Lemma 42.68.30. Let $A$ be a Noetherian local ring. Let $a, b, c \in A$. Let $M$ be a finite $A$-module with $\dim (\text{Supp}(M)) = 1$. Assume $a, b, c$ are nonzerodivisors on $M$. Then

and $d_ M(a, b)d_ M(b, a) = 1$.

Lemma 42.68.30. Let $A$ be a Noetherian local ring. Let $a, b, c \in A$. Let $M$ be a finite $A$-module with $\dim (\text{Supp}(M)) = 1$. Assume $a, b, c$ are nonzerodivisors on $M$. Then

\[ d_ M(a, bc) = d_ M(a, b) d_ M(a, c) \]

and $d_ M(a, b)d_ M(b, a) = 1$.

**Proof.**
The first statement follows from Lemma 42.68.24 applied to $M/abcM$ and endomorphisms $\alpha , \beta , \gamma $ given by multiplication by $a, b, c$. The second comes from Lemma 42.68.15.
$\square$

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