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 10.157.4. Let $\varphi : R \to S$ be a ring map. Assume

  1. $R$ is Noetherian,

  2. $S$ is Noetherian,

  3. $\varphi $ is flat,

  4. the fibre rings $S \otimes _ R \kappa (\mathfrak p)$ are $(S_ k)$, and

  5. $R$ has property $(S_ k)$.

Then $S$ has property $(S_ k)$.

Proof. Let $\mathfrak q$ be a prime of $S$ lying over a prime $\mathfrak p$ of $R$. By Lemma 10.157.2 we have

\[ \text{depth}(S_{\mathfrak q}) = \text{depth}(S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) + \text{depth}(R_{\mathfrak p}). \]

On the other hand, we have

\[ \dim (R_{\mathfrak p}) + \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) \geq \dim (S_{\mathfrak q}) \]

by Lemma 10.111.6. (Actually equality holds, by Lemma 10.111.7 but strictly speaking we do not need this.) Finally, as the fibre rings of the map are assumed $(S_ k)$ we see that $\text{depth}(S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) \geq \min (k, \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}))$. Thus the lemma follows by the following string of inequalities

\begin{eqnarray*} \text{depth}(S_{\mathfrak q}) & = & \text{depth}(S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) + \text{depth}(R_{\mathfrak p}) \\ & \geq & \min (k, \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q})) + \min (k, \dim (R_{\mathfrak p})) \\ & = & \min (2k, \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) + k, k + \dim (R_\mathfrak p), \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) + \dim (R_{\mathfrak p})) \\ & \geq & \min (k, \dim (S_{\mathfrak q})) \end{eqnarray*}

as desired. $\square$


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