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The Stacks project

Lemma 72.11.3. Let $k$ be a field. Let $X$ be an algebraic space over $k$. The following are equivalent

  1. $X$ is geometrically reduced,

  2. for some surjective étale morphism $U \to X$ where $U$ is a scheme, $U$ is geometrically reduced,

  3. for any étale morphism $U \to X$ where $U$ is a scheme, $U$ is geometrically reduced.

Proof. Immediate from the definitions and Lemma 72.11.2. $\square$

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