Lemma 81.10.10. In Situation 81.10.6 the functor $Rf_*$ induces an equivalence between $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$ and $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$ with quasi-inverse given by $Lf^*$.
Proof. Since $f$ is quasi-compact and quasi-separated we see that $Rf_*$ defines a functor from $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$ to $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$, see Derived Categories of Spaces, Lemma 75.6.1. By Derived Categories of Spaces, Lemma 75.5.5 we see that $Lf^*$ maps $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$ into $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$. In Lemma 81.10.5 we have seen that $Lf^*Rf_*Q = Q$ for $Q$ in $D_{\mathit{QCoh}, |f^{-1}Z|}(\mathcal{O}_ Y)$. By the dual of Derived Categories, Lemma 13.7.2 to finish the proof it suffices to show that $Lf^*K = 0$ implies $K = 0$ for $K$ in $D_{\mathit{QCoh}, |Z|}(\mathcal{O}_ X)$. This follows from the fact that $f$ is flat at all points of $f^{-1}Z$ and the fact that $f^{-1}Z \to Z$ is surjective. $\square$
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