Lemma 56.9.4. Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times _ S Y})$. The corresponding Fourier-Mukai functor $\Phi _ K$ sends $D_{perf}(\mathcal{O}_ X)$ into $D_{perf}(\mathcal{O}_ Y)$ if at least one of the following conditions is satisfied:

$S$ is Noetherian, $X \to S$ and $Y \to S$ are of finite type, $K \in D^ b_{\textit{Coh}}(\mathcal{O}_{X \times _ S Y})$, the support of $H^ i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\text{pr}_2^{-1}\mathcal{O}_ Y)$,

$X \to S$ is of finite presentation and $K$ can be represented by a bounded complex $\mathcal{K}^\bullet $ of finitely presented $\mathcal{O}_{X \times _ S Y}$-modules, flat over $Y$, with support proper over $Y$,

$X \to S$ is a proper flat morphism of finite presentation and $K$ is perfect,

$S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is perfect

$X \to S$ is a proper flat morphism of finite presentation and $K$ is $Y$-perfect,

$S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is $Y$-perfect.

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