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

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

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