The Stacks project

Lemma 75.22.3. Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$. Let $f : X \to B$ be a morphism of algebraic spaces which is locally of finite type and quasi-separated. 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 $B$ with support proper over $B$. Then $K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G})$ is a perfect object of $D(\mathcal{O}_ B)$.

Proof. Since $E$ is a perfect complex there exists a dual perfect complex $E^\vee $, see Cohomology on Sites, Lemma 21.48.4. 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 75.22.2. $\square$

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