Lemma 10.138.4. A polynomial ring over $R$ is formally smooth over $R$.

**Proof.**
Suppose we have a diagram as in Definition 10.138.1 with $S = R[x_ j; j \in J]$. Then there exists a dotted arrow simply by choosing lifts $a_ j \in A$ of the elements in $A/I$ to which the elements $x_ j$ map to under the top horizontal arrow.
$\square$

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