Definition 10.137.1. Let $R \to S$ be a ring map. We say $S$ is *formally smooth over $R$* if for every commutative solid diagram

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

where $I \subset A$ is an ideal of square zero, a dotted arrow exists which makes the diagram commute.

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