O-Modules

## $𝒪$-modules
Let $\left(X,𝒪\right)$ be a ringed space. An $𝒪$-module is a sheaf $ℱ$ of abelian groups on $X$ such that
1. if $U$ is open in $X$ then $ℱ\left(U\right)$ is an $𝒪\left(U\right)$-module
and the $𝒪\left(U\right)$-module structures are functional.
A sequence $ℱ\stackrel{\varphi }{\to }𝒢\stackrel{\psi }{\to }ℋ$ of $𝒪$-modules is exact if and only if ${ℱ}_{x}\stackrel{{\varphi }_{x}}{\to }{𝒢}_{x}\stackrel{{\psi }_{x}}{\to }{ℋ}_{x}$ is exact for all $x\in X$.
The global section function $\Gamma :\left\{𝒪\text{-modules}\right\}\to \left\{𝒪\left(X\right)\text{-modules}\right\}$ is defined by $Γ ℱ = ℱ X$ and the right derived functions of $\Gamma$ are ${H}^{p}\left(X,ℱ\right)={R}^{p}\Gamma \left(𝒪\right)\text{.}$