The Stacks project

Lemma 57.8.4. 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:

  1. $S$ 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 $D(\text{pr}_2^{-1}\mathcal{O}_ Y)$,

  2. $X \to S$ 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 $Y$,

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

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

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

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

Proof. If $M$ is perfect on $X$, then $L\text{pr}_1^*M$ is perfect on $X \times _ S Y$, see Cohomology, Lemma 20.49.6. We will use this without further mention below. We will also use that if $X \to S$ is of finite type, or proper, or flat, or of finite presentation, then the same thing is true for the base change $\text{pr}_2 : X \times _ S Y \to Y$, see Morphisms, Lemmas 29.15.4, 29.41.5, 29.25.8, and 29.21.4.

Part (1) follows from Derived Categories of Schemes, Lemma 36.27.1 combined with Derived Categories of Schemes, Lemma 36.11.6.

Part (2) follows from Derived Categories of Schemes, Lemma 36.30.1.

Part (3) follows from Derived Categories of Schemes, Lemma 36.30.4.

Part (4) follows from part (3) and the fact that a finite type morphism of Noetherian schemes is of finite presentation by Morphisms, Lemma 29.21.9.

Part (5) follows from Derived Categories of Schemes, Lemma 36.35.10 combined with Derived Categories of Schemes, Lemma 36.35.5.

Part (6) follows from part (5) in the same way that part (4) follows from part (3). $\square$

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