The Stacks project

Lemma 37.34.5. Let $f : X \to S$ be a morphism of affine schemes, which is smooth. Then there exists a diagram as in Lemma 37.34.1 such that in addition $f_0$ is smooth.

Proof. Write $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$, and as $f$ is smooth we see that $B$ is smooth as an $A$-algebra, see Morphisms, Lemma 29.34.2. Hence the lemma follows from Algebra, Lemma 10.138.14. $\square$

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