Lemma 33.6.5. Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. Let $x \in X$. The following are equivalent

$X$ is geometrically reduced at $x$,

$\mathcal{O}_{X_{k'}, x'}$ is reduced for every finite purely inseparable field extension $k'$ of $k$ and $x' \in X_{k'}$ the unique point lying over $x$,

$\mathcal{O}_{X_{k^{1/p}}, x'}$ is reduced for $x' \in X_{k^{1/p}}$ the unique point lying over $x$, and

$\mathcal{O}_{X_{k^{perf}}, x'}$ is reduced for $x' \in X_{k^{perf}}$ the unique point lying over $x$.

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