Proposition 35.9.4. Let $f : T \to S$ be a morphism of schemes.

The equivalences of categories of Proposition 35.8.9 are compatible with pullback. More precisely, we have $f^*(\mathcal{G}^ a) = (f^*\mathcal{G})^ a$ for any quasi-coherent sheaf $\mathcal{G}$ on $S$.

The equivalences of categories of Proposition 35.8.9 part (1) are

**not**compatible with pushforward in general.If $f$ is quasi-compact and quasi-separated, and $\tau \in \{ Zariski, {\acute{e}tale}\} $ then $f_*$ and $f_{small, *}$ preserve quasi-coherent sheaves and the diagram

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^ a} & & \mathit{QCoh}(\mathcal{O}_ S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^ a} \\ \mathit{QCoh}(T_\tau , \mathcal{O}) \ar[rr]^{f_{small, *}} & & \mathit{QCoh}(S_\tau , \mathcal{O}) } \]is commutative, i.e., $f_{small, *}(\mathcal{F}^ a) = (f_*\mathcal{F})^ a$.

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