Lemma 57.8.9 (0FYY)—The Stacks project

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 $Y \times _ S X \to X \times _ S Y$.

Proof. Part (1) is immediate from Lemma 57.8.4 part (4).

Part (2) follows from Lemma 57.8.3 and the fact that $K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S Z}}^\mathbf {L} L\text{pr}_{23}^*K')$ is perfect for example by Derived Categories of Schemes, Lemma 36.27.4.

The adjointness in part (3) on all complexes with quasi-coherent cohomology sheaves follows from Lemma 57.8.7 with $K'$ equal to the pullback of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S Y}}(K, a(\mathcal{O}_ Y))$ by $Y \times _ S X \to X \times _ S Y$ where $a$ is the right adjoint to $R\text{pr}_{2, *} : D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y}) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. Denote $f : X \to S$ the structure morphism of $X$. Since $f$ is proper the functor $f^! : D_\mathit{QCoh}^+(\mathcal{O}_ S) \to D_\mathit{QCoh}^+(\mathcal{O}_ X)$ is the restriction to $D_\mathit{QCoh}^+(\mathcal{O}_ S)$ of the right adjoint to $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ S)$, see Duality for Schemes, Section 48.16. Hence the relative dualizing complex $\omega _{X/S}^\bullet $ as defined in Duality for Schemes, Remark 48.12.5 is equal to $\omega _{X/S}^\bullet = f^!\mathcal{O}_ S$. Since formation of the relative dualizing complex commutes with base change (see Duality for Schemes, Remark 48.12.5) we see that $a(\mathcal{O}_ Y) = L\text{pr}_1^*\omega _{X/S}^\bullet $. Thus

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S Y}}(K, a(\mathcal{O}_ Y)) \cong L\text{pr}_1^*\omega _{X/S}^\bullet \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K^\vee \]

by Cohomology, Lemma 20.50.5. Finally, since $X \to S$ is assumed Gorenstein the relative dualizing complex is invertible: this follows from Duality for Schemes, Lemma 48.25.10. We conclude that $\omega _{X/S}^\bullet $ is perfect (Cohomology, Lemma 20.52.2) and hence $K'$ is perfect. Therefore $\Phi _{K'}$ does indeed map $D_{perf}(\mathcal{O}_ Y)$ into $D_{perf}(\mathcal{O}_ X)$ which finishes the proof of (3).

The proof of (4) is the same as the proof of (3) except one uses Lemma 57.8.8 instead of Lemma 57.8.7. $\square$

The Stacks project

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