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 10.124.3. 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. $R \to S$ is of finite type,

  2. $\dim _{\mathfrak q}(S/R) = n$, and

  3. $\text{trdeg}_{\kappa (\mathfrak p)}\kappa (\mathfrak q) = r$.

Then there exist $f \in R$, $f \not\in \mathfrak p$, $g \in S$, $g \not\in \mathfrak q$ and a quasi-finite ring map

\[ \varphi : R_ f[x_1, \ldots , x_ n] \longrightarrow S_ g \]

such that $\varphi ^{-1}(\mathfrak qS_ g) = (\mathfrak p, x_{r + 1}, \ldots , x_ n)R_ f[x_{r + 1}, \ldots , x_ n]$

Proof. After replacing $S$ by a principal localization we may assume there exists a quasi-finite ring map $\varphi : R[t_1, \ldots , t_ n] \to S$, see Lemma 10.124.2. Set $\mathfrak q' = \varphi ^{-1}(\mathfrak q)$. Let $\overline{\mathfrak q}' \subset \kappa (\mathfrak p)[t_1, \ldots , t_ n]$ be the prime corresponding to $\mathfrak q'$. By Lemma 10.114.6 there exists a finite ring map $\kappa (\mathfrak p)[x_1, \ldots , x_ n] \to \kappa (\mathfrak p)[t_1, \ldots , t_ n]$ such that the inverse image of $\overline{\mathfrak q}'$ is $(x_{r + 1}, \ldots , x_ n)$. Let $\overline{h}_ i \in \kappa (\mathfrak p)[t_1, \ldots , t_ n]$ be the image of $x_ i$. We can find an element $f \in R$, $f \not\in \mathfrak p$ and $h_ i \in R_ f[t_1, \ldots , t_ n]$ which map to $\overline{h}_ i$ in $\kappa (\mathfrak p)[t_1, \ldots , t_ n]$. Then the ring map

\[ R_ f[x_1, \ldots , x_ n] \longrightarrow R_ f[t_1, \ldots , t_ n] \]

becomes finite after tensoring with $\kappa (\mathfrak p)$. In particular, $R_ f[t_1, \ldots , t_ n]$ is quasi-finite over $R_ f[x_1, \ldots , x_ n]$ at the prime $\mathfrak q'R_ f[t_1, \ldots , t_ n]$. Hence, by Lemma 10.122.13 there exists a $g \in R_ f[t_1, \ldots , t_ n]$, $g \not\in \mathfrak q'R_ f[t_1, \ldots , t_ n]$ such that $R_ f[x_1, \ldots , x_ n] \to R_ f[t_1, \ldots , t_ n, 1/g]$ is quasi-finite. Thus we see that the composition

\[ R_ f[x_1, \ldots , x_ n] \longrightarrow R_ f[t_1, \ldots , t_ n, 1/g] \longrightarrow S_{\varphi (g)} \]

is quasi-finite and we win. $\square$


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