Lemma 57.8.9. Let $S$ be a Noetherian scheme.

For $X$, $Y$ proper and flat over $S$ and $K$ in $D_{perf}(\mathcal{O}_{X \times _ S Y})$ we obtain a Fourier-Mukai functor $\Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$.

For $X$, $Y$, $Z$ proper and flat over $S$, $K \in D_{perf}(\mathcal{O}_{X \times _ S Y})$, $K' \in D_{perf}(\mathcal{O}_{Y \times _ S Z})$ the composition $\Phi _{K'} \circ \Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Z)$ is equal to $\Phi _{K''}$ with $K'' \in D_{perf}(\mathcal{O}_{X \times _ S Z})$ computed as in Lemma 57.8.3,

For $X$, $Y$, $K$, $\Phi _ K$ as in (1) if $X \to S$ is Gorenstein, then $\Phi _{K'} : D_{perf}(\mathcal{O}_ Y) \to D_{perf}(\mathcal{O}_ X)$ is a right adjoint to $\Phi _ K$ where $K' \in D_{perf}(\mathcal{O}_{Y \times _ S X})$ is the pullback of $L\text{pr}_1^*\omega _{X/S}^\bullet \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K^\vee $ by $Y \times _ S X \to X \times _ S Y$.

For $X$, $Y$, $K$, $\Phi _ K$ as in (1) if $Y \to S$ is Gorenstein, then $\Phi _{K''} : D_{perf}(\mathcal{O}_ Y) \to D_{perf}(\mathcal{O}_ X)$ is a left adjoint to $\Phi _ K$ where $K'' \in D_{perf}(\mathcal{O}_{Y \times _ S X})$ is the pullback of $L\text{pr}_2^*\omega _{Y/S}^\bullet \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K^\vee $ by $Y \times _ S X \to X \times _ S Y$.

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