The Stacks project

Lemma 73.25.3. 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 pseudo-coherent object. Let $\mathcal{G}^\bullet $ be a bounded above complex of finitely presented $\mathcal{O}_ X$-modules, flat over $Y$, with support proper over $Y$. Then

\[ K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]

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

Proof. The statement on base change is Lemma 73.21.4. Thus it suffices to show that $K$ is a pseudo-coherent object. This will follow from Lemma 73.25.1 by approximation by perfect complexes. We encourage the reader to skip the rest of the proof.

The question is ├ętale local on $Y$, hence we may assume $Y$ is affine. Then $X$ is quasi-compact and quasi-separated. Moreover, there exists an integer $N$ such that total direct image $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ has cohomological dimension $N$ as explained in Lemma 73.6.1. Choose an integer $b$ such that $\mathcal{G}^ i = 0$ for $i > b$. It suffices to show that $K$ is $m$-pseudo-coherent for every $m$. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, m - N - 1 - b)$. This is possible by Theorem 73.14.7. Choose a distinguished triangle

\[ P \to E \to C \to P[1] \]

in $D_\mathit{QCoh}(\mathcal{O}_ X)$. The cohomology sheaves of $C$ are zero in degrees $\geq m - N - 1 - b$. Hence the cohomology sheaves of $C \otimes ^\mathbf {L} \mathcal{G}^\bullet $ are zero in degrees $\geq m - N - 1$. Thus the cohomology sheaves of $Rf_*(C \otimes ^\mathbf {L} \mathcal{G})$ are zero in degrees $\geq m - 1$. Hence

\[ Rf_*(P \otimes ^\mathbf {L} \mathcal{G}) \to Rf_*(E \otimes ^\mathbf {L} \mathcal{G}) \]

is an isomorphism on cohomology sheaves in degrees $\geq m$. Next, suppose that $H^ i(P) = 0$ for $i > a$. Then $ P \otimes ^\mathbf {L} \sigma _{\geq m - N - 1 - a}\mathcal{G}^\bullet \longrightarrow P \otimes ^\mathbf {L} \mathcal{G}^\bullet $ is an isomorphism on cohomology sheaves in degrees $\geq m - N - 1$. Thus again we find that

\[ Rf_*(P \otimes ^\mathbf {L} \sigma _{\geq m - N - 1 - a}\mathcal{G}^\bullet ) \to Rf_*(P \otimes ^\mathbf {L} \mathcal{G}^\bullet ) \]

is an isomorphism on cohomology sheaves in degrees $\geq m$. By Lemma 73.25.1 the source is a perfect complex. We conclude that $K$ is $m$-pseudo-coherent as desired. $\square$


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