The Stacks project

Lemma 75.5.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\mathcal{O}_ X)$. The following are equivalent

1. $E$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$,

2. for every étale morphism $\varphi : U \to X$ where $U$ is an affine scheme $\varphi ^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$,

3. for every étale morphism $\varphi : U \to X$ where $U$ is a scheme $\varphi ^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$,

4. there exists a surjective étale morphism $\varphi : U \to X$ where $U$ is a scheme such that $\varphi ^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$, and

5. there exists a surjective étale morphism of algebraic spaces $f : Y \to X$ such that $Lf^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ Y)$.