Lemma 42.68.34. Let $A$ be a Noetherian local ring. Let $a, b \in A$. Let

be a short exact sequence of $A$-modules of dimension $1$ such that $a, b$ are nonzerodivisors on all three $A$-modules. Then

in $\kappa ^*$.

Lemma 42.68.34. Let $A$ be a Noetherian local ring. Let $a, b \in A$. Let

\[ 0 \to M \to M' \to M'' \to 0 \]

be a short exact sequence of $A$-modules of dimension $1$ such that $a, b$ are nonzerodivisors on all three $A$-modules. Then

\[ d_{M'}(a, b) = d_ M(a, b) d_{M''}(a, b) \]

in $\kappa ^*$.

**Proof.**
It is easy to see that this leads to a short exact sequence of exact $(2, 1)$-periodic complexes

\[ 0 \to (M/abM, a, b) \to (M'/abM', a, b) \to (M''/abM'', a, b) \to 0 \]

Hence the lemma follows from Lemma 42.68.18. $\square$

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