Definition 37.11.1. Let $f : X \to S$ be a morphism of schemes. We say $f$ is formally smooth if given any solid commutative diagram

$\xymatrix{ X \ar[d]_ f & T \ar[d]^ i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] }$

where $T \subset T'$ is a first order thickening of affine schemes over $S$ there exists a dotted arrow making the diagram commute.

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