Lemma 75.22.2. 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 $K = Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet )$ is a perfect object of $D(\mathcal{O}_ B)$.

Proof. The object $K$ is perfect by Lemma 75.22.1. We check the lemma applies: Locally $E$ is isomorphic to a finite complex of finite free $\mathcal{O}_ X$-modules. Hence locally $E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet$ is isomorphic to a finite complex whose terms are of the form

$\bigoplus \nolimits _{i = a, \ldots , b} (\mathcal{G}^ i)^{\oplus r_ i}$

for some integers $a, b, r_ a, \ldots , r_ b$. This immediately implies the cohomology sheaves $H^ i(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G})$ are coherent. The hypothesis on the tor dimension also follows as $\mathcal{G}^ i$ is flat over $f^{-1}\mathcal{O}_ Y$. $\square$

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