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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