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\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 10.134.11. Let $R$ be a ring. Let $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$. We will find $h \in R[x_1, \ldots , x_ n]$ which maps to $g \in S$ such that

\[ S_ g = R[x_1, \ldots , x_ n, x_{n + 1}]/(f_1, \ldots , f_ c, hx_{n + 1} - 1) \]

is a relative global complete intersection over $R$ in each of the following cases:

  1. Let $I \subset R$ be an ideal. If the fibres of $\mathop{\mathrm{Spec}}(S/IS) \to \mathop{\mathrm{Spec}}(R/I)$ have dimension $n - c$, then we can find $(h, g)$ as above such that $g$ maps to $1 \in S/IS$.

  2. Let $\mathfrak p \subset R$ be a prime. If $\dim (S \otimes _ R \kappa (\mathfrak p)) = n - c$, then we can find $(h, g)$ as above such that $g$ maps to a unit of $S \otimes _ R \kappa (\mathfrak p)$.

  3. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p \subset R$. If $\dim _{\mathfrak q}(S/R) = n - c$, then we can find $(h, g)$ as above such that $g \not\in \mathfrak q$.

Proof. Ad (1). By Lemma 10.124.6 there exists an open subset $W \subset \mathop{\mathrm{Spec}}(S)$ containing $V(IS)$ such that all fibres of $W \to \mathop{\mathrm{Spec}}(R)$ have dimension $\leq n - c$. Say $W = \mathop{\mathrm{Spec}}(S) \setminus V(J)$. Then $V(J) \cap V(IS) = \emptyset $ hence we can find a $g \in J$ which maps to $1 \in S/IS$. Let $h \in R[x_1, \ldots , x_ n]$ be any preimage of $g$.

Ad (2). By Lemma 10.124.6 there exists an open subset $W \subset \mathop{\mathrm{Spec}}(S)$ containing $\mathop{\mathrm{Spec}}(S \otimes _ R \kappa (\mathfrak p))$ such that all fibres of $W \to \mathop{\mathrm{Spec}}(R)$ have dimension $\leq n - c$. Say $W = \mathop{\mathrm{Spec}}(S) \setminus V(J)$. Then $V(J \cdot S \otimes _ R \kappa (\mathfrak p)) = \emptyset $. Hence we can find a $g \in J$ which maps to a unit in $S \otimes _ R \kappa (\mathfrak p)$ (details omitted). Let $h \in R[x_1, \ldots , x_ n]$ be any preimage of $g$.

Ad (3). By Lemma 10.124.6 there exists a $g \in S$, $g \not\in \mathfrak q$ such that all nonempty fibres of $R \to S_ g$ have dimension $\leq n - c$. Let $h \in R[x_1, \ldots , x_ n]$ be any element that maps to $g$. $\square$


Comments (1)

Comment #717 by Keenan Kidwell on

In (3), "then find" should be "then we can find."

There are also:

  • 2 comment(s) on Section 10.134: Syntomic morphisms

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