I'm trying to show that the functor $\Gamma(X,-)$ from the category of quasi-coherent sheaves maps a quasi-coherent sheaf to an R-Module, and also that for coherent sheaves the same functor takes the sheaf to a finitely generated R-Module (where the sheaf is defined over a Noetherian affine scheme.)
However I'm struggling to see why this is true.
I have that for the structure sheaf we can set $X=Spec(R)=U_1=Spec(R)$ \ (V(1)). Then $O_X(X)=R_1=R.$ Where $R_1$ is the localisation at 1.
Hence $\Gamma(X, O_X)=R$ for $X=Spec(R)$
But am stuck as to where to go from here to do the case for quasi-coherent/coherent sheaves over a Noetherian affine scheme. Can anybody help?