Lemma 76.47.5. Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$. Let $f : X \to Y$ be a perfect proper morphism of algebraic spaces. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ Y)$.
Proof. We claim that Derived Categories of Spaces, Lemma 75.22.1 applies. Conditions (1) and (2) are immediate. Condition (3) is local on $X$. Thus we may assume $X$ and $Y$ affine and $E$ represented by a strictly perfect complex of $\mathcal{O}_ X$-modules. Thus it suffices to show that $\mathcal{O}_ X$ has finite tor dimension as a sheaf of $f^{-1}\mathcal{O}_ Y$-modules on the étale site. By Derived Categories of Spaces, Lemma 75.13.4 it suffices to check this on the Zariski site. This is equivalent to being perfect for finite type morphisms of schemes by More on Morphisms, Lemma 37.61.11. $\square$
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