Definition 56.9.1. Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times _ S Y})$. The exact functor

$\Phi _ K : D(\mathcal{O}_ X) \longrightarrow D(\mathcal{O}_ Y),\quad M \longmapsto R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K)$

of triangulated categories is called a Fourier-Mukai functor and $K$ is called a Fourier-Mukai kernel for this functor. Moreover,

1. if $\Phi _ K$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ then the resulting exact functor $\Phi _ K : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is called a Fourier-Mukai functor,

2. if $\Phi _ K$ sends $D_{perf}(\mathcal{O}_ X)$ into $D_{perf}(\mathcal{O}_ Y)$ then the resulting exact functor $\Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ is called a Fourier-Mukai functor, and

3. if $X$ and $Y$ are Noetherian and $\Phi _ K$ sends $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ then the resulting exact functor $\Phi _ K : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ is called a Fourier-Mukai functor. Similarly for $D_{\textit{Coh}}$, $D^+_{\textit{Coh}}$, $D^-_{\textit{Coh}}$.

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