DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-27

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Gamma-acyclic abelian sheaf

Definition

Assume the Axiom of Choice, let $X$ be a topological space and let $H^{q} (X, -)$ be its sheaf cohomology (Sheaf cohomology as right derived global sections).

A sheaf of abelian groups $F$ on $X$ is $Γ$ -acyclic, or acyclic for global sections, when $H^{q} (X, F) = 0 for every q > 0.$ Equivalently, since $H^{q} (X, -) = R_{I}^{q} Γ (X, -)$ and $Γ (X, -)$ is additive and left exact (Global sections of an abelian sheaf, An acyclic object for a left exact functor), $F$ is acyclic for the left exact functor $Γ (X, -)$ in the sense of the general definition of $F$ -acyclicity.

For an open subspace $U \subseteq X$ (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) we say that $F$ is $Γ_{U}$ -acyclic when the restriction $F ∣_{U}$ (Restriction of a sheaf to an open subspace) is $Γ$ -acyclic on the space $U$ , that is, when $H^{q} (U, F ∣_{U}) = 0$ for every $q > 0$ .

Gamma-acyclic abelian sheaf

Definition

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