The Stacks project

Lemma 10.52.12. 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

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

by $B$-submodules as in Lemma 10.52.11. Then each quotient $M_ j/M_{j - 1}$ is isomorphic to $\kappa (\mathfrak m_{i(j)})$ for some $i(j) \in \{ 1, \ldots , n\} $. Moreover $\text{length}_ A(\kappa (\mathfrak m_ i)) = [\kappa (\mathfrak m_ i) : \kappa (\mathfrak m)]$ by Lemma 10.52.6. The lemma follows by additivity of lengths (Lemma 10.52.3). $\square$

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