The Stacks project

Lemma 36.27.3. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_ X)$ be perfect. Let $\mathcal{G}^\bullet $ be a bounded complex of coherent $\mathcal{O}_ X$-modules flat over $S$ with support proper over $S$. Then $K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet )$ is a perfect object of $D(\mathcal{O}_ S)$.

Proof. Since $E$ is a perfect complex there exists a dual perfect complex $E^\vee $, see Cohomology, Lemma 20.50.5. Observe that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet ) = E^\vee \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet $. Thus the perfectness of $K$ follows from Lemma 36.27.2. $\square$

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