Lemma 42.68.10. Let $R$ be a local ring with residue field $\kappa $. Let $u \in R^*$ be a unit. Let $M$ be a module of finite length over $R$. Denote $u_ M : M \to M$ the map multiplication by $u$. Then
is multiplication by $\overline{u}^ l$ where $l = \text{length}_ R(M)$ and $\overline{u} \in \kappa ^*$ is the image of $u$.
Proof. Denote $f_ M \in \kappa ^*$ the element such that $\det \nolimits _\kappa (u_ M) = f_ M \text{id}_{\det \nolimits _\kappa (M)}$. Suppose that $0 \to K \to L \to M \to 0$ is a short exact sequence of finite $R$-modules. Then we see that $u_ k$, $u_ L$, $u_ M$ give an isomorphism of short exact sequences. Hence by Lemma 42.68.6 (1) we conclude that $f_ K f_ M = f_ L$. This means that by induction on length it suffices to prove the lemma in the case of length $1$ where it is trivial. $\square$
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