The Stacks project

Proof. Let $\mathcal{F}$ be an $\mathcal{A}$-module. If $\mathcal{F}$ is quasi-coherent as an $\mathcal{A}$-module, then for every $s \in S$ there exists an open neighbourhood $U$ of $s$ and an exact sequence

of $\mathcal{A}|_ U$-modules. Then this is also an exact sequence of $\mathcal{O}_ U$-modules. Hence $\mathcal{F}|_ U$ is quasi-coherent as the cokernel of a morphism of quasi-coherent $\mathcal{O}_ U$-modules on a scheme. It follows that $\mathcal{F}$ is quasi-coherent as an $\mathcal{O}_ X$-module.

Conversely, assume $\mathcal{F}$ is quasi-coherent as an $\mathcal{O}_ X$-module. Pick an open affine $\mathop{\mathrm{Spec}}(R) = U \subset S$. We have isomorphisms of $\mathcal{O}_ U$-modules $\mathcal{A}|_ U \cong \widetilde{A}$ and $\mathcal{F}|_ U \cong \widetilde{M}$, for some $R$-algebra $A$ and some $R$-module $M$. The $\mathcal{A}$-module structure on $\mathcal{F}$ translates into an $A$-module structure on $M$ compatible with the given $R$-module structure (details omitted). Choose an exact sequence

of $A$-modules. Since the functor $\widetilde{\ }$ is exact, this produces an exact sequence

of $\widetilde{A}$-modules. This means that $\mathcal{F}$ is quasi-coherent as an $\mathcal{A}$-module. $\square$