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The Stacks project

Lemma 29.11.6. Let f : X \to S be an affine morphism of schemes. Let \mathcal{A} = f_*\mathcal{O}_ X. The functor \mathcal{F} \mapsto f_*\mathcal{F} induces an equivalence of categories

\left\{ \begin{matrix} \text{category of quasi-coherent} \\ \mathcal{O}_ X\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}_ S-module if and only if it is quasi-coherent as an \mathcal{A}-module.

Proof. Omitted. \square


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