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The Stacks project

Lemma 57.8.2. Let S be a scheme. Let X and Y be schemes over S. Let K \in D(\mathcal{O}_{X \times _ S Y}). The corresponding Fourier-Mukai functor \Phi _ K sends D_\mathit{QCoh}(\mathcal{O}_ X) into D_\mathit{QCoh}(\mathcal{O}_ Y) if K is in D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y}) and X \to S is quasi-compact and quasi-separated.

Proof. This follows from the fact that derived pullback preserves D_\mathit{QCoh} (Derived Categories of Schemes, Lemma 36.3.8), derived tensor products preserve D_\mathit{QCoh} (Derived Categories of Schemes, Lemma 36.3.9), the projection \text{pr}_2 : X \times _ S Y \to Y is quasi-compact and quasi-separated (Schemes, Lemmas 26.19.3 and 26.21.12), and total direct image along a quasi-separated and quasi-compact morphism preserves D_\mathit{QCoh} (Derived Categories of Schemes, Lemma 36.4.1). \square


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