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*}

Lemma 28.34.15. 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:

  1. The morphism $f$ is étale at $x$.

  2. 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 étale at $x$.

  3. 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 unramified at $x$.

  4. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\mathcal{O}_{X, x}$-module $\Omega _{X/S, x}$ is zero.

  5. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\kappa (x)$-vector space

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

    is zero.

  6. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat, we have $\mathfrak m_ s\mathcal{O}_{X, x} = \mathfrak m_ x$ and the field extension $\kappa (s) \subset \kappa (x)$ is finite separable.

  7. 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 of relative dimension $0$.

  8. 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

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

    with

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

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

  9. 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 étale.

  10. 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

    \[ A = R[x]_ Q/(P) = R[x, 1/Q]/(P) \]

    with $P, Q \in R[x]$, $P$ monic and $P' = \text{d}P/\text{d}x$ mapping to an element of $A$ not in $\mathfrak q$.

Proof. Use Lemma 28.34.14 and the definitions to see that (1) implies all of the other conditions. For each of the conditions (2) – (10) combine Lemmas 28.32.14 and 28.33.14 to see that (1) holds by showing $f$ is both smooth and unramified at $x$ and applying Lemma 28.34.5. Some details omitted. $\square$


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