The Stacks project

Lemma 72.10.1. Let $k$ be a field. Let $X$ be an algebraic space over $k$. If there exists a purely inseparable field extension $k'/k$ such that $X_{k'}$ is a scheme, then $X$ is a scheme.

Proof. The morphism $X_{k'} \to X$ is integral, surjective, and universally injective. Hence this lemma follows from Limits of Spaces, Lemma 70.15.4. $\square$

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