 Solution

An $A$-module homomorphism $h:M\to N$ induces, by localization of a module, compatible mapson all principal open subsets $D(a)$. These glue to an ${\mathcal{O}}_X$-linear morphism of sheaves $\tilde{h}:\tilde{M}\to \tilde{N}$.

Conversely, taking global sections of an ${\mathcal{O}}_X$-linear morphism of sheaves gives an $A$-module homomorphism $h:M\to N$, since $\Gamma (X,\tilde{M})=M$ and $\Gamma (X,\tilde{N})=N$. On $D(a)$, compatibility with restriction determines the image of $m/1$, and ${\mathcal{O}}_X$-linearity determines the image of $a^{-r}m$. It must be $h(m)/a^r$. Thus the morphism of sheaves is uniquely recovered from $h$. Both constructions preserve addition and composition, proving the affine module-sheaf equivalence on morphisms:No finite-generation assumption is used.