Lemma 29.32.14. 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 $\mathfrak q$.

## Comments (2)

Comment #4594 by James Waldron on

Comment #4771 by Johan on

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