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 10.124.8. Let $R \to S$ be a ring homomorphism of finite presentation. Let $n \geq 0$. The set

\[ V_ n = \{ \mathfrak q \in \mathop{\mathrm{Spec}}(S) \mid \dim _{\mathfrak q}(S/R) \leq n\} \]

is a quasi-compact open subset of $\mathop{\mathrm{Spec}}(S)$.

Proof. It is open by Lemma 10.124.6. Let $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ be a presentation of $S$. Let $R_0$ be the $\mathbf{Z}$-subalgebra of $R$ generated by the coefficients of the polynomials $f_ i$. Let $S_0 = R_0[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. Then $S = R \otimes _{R_0} S_0$. By Lemma 10.124.7 $V_ n$ is the inverse image of an open $V_{0, n}$ under the quasi-compact continuous map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(S_0)$. Since $S_0$ is Noetherian we see that $V_{0, n}$ is quasi-compact. $\square$


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