The Stacks project

Lemma 36.35.10. Let $f : X \to S$ be a morphism of schemes which is flat, proper, and of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be $S$-perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma 36.26.4 (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 $S$ is Noetherian affine by a limit argument.

The question is local on $S$, hence we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits, Lemma 32.10.1 there exists an $i$ and a scheme $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits, Lemmas 32.13.1 and 32.8.7 we may assume $X_ i$ is proper and flat over $R_ i$. By Lemma 36.35.9 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 Lemma 36.27.1 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ and $E_ i$ and using the base change property already shown we obtain the result. $\square$


Comments (1)

Comment #6023 by Noah Olander on

Since is flat it seems like "Tor-independent base change" is a better reference to get the base change than 0A1D.


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