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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