Lemma 57.15.3. With notation as in Definition 57.15.1 let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Let $S_1 \to S$ be a morphism of schemes. Let $X_1 = S_1 \times _ S X$ and $Y_1 = S_1 \times _ S Y$. Then the pullback $K_1 = L(X_1 \times _{S_1} Y_1 \to X \times _ S Y)^*K$ is the Fourier-Mukai kernel of a relative equivalence from $X_1$ to $Y_1$ over $S_1$.
Proof. Let $K' \in D_{perf}(\mathcal{O}_{Y \times _ S X})$ be the object assumed to exist in Definition 57.15.1. Denote $K'_1$ the pullback of $K'$ by $Y_1 \times _{S_1} X_1 \to Y \times _ S X$. Then it suffices to prove that we have
\[ \Delta _{X_1/S_1, *}\mathcal{O}_ X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K_1 \otimes _{\mathcal{O}_{X_1 \times _{S_1} Y_1 \times _{S_1} X_1}}^\mathbf {L} L\text{pr}_{23}^*K_1') \]
in $D(\mathcal{O}_{X_1 \times _{S_1} X_1})$ and similarly for the other condition. Since
\[ \xymatrix{ X_1 \times _{S_1} Y_1 \times _{S_1} X_1 \ar[r] \ar[d]_{\text{pr}_{13}} & X \times _ S Y \times _ S X \ar[d]^{\text{pr}_{13}} \\ X_1 \times _{S_1} X_1 \ar[r] & X \times _ S X } \]
is cartesian it suffices by Derived Categories of Schemes, Lemma 36.30.4 to prove that
\[ \Delta _{X_1/S_1, *}\mathcal{O}_{X_1} \cong L(X_1 \times _{S_1} X_1 \to X \times _ S X)^*\Delta _{X/S, *}\mathcal{O}_ X \]
This in turn will be true if $X$ and $X_1 \times _{S_1} X_1$ are tor independent over $X \times _ S X$, see Derived Categories of Schemes, Lemma 36.22.5. This tor independence can be seen directly but also follows from the more general More on Morphisms, Lemma 37.69.1 applied to the square with corners $X, X, X, S$ and its base change by $S_1 \to S$. $\square$
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