The Stacks project

Lemma 42.68.33. Let $A$ be a Noetherian local ring. Let $M$ be a finite $A$-module of dimension $1$. Let $b \in A$ be a nonzerodivisor on $M$, and let $u \in A^*$. Then

\[ d_ M(u, b) = u^{\text{length}_ A(M/bM)} \bmod \mathfrak m_ A. \]

In particular, if $M = A$, then $d_ A(u, b) = u^{\text{ord}_ A(b)} \bmod \mathfrak m_ A$.

Proof. Note that in this case $M/ubM = M/bM$ on which multiplication by $b$ is zero. Hence $d_ M(u, b) = \det _\kappa (u|_{M/bM})$ by Lemma 42.68.17. The lemma then follows from Lemma 42.68.10. $\square$

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