Lemma 33.6.8. Let $k$ be a field. Let $X$ be a scheme over $k$.

If $x' \leadsto x$ is a specialization and $X$ is geometrically reduced at $x$, then $X$ is geometrically reduced at $x'$.

If $x \in X$ such that (a) $\mathcal{O}_{X, x}$ is reduced, and (b) for each specialization $x' \leadsto x$ where $x'$ is a generic point of an irreducible component of $X$ the scheme $X$ is geometrically reduced at $x'$, then $X$ is geometrically reduced at $x$.

If $X$ is reduced and geometrically reduced at all generic points of irreducible components of $X$, then $X$ is geometrically reduced.

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