Lemma 57.8.3. Let $S$ be a scheme. Let $X, Y, Z$ be schemes over $S$. Assume $X \to S$, $Y \to S$, and $Z \to S$ are quasi-compact and quasi-separated. Let $K \in D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$. Let $K' \in D_\mathit{QCoh}(\mathcal{O}_{Y \times _ S Z})$. Consider the Fourier-Mukai functors $\Phi _ K : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ and $\Phi _{K'} : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ Z)$. If $X$ and $Z$ are tor independent over $S$ and $Y \to S$ is flat, then
\[ \Phi _{K'} \circ \Phi _ K = \Phi _{K''} : D_\mathit{QCoh}(\mathcal{O}_ X) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ Z) \]
where
\[ K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S Z}}^\mathbf {L} L\text{pr}_{23}^*K') \]
in $D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Z})$.
Proof.
The statement makes sense by Lemma 57.8.2. We are going to use Derived Categories of Schemes, Lemmas 36.3.8, 36.3.9, and 36.4.1 and Schemes, Lemmas 26.19.3 and 26.21.12 without further mention. By Derived Categories of Schemes, Lemma 36.22.4 we see that $X \times _ S Y$ and $Y \times _ S Z$ are tor independent over $Y$. This means that we have base change for the cartesian diagram
\[ \xymatrix{ X \times _ S Y \times _ S Z \ar[d] \ar[r] & Y \times _ S Z \ar[d]^{p^{YZ}_ Y} \\ X \times _ S Y \ar[r]^{p^{XY}_ Y} & Y } \]
for complexes with quasi-coherent cohomology sheaves, see Derived Categories of Schemes, Lemma 36.22.5. Abbreviating $p^* = Lp^*$, $p_* = Rp_*$ and $\otimes = \otimes ^\mathbf {L}$ we have for $M \in D_\mathit{QCoh}(\mathcal{O}_ X)$ the sequence of equalities
\begin{align*} \Phi _{K'}(\Phi _ K(M)) & = p^{YZ}_{Z, *}(p^{YZ, *}_ Y p^{XY}_{Y, *}(p^{XY, *}_ X M \otimes K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *} \text{pr}_{12}^*(p^{XY, *}_ X M \otimes K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K')) \\ & = \text{pr}_{3, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K') \\ & = p^{XZ}_{Z, *}\text{pr}_{13, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K') \\ & = p^{XZ}_{Z, *} (p^{XZ, *}_ X M \otimes \text{pr}_{13, *}(\text{pr}_{12}^*K \otimes \text{pr}_{23}^*K')) \end{align*}
as desired. Here we have used the remark on base change in the second equality and we have use Derived Categories of Schemes, Lemma 36.22.1 in the $4$th and last equality.
$\square$
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