Lemma 37.12.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Assume that $S$ is locally Noetherian and $f$ locally of finite type. The following are equivalent:

$f$ is smooth at $x$,

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 surjection of local rings with $\mathop{\mathrm{Ker}}(B' \to B)$ of square zero, and $\alpha $ mapping the closed point of $\mathop{\mathrm{Spec}}(B)$ to $x$ there exists a dotted arrow making the diagram commute,

same as in (2) but with $B' \to B$ ranging over small extensions (see Algebra, Definition 10.140.1), and

same as in (2) but with $B' \to B$ ranging over small extensions such that $\alpha $ induces an isomorphism $\kappa (x) \to \kappa (\mathfrak m)$ where $\mathfrak m \subset B$ is the maximal ideal.

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