Lemma 32.8.9. Notation and assumptions as in Situation 32.8.1. If

$f$ is smooth,

$f_0$ is locally of finite presentation,

then $f_ i$ is smooth for some $i \geq 0$.

Lemma 32.8.9. Notation and assumptions as in Situation 32.8.1. If

$f$ is smooth,

$f_0$ is locally of finite presentation,

then $f_ i$ is smooth for some $i \geq 0$.

**Proof.**
Being smooth is local on the source and the target (Morphisms, Lemma 29.34.2) hence we may assume $S_0, X_0, Y_0$ affine (details omitted). The corresponding algebra fact is Algebra, Lemma 10.167.8.
$\square$

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