Lemma 29.35.20. Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. Let $V \subset Y$ be an affine open neighbourhood of $f(x)$. If $\varphi $ is smooth at $x$, then there exists an integer $d \geq 0$ and an affine open $U \subset X$ with $x \in U$ and $\varphi (U) \subset V$ such that there exists a commutative diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^ d_ V \ar[ld] \\ Y & V \ar[l] } \]

where $\pi $ is étale.

**Proof.**
By Lemma 29.33.11 we can find an affine open $U$ as in the lemma such that $\varphi |_ U : U \to V$ is standard smooth. Write $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(R)$ so that we can write

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

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 invertible element of $A$. Then it is clear that $R[x_{c + 1}, \ldots , x_ n] \to A$ is standard smooth of relative dimension $0$. Hence it is smooth of relative dimension $0$. In other words the ring map $R[x_{c + 1}, \ldots , x_ n] \to A$ is étale. As $\mathbf{A}^{n - c}_ V = \mathop{\mathrm{Spec}}(R[x_{c + 1}, \ldots , x_ n])$ the lemma with $d = n - c$.
$\square$

## Comments (2)

Comment #1364 by Konrad Voelkel on

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