The Stacks project

Lemma 65.6.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a quasi-separated scheme $U$ and a surjective ├ętale morphism $U \to X$ such that either of the projections $U \times _ X U \to U$ is quasi-compact, then $X$ is quasi-separated.

Proof. We may think of $X$ as an algebraic space over $\mathbf{Z}$. Consider the cartesian diagram

\[ \xymatrix{ U \times _ X U \ar[r] \ar[d]_ j & X \ar[d]^\Delta \\ U \times U \ar[r] & X \times X } \]

Since $U$ is quasi-separated the projection $U \times U \to U$ is quasi-separated (as a base change of a quasi-separated morphism of schemes, see Schemes, Lemma 26.21.12). Hence the assumption in the lemma implies $j$ is quasi-compact by Schemes, Lemma 26.21.14. By Spaces, Lemma 64.11.4 we see that $\Delta $ is quasi-compact as desired. $\square$

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