The Stacks project

Lemma 57.9.2. 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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