Show that $\oplus\widetilde{M}_{\alpha}\cong \widetilde{\oplus M_{\alpha}}$ with $M_{\alpha}$ $\mathcal{O}_{X}(X)$-modules.

Question

Let $X=\operatorname{Spec}(A)$ be an affine scheme, and let $M_{\alpha}$-be $A$-modules. I want to show that $\oplus\widetilde{M}_{\alpha}\cong\widetilde{\oplus M_{\alpha}}$.

Let $D(f)$ be a distinguished open in $X$, then notice by construction we have $$\widetilde{\oplus M_{\alpha}}(D(f)) = (\oplus M_{\alpha})_{f} \cong \oplus M_{\alpha,f} = \oplus\widetilde{M_{\alpha}}(D(f)).$$

Are we then able to conclude that since the two sheaves on $X$ agree on the distinguished opens (which form a basis), that we actually have $\oplus\widetilde{M}_{\alpha}\cong\widetilde{\oplus M_{\alpha}}$?

Answer 1

$\widetilde{-}$ is left adjoint to the global sections functor. See Hartshorne exercise II.5.3. Then use that left adjoints preserve coproducts.

For the isomorphism on distinguished open sets, you need to find a morphism of sheaves that induces those isomorphisms on distinguished open sets.