Lemma 29.34.14 (01V9)—The Stacks project

Lemma 29.34.14. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume $f$ is locally of finite presentation. The following are equivalent:

The morphism $f$ is smooth at $x$.
The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $X_ s \to \mathop{\mathrm{Spec}}(\kappa (s))$ is smooth at $x$.
The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\mathcal{O}_{X, x}$-module $\Omega _{X/S, x}$ can be generated by at most $\dim _ x(X_{f(x)})$ elements.
The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\kappa (x)$-vector space

\[ \Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) = \Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]

can be generated by at most $\dim _ x(X_{f(x)})$ elements.
There exist affine opens $U \subset X$, and $V \subset S$ such that $x \in U$, $f(U) \subset V$ and the induced morphism $f|_ U : U \to V$ is standard smooth.
There exist affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$ and $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $x \in U$ corresponding to $\mathfrak q \subset A$, and $f(U) \subset V$ such that there exists a presentation

\[ A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c) \]

with

\[ g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_ c/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_ c/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_ c & \partial f_2/\partial x_ c & \ldots & \partial f_ c/\partial x_ c \end{matrix} \right) \]

mapping to an element of $A$ not in $\mathfrak q$.

Proof. Note that if $f$ is smooth at $x$, then we see from Lemma 29.34.11 that (5) holds, and (6) is a slightly weakened version of (5). Moreover, $f$ smooth implies that the ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat (see Lemma 29.34.9) and that $\Omega _{X/S}$ is finite locally free of rank equal to $\dim _ x(X_ s)$ (see Lemma 29.34.12). Thus (1) implies (3) and (4). By Lemma 29.34.5 we also see that (1) implies (2).

By Lemma 29.32.10 the module of differentials $\Omega _{X_ s/s}$ of the fibre $X_ s$ over $\kappa (s)$ is the pullback of the module of differentials $\Omega _{X/S}$ of $X$ over $S$. Hence the displayed equality in part (4) of the lemma. By Lemma 29.32.12 these modules are of finite type. Hence the minimal number of generators of the modules $\Omega _{X/S, x}$ and $\Omega _{X_ s/s, x}$ is the same and equal to the dimension of this $\kappa (x)$-vector space by Nakayama's Lemma (Algebra, Lemma 10.20.1). This in particular shows that (3) and (4) are equivalent.

Algebra, Lemma 10.137.17 shows that (2) implies (1). Algebra, Lemma 10.140.3 shows that (3) and (4) imply (2). Finally, (6) implies (5) see for example Algebra, Example 10.137.8 and (5) implies (1) by Algebra, Lemma 10.137.7. $\square$

Comments (2)

Comment #4594 by James Waldron on October 13, 2019 at 09:26

Typo in statement (6): the denominators in the bottom row of the matrix should be $x_n$ instead of $x_c$ .

Comment #4771 by Johan on December 09, 2019 at 14:11

Hmmm... actually no that would be the etale case.

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