supported first cohomology on opens characterises c-softness

Statement

Assume the Axiom of Choice. Let $X$ be a locally compact Hausdorff space (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and let $E$ be a sheaf of abelian groups on $X$ (A sheaf on a topological space); every clause applies in particular to sheaves of $k$ -vector spaces. Suppose that $H_{c}^{1} (V, E ∣_{V}) = 0 for every open subspace V \subseteq X,$ where $E ∣_{V}$ is the restriction (Restriction of a sheaf to an open subspace) and $H_{c}^{1}$ is the first compactly supported cohomology (Compactly supported global sections and their right derived functors). Then $E$ is c-soft (c-soft sheaves); more precisely, every section of $E$ over a compact subset $K \subseteq X$ extends to a compactly supported global section of $E$ . Conversely, if $E$ is c-soft, then $H_{c}^{q} (V, E ∣_{V}) = 0$ for every $q > 0$ and every open subspace $V \subseteq X$ . Hence c-softness of $E$ is equivalent to $R Γ_{c} (V, E ∣_{V})$ being concentrated in degree $0$ for every open subspace $V \subseteq X$ .

Facts & Assumptions

Given: A locally compact Hausdorff space $X$ , a sheaf $E$ of abelian groups on $X$ with $H_{c}^{1} (V, E ∣_{V}) = 0$ for every open $V \subseteq X$ , and the Axiom of Choice.

[F1]

Short exact sequence of the closed complement: for an open subspace $j : U ↪ X$ with closed complement $Z = X ∖ U$ and closed inclusion $i : Z ↪ X$ , and for every sheaf $F$ of abelian groups on $X$ , there is a short exact sequence $0 \to j_{!} (F ∣_{U}) \to F \to i_{*} (F ∣_{Z}) \to 0$ , where $j_{!}$ is extension by zero and $i_{*}$ is the direct image (Extension by zero and the closed complement: a short exact sequence, Extension by zero for abelian sheaves on an open subspace, Direct image of a sheaf along a continuous map, Direct image preserves sheaves and objectwise algebraic structure).

[F2]

The compact-support calculus: for $j : U ↪ X$ open and $i : Z ↪ X$ closed, the canonical maps $Γ_{c} (X, j_{!} A) \to Γ_{c} (U, A)$ and $Γ_{c} (X, i_{*} B) \to Γ_{c} (Z, B)$ are natural isomorphisms, the functors $j_{!}$ and $i_{*}$ carry c-soft sheaves to c-soft sheaves, and the induced isomorphisms $R^{q} Γ_{c} (X, j_{!} A) ≅ R^{q} Γ_{c} (U, A)$ and $R^{q} Γ_{c} (X, i_{*} B) ≅ R^{q} Γ_{c} (Z, B)$ hold for all $q \geq 0$ ; for compact $Z$ one has $Γ_{c} (Z, B) = Γ (Z, B)$ (Compact-support calculus along open and closed embeddings, Compactly supported global sections and their right derived functors).

[F3]

Long exact sequences: the functors $R^{q} Γ_{c} (X, -)$ form a cohomological delta functor with $R^{0} Γ_{c} (X, -) ≅ Γ_{c} (X, -)$ , and every short exact sequence of abelian sheaves on $X$ has a natural long exact compactly supported cohomology sequence, so a segment $H_{c}^{0} (X, A) \to H_{c}^{0} (X, B) \to H_{c}^{0} (X, C) \overset{\partial}{\to} H_{c}^{1} (X, A)$ is exact; here $H_{c}^{0} (X, F) = Γ_{c} (X, F)$ (Right derived functors form a cohomological delta functor, Compactly supported global sections and their right derived functors, Enough injective abelian sheaves).

[F4]

c-softness: a sheaf is c-soft when every section over a compact set extends to a global section; restriction of a c-soft sheaf to an open subspace is c-soft; and c-soft sheaves are acyclic, $H_{c}^{q} (X, F) = 0$ for $q > 0$ (c-soft sheaves, Compact-support acyclicity of c-soft sheaves).

[F5]

Topology: a compact subset $K$ of the Hausdorff space $X$ is closed, so its complement $X ∖ K$ is an open subspace and is itself locally compact Hausdorff; compactness of a subset may be read in the subspace or in $X$ ; and $i^{- 1} E$ over $K$ is the sheaf of sections of $E$ over $K$ (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, In a locally compact Hausdorff space every point has a neighbourhood base of compact sets, every open subspace and every closed subspace is locally compact, every open set around a point contains an open set with compact closure inside it, and every compact set sits inside an open set with compact closure, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Inverse image presheaf and inverse image sheaf, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[F6]

The Axiom of Choice, from which dependent choice follows; dependent choice is used by the delta-functor and derived-functor theorems invoked in fact F3 (The Axiom of Choice, AC implies DC implies countable choice).

