Lemma 37.12.2 (02HY)—The Stacks project

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.

Proof. If $f$ is smooth, then the infinitesimal lifting criterion (Lemma 37.11.7) says $f$ is formally smooth and (2) holds.

Assume (2). The set of points $x \in X$ where $f$ is not smooth forms a closed subset $T$ of $X$. By the discussion in Morphisms, Section 29.16, if $T \not= \emptyset $ there exists a point $x \in T \subset X$ such that the morphism

\[ \mathop{\mathrm{Spec}}(\kappa (x)) \to X \to S \]

is of finite type (namely, pick any point $x$ of $T$ which is closed in an affine open of $X$). By Morphisms, Lemma 29.16.2 given any local Artinian ring $B'$ with residue field $\kappa (x)$ then any morphism $\beta : \mathop{\mathrm{Spec}}(B') \to S$ is of finite type. Thus we see that all the diagrams used in Lemma 37.12.1 (4) correspond to diagrams as in the current lemma (2). Whence $X \to S$ is smooth a $x$ a contradiction. $\square$

The Stacks project

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