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 10.138.9. Let $R \to S$ be an injective finite type ring map with $R$ and $S$ domains. Then $R \to S$ is smooth at $\mathfrak q = (0)$ if and only if the induced extension $L/K$ of fraction fields is separable.

Proof. Assume $R \to S$ is smooth at $(0)$. We may replace $S$ by $S_ g$ for some nonzero $g \in S$ and assume that $R \to S$ is smooth. Then $K \to S \otimes _ R K$ is smooth (Lemma 10.135.4). Moreover, for any field extension $K \subset K'$ the ring map $K' \to S \otimes _ R K'$ is smooth as well. Hence $S \otimes _ R K'$ is a regular ring by Lemma 10.138.3, in particular reduced. It follows that $S \otimes _ R K$ is a geometrically reduced over $K$. Hence $L$ is geometrically reduced over $K$, see Lemma 10.42.3. Hence $L/K$ is separable by Lemma 10.43.1.

Conversely, assume that $L/K$ is separable. We may assume $R \to S$ is of finite presentation, see Lemma 10.29.1. It suffices to prove that $K \to S \otimes _ R K$ is smooth at $(0)$, see Lemma 10.135.17. This follows from Lemma 10.138.5, the fact that a field is a regular ring, and the assumption that $L/K$ is separable. $\square$


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