Proof

technique · direct

1.1F1F5

Setup. Let $K \subseteq X$ be compact; by [F5] the set $K$ is closed, so $U : = X ∖ K$ is open and the pair $(U, K)$ is the open/closed decomposition of [F1] with closed inclusion $i : K ↪ X$ . The restriction $E ∣_{K} = i^{- 1} E$ is the sheaf of sections of $E$ over $K$ [F5], and [F1] applied to $F = E$ produces the short exact sequence of sheaves on $X$ $0 \to j_{!} (E ∣_{U}) \to E \to i_{*} (E ∣_{K}) \to 0$ with $j : U ↪ X$ the open inclusion.

1.2F4

The converse. Suppose $E$ is c-soft. For every open $V \subseteq X$ the restriction $E ∣_{V}$ is c-soft on $V$ by [F4], and c-soft sheaves are acyclic, so $H_{c}^{q} (V, E ∣_{V}) = 0$ for every $q > 0$ ; thus $R Γ_{c} (V, E ∣_{V})$ is concentrated in degree $0$ with value $Γ_{c} (V, E ∣_{V})$ . This proves the converse direction and, taking $q > 0$ arbitrary, shows that $R Γ_{c} (V, E ∣_{V})$ concentrated in degree $0$ for all opens $V$ implies in particular the hypothesis $H_{c}^{1} (V, E ∣_{V}) = 0$ used below.

2.1step 1.1F2F3

The long exact sequence of the pair. Apply the delta-functor machinery of [F3] to the short exact sequence of [step 1.1]; the calculus of [F2] identifies its compactly supported cohomology with that of the open and closed pieces: $H_{c}^{0} (X, j_{!} (E ∣_{U})) ≅ Γ_{c} (U, E)$ and $H_{c}^{1} (X, j_{!} (E ∣_{U})) ≅ H_{c}^{1} (U, E)$ , while $H_{c}^{0} (X, i_{*} (E ∣_{K})) ≅ Γ_{c} (K, E ∣_{K}) = Γ (K, E ∣_{K})$ because $K$ is compact. The exact sequence therefore reads $0 \to Γ_{c} (U, E) \to Γ_{c} (X, E) \to Γ (K, E ∣_{K}) \overset{\partial}{\to} H_{c}^{1} (U, E) \to \dots,$ and the hypothesis of the statement applies to the open subspace $U = X ∖ K$ , giving $H_{c}^{1} (U, E) = 0$ . Hence the kernel of $\partial$ is all of $Γ (K, E ∣_{K})$ , that is, the restriction map $Γ_{c} (X, E) \to Γ (K, E ∣_{K})$ is surjective.

3.1step 2.1F4

Conclusion of the direct direction. Since $Γ_{c} (X, E) \subseteq Γ (X, E)$ , the surjectivity of [step 2.1] shows that every section of $E$ over $K$ extends to a compactly supported global section, hence in particular to a global section; as $K$ was an arbitrary compact subset, $E$ is c-soft by [F4].

4.1step 1.2step 3.1F2F3F6∎

The equivalence. The direct direction is [step 3.1], and it produces the sharper statement that sections extend with compact support; the converse direction is [step 1.2]. Consequently c-softness of $E$ is equivalent to the vanishing of $H_{c}^{q} (V, E ∣_{V})$ for all $q > 0$ and all open $V \subseteq X$ , i.e. to $R Γ_{c} (V, E ∣_{V})$ being concentrated in degree $0$ for every open $V \subseteq X$ . The Axiom of Choice enters only through the injective resolution data and the delta-functor and calculus theorems of [F2] and [F3], whose comparison steps use dependent choice, a consequence of AC [F6].

Remarks

Why the hypothesis is stated for all opens. Only the complements $X ∖ K$ of compact sets are used in the proof, but an arbitrary open subspace of a locally compact Hausdorff space need not be a compact complement, and the criterion is stated in the form needed by its consumers, where the vanishing is available for every open subspace of a fibre. The proof shows that the weaker hypothesis " $H_{c}^{1} (X ∖ K, E) = 0$ for every compact $K$ " already implies c-softness.
Relation to the acyclicity lemma. The acyclicity lemma Compact-support acyclicity of c-soft sheaves proves that c-softness implies vanishing on the whole space and on open and closed restrictions; the present lemma is the converse, and the two together identify c-softness with a vanishing condition on all open subspaces, which is the form in which the finite-fibre-dimension theorem of this page verifies fibrewise c-softness.

Depends on

Used by

Bounded fibrewise-soft resolutions from a uniform all-sheaf dimension bound Lemma

Dependency tree · two levels

105 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

Pierre Schapira, An Introduction to Categories and Sheaves, lecture notes, 2023 (standard reference, not scraped)
Jean-Louis Verdier, Dualite dans la cohomologie des espaces localement compacts, Seminaire N. Bourbaki 1964-1966, expose 300, pp. 337-349 (Novembre 1965) (standard reference, not scraped)