Lemma 10.137.19. Let $K/k$ 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 $K$.

**Proof.**
This is a special case of Lemma 10.137.18.
$\square$

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