The Stacks project

Lemma 70.11.5. Let $k$ be a field of characteristic $p > 0$. Let $X$ be an algebraic space over $k$. The following are equivalent

  1. $X$ is geometrically reduced over $k$,

  2. $X_{k'}$ is reduced for every field extension $k'/k$,

  3. $X_{k'}$ is reduced for every finite purely inseparable field extension $k'/k$,

  4. $X_{k^{1/p}}$ is reduced,

  5. $X_{k^{perf}}$ is reduced, and

  6. $X_{\bar k}$ is reduced.

Proof. Choose a surjective ├ętale morphism $U \to X$ where $U$ is a scheme. Via Lemma 70.11.3 the lemma follows from the result for $U$ over $k$. See Varieties, Lemma 33.6.4. $\square$

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