Lemma 86.2.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent
For every étale morphism $U \to X$ where $U$ is a scheme the restriction $K|_ U$ is a dualizing complex for $U$ (as discussed above).
There exists a surjective étale morphism $U \to X$ where $U$ is a scheme such that $K|_ U$ is a dualizing complex for $U$.
Proof. Assume $U \to X$ is surjective étale where $U$ is a scheme. Let $V \to X$ be an étale morphism where $V$ is a scheme. Then
are étale morphisms of schemes with the arrow to $V$ surjective. Hence we can use Duality for Schemes, Lemma 48.26.1 to see that if $K|_ U$ is a dualizing complex for $U$, then $K|_ V$ is a dualizing complex for $V$. $\square$
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