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 10.71.10. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $R \to S$ be a finite ring map. Let $\mathfrak m_1, \ldots , \mathfrak m_ n$ be the maximal ideals of $S$. Let $N$ be a finite $S$-module. Then

\[ \min \nolimits _{i = 1, \ldots , n} \text{depth}(N_{\mathfrak m_ i}) = \text{depth}(N) \]

Proof. By Lemmas 10.35.20, 10.35.22, and Lemma 10.35.21 the maximal ideals of $S$ are exactly the primes of $S$ lying over $\mathfrak m$ and there are finitely many of them. Hence the statement of the lemma makes sense. We will prove the lemma by induction on $k = \min \nolimits _{i = 1, \ldots , n} \text{depth}(N_{\mathfrak m_ i})$. If $k = 0$, then $\text{depth}(N_{\mathfrak m_ i}) = 0$ for some $i$. By Lemma 10.71.5 this means $\mathfrak m_ i S_{\mathfrak m_ i}$ is an associated prime of $N_{\mathfrak m_ i}$ and hence $\mathfrak m_ i$ is an associated prime of $N$ (Lemma 10.62.16). By Lemma 10.62.13 we see that $\mathfrak m$ is an associated prime of $N$ as an $R$-module. Whence $\text{depth}(N) = 0$. This proves the base case. If $k > 0$, then we see that $\mathfrak m_ i \not\in \text{Ass}_ S(N)$. Hence $\mathfrak m \not\in \text{Ass}_ R(N)$, again by Lemma 10.62.13. Thus we can find $f \in \mathfrak m$ which is not a zerodivisor on $N$, see Lemma 10.62.18. By Lemma 10.71.7 all the depths drop exactly by $1$ when passing from $N$ to $N/fN$ and the induction hypothesis does the rest. $\square$


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