Lemma 57.8.8. Let $S$ be a Noetherian scheme. Let $Y \to S$ be a flat proper Gorenstein morphism and let $X \to S$ be a finite type morphism. Denote $\omega ^\bullet _{Y/S}$ the relative dualizing complex of $Y$ over $S$. Let $\Phi : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ be a Fourier-Mukai functor with perfect kernel $K \in D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$. Denote

\[ K' = (Y \times _ S X \to X \times _ S Y)^*(K^\vee \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} L\text{pr}_2^*\omega ^\bullet _{Y/S}) \in D_\mathit{QCoh}(\mathcal{O}_{Y \times _ S X}) \]

and denote $\Phi ' : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ the corresponding Fourier-Mukai transform. There is a canonical isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _ Y(N, \Phi (M)) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ X(\Phi '(N), M) \]

functorial in $M$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $N$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

**Proof.**
By Lemma 57.8.2 we obtain a functor $\Phi $ as in the statement.

Observe that formation of the relative dualizing complex commutes with base change in our setting, see Duality for Schemes, Remark 48.12.5. Thus $L\text{pr}_2^*\omega ^\bullet _{Y/S} = \omega ^\bullet _{X \times _ S Y/X}$. Moreover, we observe that $\omega ^\bullet _{Y/S}$ is an invertible object of the derived category, see Duality for Schemes, Lemma 48.25.10, and a fortiori perfect.

To actually prove the lemma we're going to cheat. Namely, we will show that if we replace the roles of $X$ and $Y$ and $K$ and $K'$ then these are as in Lemma 57.8.7 and we get the result. It is clear that $K'$ is perfect as a tensor product of perfect objects so that the discussion in Lemma 57.8.7 applies to it. To show that the procedure of Lemma 57.8.7 applied to $K'$ on $Y \times _ S X$ produces a complex isomorphic to $K$ it suffices (details omitted) to show that

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, \omega ^\bullet _{X \times _ S Y/X}), \omega ^\bullet _{X \times _ S Y/X}) = K \]

This is clear because $K$ is perfect and $\omega ^\bullet _{X \times _ S Y/X}$ is invertible; details omitted. Thus Lemma 57.8.7 produces a map

\[ \mathop{\mathrm{Hom}}\nolimits _ Y(N, \Phi (M)) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ X(\Phi '(N), M) \]

functorial in $M$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $N$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ which is an isomorphism because $K'$ is perfect. This finishes the proof.
$\square$

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