The Stacks project

Lemma 75.25.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Let $\mathcal{G}^\bullet $ be a bounded complex of finitely presented $\mathcal{O}_ X$-modules, flat over $Y$, with support proper over $Y$. Then

\[ K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet ) \]

is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma 75.21.5. Thus it suffices to show that $K$ is a perfect object. If $Y$ is Noetherian, then this follows from Lemma 75.22.3. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is 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, Lemma 70.7.2 we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_ i}$-module $\mathcal{G}_ i^\bullet $ whose pullback to $X$ is $\mathcal{G}$. After increasing $i$ we may assume $\mathcal{G}_ i^ n$ is flat over $R_ i$, see Limits of Spaces, Lemma 70.6.12. After increasing $i$ we may assume the support of $\mathcal{G}_ i^ n$ is proper over $R_ i$, see Limits of Spaces, Lemma 70.12.3. Finally, by Lemma 75.13.5 we may, after increasing $i$, assume there exists a perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. Applying Lemma 75.23.2 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$, $E_ i$, $\mathcal{G}_ i^\bullet $ and using the base change property already shown we obtain the result. $\square$

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