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 10.122.13. Let $R \to S$ be a finite type ring map. The set of points $\mathfrak q$ of $\mathop{\mathrm{Spec}}(S)$ at which $S/R$ is quasi-finite is open in $\mathop{\mathrm{Spec}}(S)$.

Proof. Let $\mathfrak q \subset S$ be a point at which the ring map is quasi-finite. By Theorem 10.122.12 there exists an integral ring extension $R \to S'$, $S' \subset S$ and an element $g \in S'$, $g\not\in \mathfrak q$ such that $S'_ g \cong S_ g$. Since $S$ and hence $S_ g$ are of finite type over $R$ we may find finitely many elements $y_1, \ldots , y_ N$ of $S'$ such that $S''_ g \cong S_ g$ where $S'' \subset S'$ is the sub $R$-algebra generated by $g, y_1, \ldots , y_ N$. Since $S''$ is finite over $R$ (see Lemma 10.35.4) we see that $S''$ is quasi-finite over $R$ (see Lemma 10.121.4). It is easy to see that this implies that $S''_ g$ is quasi-finite over $R$, for example because the property of being quasi-finite at a prime depends only on the local ring at the prime. Thus we see that $S_ g$ is quasi-finite over $R$. By the same token this implies that $R \to S$ is quasi-finite at every prime of $S$ which lies in $D(g)$. $\square$


Comments (2)

Comment #3242 by Dario WeiƟmann on

Typo in the proof: should be

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  • 3 comment(s) on Section 10.122: Zariski's Main Theorem

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