Lemma 10.137.15. Let R be a ring. Let S = S' \times S'' be a product of R-algebras. Then S is smooth over R if and only if both S' and S'' are smooth over R.
Proof. Omitted. Hints: By Lemma 10.137.13 we can check smoothness one prime at a time. Since \mathop{\mathrm{Spec}}(S) is the disjoint union of \mathop{\mathrm{Spec}}(S') and \mathop{\mathrm{Spec}}(S'') by Lemma 10.21.2 we find that smoothness of R \to S at \mathfrak q corresponds to either smoothness of R \to S' at the corresponding prime or smoothness of R \to S'' at the corresponding prime. \square
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