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 10.135.15. Let $R$ be a ring. Let $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ be a relative global complete intersection. Let $\mathfrak q \subset S$ be a prime. Then $R \to S$ is smooth at $\mathfrak q$ if and only if there exists a subset $I \subset \{ 1, \ldots , n\} $ of cardinality $c$ such that the polynomial

\[ g_ I = \det (\partial f_ j/\partial x_ i)_{j = 1, \ldots , c, \ i \in I}. \]

does not map to an element of $\mathfrak q$.

Proof. By Lemma 10.134.13 we see that the naive cotangent complex associated to the given presentation of $S$ is the complex

\[ \bigoplus \nolimits _{j = 1}^ c S \cdot f_ j \longrightarrow \bigoplus \nolimits _{i = 1}^ n S \cdot \text{d}x_ i, \quad f_ j \longmapsto \sum \frac{\partial f_ j}{\partial x_ i} \text{d}x_ i. \]

The maximal minors of the matrix giving the map are exactly the polynomials $g_ I$.

Assume $g_ I$ maps to $g \in S$, with $g \not\in \mathfrak q$. Then the algebra $S_ g$ is smooth over $R$. Namely, its naive cotangent complex is quasi-isomorphic to the complex above localized at $g$, see Lemma 10.132.13. And by construction it is quasi-isomorphic to a free rank $n - c$ module in degree $0$.

Conversely, suppose that all $g_ I$ end up in $\mathfrak q$. In this case the complex above tensored with $\kappa (\mathfrak q)$ does not have maximal rank, and hence there is no localization by an element $g \in S$, $g \not\in \mathfrak q$ where this map becomes a split injection. By Lemma 10.132.13 again there is no such localization which is smooth over $R$. $\square$


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