37.12 Smoothness over a Noetherian base
It turns out that if the base is Noetherian then we can get away with less in the formulation of formal smoothness. In some sense the following lemmas are the beginning of deformation theory.
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.
$\square$
Sometimes it is useful to know that one only needs to check the lifting criterion for small extensions “centered” at points of finite type (see Morphisms, Section 29.16).
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.
$\square$
Here is a useful application.
Lemma 37.12.3. Let $f : X \to S$ be a finite type morphism of locally Noetherian schemes. Let $Z \subset S$ be a closed subscheme with $n$th infinitesimal neighbourhood $Z_ n \subset S$. Set $X_ n = Z_ n \times _ S X$.
If $X_ n \to Z_ n$ is smooth for all $n$, then $f$ is smooth at every point of $f^{-1}(Z)$.
If $X_ n \to Z_ n$ is étale for all $n$, then $f$ is étale at every point of $f^{-1}(Z)$.
Proof.
Assume $X_ n \to Z_ n$ is smooth for all $n$. Let $x \in X$ be a point lying over a point of $Z$. Given a small extension $B' \to B$ and morphisms $\alpha $, $\beta $ as in Lemma 37.12.1 part (3) the maximal ideal of $B'$ is nilpotent (as $B'$ is Artinian) and hence the morphism $\beta $ factors through $Z_ n$ and $\alpha $ factors through $X_ n$ for a suitable $n$. Thus the lifting property for $X_ n \to Z_ n$ kicks in to get the desired dotted arrow in the diagram. This proves (1). Part (2) follows from (1) and the fact that a morphism is étale if and only if it is smooth of relative dimension $0$.
$\square$
Lemma 37.12.4. Let $f : X \to S$ be a morphism of locally Noetherian schemes. Let $Z \subset S$ be a closed subscheme with $n$th infinitesimal neighbourhood $Z_ n \subset S$. Set $X_ n = Z_ n \times _ S X$. If $X_ n \to Z_ n$ is flat for all $n$, then $f$ is flat at every point of $f^{-1}(Z)$.
Proof.
This is a translation of Algebra, Lemma 10.99.11 into the language of schemes.
$\square$
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