Remark 1

Tag 0D8W, permanent. Cite it as [Bourbaki, Tag 0D8W].

Commutative Algebra, chapter X, § 7. Algèbres lisses

Let $k$ be a ring, $A$ a $k$ -algebra and $J$ an ideal of $A$ . If the $k$ -algebra $A/J$ is formally smooth (for the discrete topology), the identity mapping of $A/J$ admits a lifting to $A/J^2$ ; consequently the sets described in Prop. 2 are non-empty. In particular, the sequence

0 \to J/J^2 \xrightarrow{\bar{d}} (A/J) \otimes_A \Omega_k(A) \longrightarrow \Omega_k(A/J) \to 0

is exact and split.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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