Lemma 37.12.2. Let $f : X \to S$ be a morphism of schemes. Assume that $S$ is locally Noetherian and $f$ locally of finite type. The following are equivalent:

$f$ is smooth,

for every solid commutative diagram

\[ \xymatrix{ X \ar[d]_ f & \mathop{\mathrm{Spec}}(B) \ar[d]^ i \ar[l]^-\alpha \\ S & \mathop{\mathrm{Spec}}(B') \ar[l]_-{\beta } \ar@{-->}[lu] } \]where $B' \to B$ is a small extension of Artinian local rings and $\beta $ of finite type (!) there exists a dotted arrow making the diagram commute.

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