The Stacks project

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

  1. $X$ is geometrically reduced at $x$,

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

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

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

Proof. Note that if $k'/k$ is purely inseparable, then $X_{k'} \to X$ induces a homeomorphism on underlying topological spaces, see Algebra, Lemma 10.46.7. Whence the uniqueness of $x'$ lying over $x$ mentioned in the statement. Moreover, in this case $\mathcal{O}_{X_{k'}, x'} = \mathcal{O}_{X, x} \otimes _ k k'$. Hence the lemma follows from Lemma 33.6.2 above and Algebra, Lemma 10.44.4. $\square$

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