\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*}

The Stacks project

Lemma 10.51.11. Let $R$ be a ring. Let $M$ be a finite length $R$-module. Let $\ell = \text{length}_ R(M)$. Choose any maximal chain of submodules

\[ 0 = M_0 \subset M_1 \subset M_2 \subset \ldots \subset M_ n = M \]

with $M_ i \not= M_{i-1}$, $i = 1, \ldots , n$. Then

  1. $n = \ell $,

  2. each $M_ i/M_{i-1}$ is simple,

  3. each $M_ i/M_{i-1}$ is of the form $R/\mathfrak m_ i$ for some maximal ideal $\mathfrak m_ i$,

  4. given a maximal ideal $\mathfrak m \subset R$ we have

    \[ \# \{ i \mid \mathfrak m_ i = \mathfrak m\} = \text{length}_{R_{\mathfrak m}} (M_{\mathfrak m}). \]

Proof. If $M_ i/M_{i-1}$ is not simple then we can refine the filtration and the filtration is not maximal. Thus we see that $M_ i/M_{i-1}$ is simple. By Lemma 10.51.10 the modules $M_ i/M_{i-1}$ have length $1$ and are of the form $R/\mathfrak m_ i$ for some maximal ideals $\mathfrak m_ i$. By additivity of length, Lemma 10.51.3, we see $n = \ell $. Since localization is exact, we see that

\[ 0 = (M_0)_{\mathfrak m} \subset (M_1)_{\mathfrak m} \subset (M_2)_{\mathfrak m} \subset \ldots \subset (M_ n)_{\mathfrak m} = M_{\mathfrak m} \]

is a filtration of $M_{\mathfrak m}$ with successive quotients $(M_ i/M_{i-1})_{\mathfrak m}$. Thus the last statement follows directly from the fact that given maximal ideals $\mathfrak m$, $\mathfrak m'$ of $R$ we have

\[ (R/\mathfrak m')_{\mathfrak m} \cong \left\{ \begin{matrix} 0 & \text{if } \mathfrak m \not= \mathfrak m', \\ R_{\mathfrak m}/\mathfrak m R_{\mathfrak m} & \text{if } \mathfrak m = \mathfrak m' \end{matrix} \right. \]

This we leave to the reader. $\square$


Comments (1)

Comment #3416 by Jonas Ehrhard on

I think it would improve readability, if we omit the variable and just use , because only appears twice.


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