The Stacks project

Proof. The statement on base change is Lemma 36.26.4. Thus it suffices to show that $K$ is a pseudo-coherent object. This will follow from Lemma 36.30.1 by approximation by perfect complexes. We encourage the reader to skip the rest of the proof.

The question is local on $S$, hence we may assume $S$ is affine. Then $X$ is quasi-compact and quasi-separated. Moreover, there exists an integer $N$ such that total direct image $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ S)$ has cohomological dimension $N$ as explained in Lemma 36.4.1. 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 36.14.6. Choose a distinguished triangle

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 $Rf_*(C \otimes ^\mathbf {L} \mathcal{G}^\bullet )$ are zero in degrees $\geq m - 1$. Hence

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

is an isomorphism on cohomology sheaves in degrees $\geq m$. By Lemma 36.30.1 the source is a perfect complex. We conclude that $K$ is $m$-pseudo-coherent as desired. $\square$