Lemma 37.12.1 (02HX)—The Stacks project

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 10.141.1), 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.

Proof. Choose an affine neighbourhood $V \subset S$ of $f(x)$ and choose an affine neighbourhood $U \subset X$ of $x$ such that $f(U) \subset V$. For any “test” diagram as in (2) the morphism $\alpha $ will map $\mathop{\mathrm{Spec}}(B)$ into $U$ and the morphism $\beta $ will map $\mathop{\mathrm{Spec}}(B')$ into $V$ (see Schemes, Section 26.13). Hence the lemma reduces to the morphism $f|_ U : U \to V$ of affines. (Indeed, $V$ is Noetherian and $f|_ U$ is of finite type, see Properties, Lemma 28.5.2 and Morphisms, Lemma 29.15.2.) In this affine case the lemma is identical to Algebra, Lemma 10.141.2. $\square$

The Stacks project

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