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 10.51.12. Let $A$ be a local ring with maximal ideal $\mathfrak m$. Let $B$ be a semi-local ring with maximal ideals $\mathfrak m_ i$, $i = 1, \ldots , n$. Suppose that $A \to B$ is a homomorphism such that each $\mathfrak m_ i$ lies over $\mathfrak m$ and such that

\[ [\kappa (\mathfrak m_ i) : \kappa (\mathfrak m)] < \infty . \]

Let $M$ be a $B$-module of finite length. Then

\[ \text{length}_ A(M) = \sum \nolimits _{i = 1, \ldots , n} [\kappa (\mathfrak m_ i) : \kappa (\mathfrak m)] \text{length}_{B_{\mathfrak m_ i}}(M_{\mathfrak m_ i}), \]

in particular $\text{length}_ A(M) < \infty $.

Proof. Choose a maximal chain

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

by $B$-submodules as in Lemma 10.51.11. Then each quotient $M_ i/M_{i - 1}$ is isomorphic to $\kappa (\mathfrak m_{j(i)})$ for some $j(i) \in \{ 1, \ldots , n\} $. Moreover $\text{length}_ A(\kappa (\mathfrak m_ i)) = [\kappa (\mathfrak m_ i) : \kappa (\mathfrak m)]$ by Lemma 10.51.6. The lemma follows by additivity of lengths (Lemma 10.51.3). $\square$


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