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 10.135.16. Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime lying over the prime $\mathfrak p$ of $R$. Assume

  1. there exists a $g \in S$, $g \not\in \mathfrak q$ such that $R \to S_ g$ is of finite presentation,

  2. the local ring homomorphism $R_{\mathfrak p} \to S_{\mathfrak q}$ is flat,

  3. the fibre $S \otimes _ R \kappa (\mathfrak p)$ is smooth over $\kappa (\mathfrak p)$ at the prime corresponding to $\mathfrak q$.

Then $R \to S$ is smooth at $\mathfrak q$.

Proof. By Lemmas 10.134.15 and 10.135.5 we see that there exists a $g \in S$ such that $S_ g$ is a relative global complete intersection. Replacing $S$ by $S_ g$ we may assume $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a relative global complete intersection. For any subset $I \subset \{ 1, \ldots , n\} $ of cardinality $c$ consider the polynomial $g_ I = \det (\partial f_ j/\partial x_ i)_{j = 1, \ldots , c, i \in I}$ of Lemma 10.135.15. Note that the image $\overline{g}_ I$ of $g_ I$ in the polynomial ring $\kappa (\mathfrak p)[x_1, \ldots , x_ n]$ is the determinant of the partial derivatives of the images $\overline{f}_ j$ of the $f_ j$ in the ring $\kappa (\mathfrak p)[x_1, \ldots , x_ n]$. Thus the lemma follows by applying Lemma 10.135.15 both to $R \to S$ and to $\kappa (\mathfrak p) \to S \otimes _ R \kappa (\mathfrak p)$. $\square$


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