Lemma 37.11.6 (02H4)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.11: Formally smooth morphisms
Lemma 37.11.6 (cite)

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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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