Let $X$ be a Noetherian scheme and let $f:X\to Y=\operatorname{Spec} A $ a morphism onto an affine scheme.
Proposition III.8.5 in Hartshorne book says that for any quasi-coherent sheaf $\mathcal F$ on $X$ we have:
$$R^if_\ast(\mathcal F)=H^i(X,\mathcal F)^\sim$$
Here $R^if_\ast$ is the $i$-th right derived functor associated to $f_\ast$.
My question is probably very stupid but I'm not able to answer by myself:
$H^i(X,\mathcal F)$ is actually a $\mathcal O_X(X)$-module and not an $A$-module; so how can we construct the sheaf $H^i(X,\mathcal F)^\sim$ on $Y$? I mean in order to use the "$\sim$ construction" on affine schemes we need to start with an $A$-module.