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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 $q > 0$ .

Depends on

Used by

Acyclic open cover for a sheaf Definition
Flasque abelian sheaves are Γ-acyclic Theorem
Godement terms are flasque and compute cohomology Theorem

Dependency tree · two levels

23 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, Cohomology of Sheaves (standard reference, not scraped)
Jiahui Gao and Shuwu Zhang, Lectures on Algebraic Geometry (standard reference, not scraped)