Lemma 37.11.6. Let $f : X \to S$ be a morphism of schemes. Assume $X$ and $S$ are affine. Then $f$ is formally smooth if and only if $\mathcal{O}_ S(S) \to \mathcal{O}_ X(X)$ is a formally smooth ring map.

**Proof.**
This is immediate from the definitions (Definition 37.11.1 and Algebra, Definition 10.138.1) by the equivalence of categories of rings and affine schemes, see Schemes, Lemma 26.6.5.
$\square$

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