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 10.128.2. Suppose that $R \to S$ is a ring map which is finite type, flat. Let $d$ be an integer such that all fibres $S \otimes _ R \kappa (\mathfrak p)$ are Cohen-Macaulay and equidimensional of dimension $d$. Let $f_1, \ldots , f_ i$ be elements of $S$. The set

\[ \{ \mathfrak q \in V(f_1, \ldots , f_ i) \mid f_1, \ldots , f_ i \text{ are a regular sequence in } S_{\mathfrak q}/\mathfrak p S_{\mathfrak q} \text{ where }\mathfrak p = R \cap \mathfrak q \} \]

is open in $V(f_1, \ldots , f_ i)$.

Proof. Write $\overline{S} = S/(f_1, \ldots , f_ i)$. Suppose $\mathfrak q$ is an element of the set defined in the lemma, and $\mathfrak p$ is the corresponding prime of $R$. We will use relative dimension as defined in Definition 10.124.1. First, note that $d = \dim _{\mathfrak q}(S/R) = \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) + \text{trdeg}_{\kappa (\mathfrak p)}\ \kappa (\mathfrak q)$ by Lemma 10.115.3. Since $f_1, \ldots , f_ i$ form a regular sequence in the Noetherian local ring $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}$ general dimension theory tells us that $\dim (\overline{S}_{\mathfrak q}/\mathfrak p\overline{S}_{\mathfrak q}) = \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) - i$. By the same Lemma 10.115.3 we then conclude that $\dim _{\mathfrak q}(\overline{S}/R) = \dim (\overline{S}_{\mathfrak q}/\mathfrak p\overline{S}_{\mathfrak q}) + \text{trdeg}_{\kappa (\mathfrak p)}\ \kappa (\mathfrak q) = d - i$. By Lemma 10.124.6 we have $\dim _{\mathfrak q'}(\overline{S}/R) \leq d - i$ for all $\mathfrak q' \in V(f_1, \ldots , f_ i) = \mathop{\mathrm{Spec}}(\overline{S})$ in a neighbourhood of $\mathfrak q$. Thus after replacing $S$ by $S_ g$ for some $g \in S$, $g \not\in \mathfrak q$ we may assume that the inequality holds for all $\mathfrak q'$. The result follows from Lemma 10.128.1. $\square$

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