Definition:Sheaf Cohomology

Definition

Let $X$ be a topological space.

Let $\FF$ be an abelian sheaf on $X$.

Let $\map {\mathbf {Ab} } X$ be the category of abelian sheaves on $X$.

The sheaf cohomology (group) $\map {H^i} {X, \FF}$ for $i \in \Z$, $i \ge 0$ is defined as the $i$-th right derived functor $\mathrm R^i \map \Gamma {X, -}$ of the global sections functor

$\map \Gamma {X, -} : \map {\mathbf {Ab} } X \to \mathbf {Ab}$

applied to $\FF$.

Also see

• Definition:Čech Cohomology
• Definition:Singular Cohomology

• Results about sheaf cohomologies can be found here.

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