Lemma 35.11.5. Let

be a commutative diagram of morphisms of schemes. Assume that

$f$ is surjective, flat, and locally of finite presentation,

$p$ is smooth (resp. étale).

Then $q$ is smooth (resp. étale).

Lemma 35.11.5. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

$f$ is surjective, flat, and locally of finite presentation,

$p$ is smooth (resp. étale).

Then $q$ is smooth (resp. étale).

**Proof.**
Assume (1) and that $p$ is smooth. By Lemma 35.11.3 we see that $q$ is locally of finite presentation. By Morphisms, Lemma 29.25.13 we see that $q$ is flat. Hence now it suffices to show that the fibres of $q$ are smooth, see Morphisms, Lemma 29.34.3. Apply Varieties, Lemma 33.25.9 to the flat surjective morphisms $X_ s \to Y_ s$ for $s \in S$ to conclude. We omit the proof of the étale case.
$\square$

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## Comments (2)

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