Definition 57.16.1. 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

in $D(\mathcal{O}_{X \times _ S X})$ and

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 57.15.

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