The Stacks project

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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