Lemma 76.52.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat, proper, and of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be $Y$-perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Derived Categories of Spaces, Lemma 75.21.4 (with $\mathcal{G}^\bullet$ equal to $\mathcal{O}_ X$ in degree $0$). Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce to the case where $Y$ is Noetherian affine by a limit argument.

The question is étale local on $Y$, hence we may assume $Y$ is affine. Say $Y = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits of Spaces, Lemma 70.7.1 there exists an $i$ and an algebraic space $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemmas 70.6.13 and 70.6.12 we may assume $X_ i$ is proper and flat over $R_ i$. By Lemma 76.52.9 we may assume there exists a $R_ i$-perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. Applying Derived Categories of Spaces, Lemma 75.22.1 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ and $E_ i$ and using the base change property already shown we obtain the result. $\square$

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