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The Stacks project

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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