Lemma 29.11.7. 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.
The final statement is Lemma 29.11.6. By Lemma 29.11.2 and Schemes, Lemma 26.24.1 the pushforward $f_*\mathcal{F}$ of a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ S$-module. Hence a functor as in the statement of the lemma.
We will construct an quasi-inverse $h$ to this functor. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{A}$-module. Then we set
\[ h(\mathcal{G}) = f^*\mathcal{G} \otimes _{f^*\mathcal{A}} \mathcal{O}_ X = f^*\mathcal{G} \otimes _{f^*f_*\mathcal{O}_ X} \mathcal{O}_ X \]
Elucidation: the pullback $f^*\mathcal{A} = f^*f_*\mathcal{O}_ X$ is an $\mathcal{O}_ X$-algebra, the adjunction map $f^*f_*\mathcal{O}_ X \to \mathcal{O}_ X$ is an algebra homomorphism, and the pullback $f^*\mathcal{G}$ is an $f^*\mathcal{A}$-module. Observe that $h(\mathcal{G})$ is quasi-coherent as quasi-coherence is preserved by pullbacks and change of rings. Observe that there is a functorial map
\[ h(f_*\mathcal{F}) = f^*f_*\mathcal{F} \otimes _{f^*f_*\mathcal{O}_ X} \mathcal{O}_ X \to \mathcal{F} \]
coming from the adjunction map $f^*f_*\mathcal{F} \to \mathcal{F}$ and a functorial map
\[ \mathcal{G} \to f_*h(\mathcal{G}) \quad \text{adjoint to the map}\quad f^*\mathcal{G} \to f^*\mathcal{G} \otimes _{f^*f_*\mathcal{O}_ X} \mathcal{O}_ X \]
which sends a local section $s$ of $f^*\mathcal{F}$ to $s \otimes 1$. To finish the proof it suffices to show that these maps are isomorphisms for $\mathcal{F}$ and $\mathcal{G}$ as above. This may be checked on the members of an affine covering, i.e., when $X$ and $S$ are affine.
The key algebra observation which makes this work is the following: Let $R \to A$ be a ring map. Let $N$ be an $A$-module. Then
\[ (N \otimes _ R A) \otimes _{(A \otimes _ R A)} A = N \]
Namely, the left hand side of this equality is the effect of applying $h$ to the quasi-coherent $\widetilde{A}$-module $\widetilde{N}$ on $\mathop{\mathrm{Spec}}(R)$. We omit the details.
$\square$
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