Global section functor ($\Gamma (X,-)$)

The global section functor sends a sheaf $\mathcal{F}$ on $X$ to $\Gamma (X,\mathcal{F})$. It is left exact, and its right derived functors are sheaf cohomology.

For a closed subset $Z\subseteq X$, the section functor with support isIt is left exact.

Local cohomology consists of the right derived functors of the section functor with support. It measures sections and cohomology concentrated along a closed subset $Z$.

For $U=X\setminus Z$, local and ordinary sheaf cohomology fit into the natural sequence

For $X=\mathrm{Spec}k[x,y]$ and $Z=V(x,y)$, the only nonzero local cohomology group of the structure sheaf isIt has basis $x^{-a}{y}^{-b}$ for $a,b\ge 1$. The two-standard-open Čech cochain complex of the punctured affine plane proves the formula.