The Stacks project

Proof. Pullback always preserves $m$-pseudo-coherence, see Cohomology on Sites, Lemma 21.45.3. Thus it suffices to assume $Lf_ i^*E$ is $m$-pseudo-coherent and to prove that $E$ is $m$-pseudo-coherent. Then first we may assume $X_ i$ is a scheme for all $i$, see Topologies on Spaces, Lemma 73.9.5. Next, choose a surjective étale morphism $U \to X$ where $U$ is a scheme. Then $U_ i = U \times _ X X_ i$ is a scheme and we obtain an fpqc covering $\{ U_ i \to U\} $ of schemes, see Topologies on Spaces, Lemma 73.9.4. We know the result is true for $\{ U_ i \to U\} _{i \in I}$ by the case for schemes, see Derived Categories of Schemes, Lemma 36.12.2. On the other hand, the restriction $E|_ U$ comes from an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$ (defined using the Zariski topology and the “usual” structure sheaf of $U$), see Derived Categories of Spaces, Lemma 75.4.2. The lemma follows as the two notions of pseudo-coherent (étale and Zariski) agree by Derived Categories of Spaces, Lemma 75.13.2. $\square$