The Stacks project

Lemma 15.37.8. Let $R$, $S$ be rings. Let $\mathfrak n \subset S$ be an ideal. Let $R \to S$ be formally smooth for the $\mathfrak n$-adic topology. Let $R \to R'$ be any ring map. Then $R' \to S' = S \otimes _ R R'$ is formally smooth in the $\mathfrak n' = \mathfrak nS'$-adic topology.

Proof. Let a solid diagram

\[ \xymatrix{ S \ar[r] \ar@{-->}[rrd] & S' \ar[r] \ar@{-->}[rd] & A/J \\ R \ar[u] \ar[r] & R' \ar[r] \ar[u] & A \ar[u] } \]

as in Definition 15.37.1 be given. Then the composition $S \to S' \to A/J$ is continuous. By assumption the longer dotted arrow exists. By the universal property of tensor product we obtain the shorter dotted arrow. $\square$

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