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 10.138.5. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $\mathfrak q \subset S$ be a prime. Assume $\kappa (\mathfrak q)$ is separable over $k$. The following are equivalent:

  1. The algebra $S$ is smooth at $\mathfrak q$ over $k$.

  2. The ring $S_{\mathfrak q}$ is regular.

Proof. Denote $R = S_{\mathfrak q}$ and denote its maximal by $\mathfrak m$ and its residue field $\kappa $. By Lemma 10.138.4 and 10.130.9 we see that there is a short exact sequence

\[ 0 \to \mathfrak m/\mathfrak m^2 \to \Omega _{R/k} \otimes _ R \kappa \to \Omega _{\kappa /k} \to 0 \]

Note that $\Omega _{R/k} = \Omega _{S/k, \mathfrak q}$, see Lemma 10.130.8. Moreover, since $\kappa $ is separable over $k$ we have $\dim _{\kappa } \Omega _{\kappa /k} = \text{trdeg}_ k(\kappa )$. Hence we get

\[ \dim _{\kappa } \Omega _{R/k} \otimes _ R \kappa = \dim _\kappa \mathfrak m/\mathfrak m^2 + \text{trdeg}_ k (\kappa ) \geq \dim R + \text{trdeg}_ k (\kappa ) = \dim _{\mathfrak q} S \]

(see Lemma 10.115.3 for the last equality) with equality if and only if $R$ is regular. Thus we win by applying Lemma 10.138.3. $\square$


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