Lemma 66.20.10. Let $S$ be a scheme. Let $f : Y \to X$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{A} = f_*\mathcal{O}_ Y$. The functor $\mathcal{F} \mapsto f_*\mathcal{F}$ induces an equivalence of categories

$\left\{ \begin{matrix} \text{category of quasi-coherent} \\ \mathcal{O}_ Y\text{-modules} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of quasi-coherent} \\ \mathcal{A}\text{-modules} \end{matrix} \right\}$

Moreover, an $\mathcal{A}$-module is quasi-coherent as an $\mathcal{O}_ X$-module if and only if it is quasi-coherent as an $\mathcal{A}$-module.

Proof. Omitted. $\square$

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