Definition 57.15.1. Let $S$ be a scheme. Let $X \to S$ and $Y \to S$ be smooth proper morphisms. An object $K \in D_{perf}(\mathcal{O}_{X \times _ S Y})$ is said to be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$ if there exist an object $K' \in D_{perf}(\mathcal{O}_{X \times _ S Y})$ such that
\[ \Delta _{X/S, *}\mathcal{O}_ X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K \otimes _{\mathcal{O}_{X \times _ S Y \times _ S X}}^\mathbf {L} L\text{pr}_{23}^*K') \]
in $D(\mathcal{O}_{X \times _ S X})$ and
\[ \Delta _{Y/S, *}\mathcal{O}_ Y \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K' \otimes _{\mathcal{O}_{Y \times _ S X \times _ S Y}}^\mathbf {L} L\text{pr}_{23}^*K) \]
in $D(\mathcal{O}_{Y \times _ S Y})$. In other words, the isomorphism class of $K$ defines an invertible arrow in the category defined in Section 57.14.
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