The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Smooth schemes are étale-locally like affine spaces.

Lemma 28.34.20. 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 28.32.11 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 on

Suggested slogan: Smooth schemes are étale-locally like affine spaces.

This is problematic, as étale-locally could be understood to mean an étale cover by affine spaces. It would be more precise to say "smooth schemes have étale coordinates" but the first suggestion is already colloquially used, so it may be useful to record it.

Comment #1383 by on

OK, I agree with what you say and added your slogan. Thanks.


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