Lemma 10.135.18. Let $k \subset K$ be a field extension. Let $S$ be a finite type algebra over $k$. Let $\mathfrak q_ K$ be a prime of $S_ K = K \otimes _ k S$ and let $\mathfrak q$ be the corresponding prime of $S$. Then $S$ is smooth over $k$ at $\mathfrak q$ if and only if $S_ K$ is smooth at $\mathfrak q_ K$ over $K$.
This is a special case of Lemma 10.135.17.
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