Lemma 72.11.3 (0DMS)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.11: Geometrically reduced algebraic spaces
Lemma 72.11.3 (cite)

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

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