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 quasi-compact separated scheme and a global regular function $f$, global sections on $X_f$ are the localization of global sections by $f$. Choose a finite affine open cover. Its intersections are affine, so the sheaf gluing axiom describes global sections as the kernel of the difference map between two finite products of affine coordinate rings. Exactness of localization preserves this kernel, and localization commutes with those finite products. Localizing the same cover gives the claimed isomorphism. This applies before knowing whether $X$ itself is affine.

A global section of a sheaf of modules $\mathcal{F}$ is an element of $\Gamma (X,\mathcal{F})$, meaning a section defined over the whole space $X$.

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.