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

$X$ is geometrically reduced over $k$,

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

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

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

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

$X_{\bar k}$ is reduced.

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