The Stacks project

Proof. If $a \in A^*$, then the equality follows from the equality $\text{length}(M/bM) = \text{length}(M'/bM')$ and Lemma 42.68.33. Similarly if $b$ is a unit the lemma holds as well (by the symmetry of Lemma 42.68.30). Hence we may assume that $a, b \in \mathfrak m_ A$. This in particular implies that $\mathfrak m$ is not an associated prime of $M$, and hence $\alpha : M \to M'$ is injective. This permits us to think of $M$ as a submodule of $M'$. By assumption $M'/M$ is a finite $A$-module with support $\{ \mathfrak m_ A\}$ and hence has finite length. Note that for any third module $M''$ with $M \subset M'' \subset M'$ the maps $M \to M''$ and $M'' \to M'$ satisfy the assumptions of the lemma as well. This reduces us, by induction on the length of $M'/M$, to the case where $\text{length}_ A(M'/M) = 1$. Finally, in this case consider the map

By construction the cokernel $Q$ of $\overline{\alpha }$ has length $1$. Since $a, b \in \mathfrak m_ A$, they act trivially on $Q$. It also follows that the kernel $K$ of $\overline{\alpha }$ has length $1$ and hence also $a$, $b$ act trivially on $K$. Hence we may apply Lemma 42.68.25. Thus it suffices to see that the two maps $\alpha _ i : Q \to K$ are the same. In fact, both maps are equal to the map $q = x' \bmod \mathop{\mathrm{Im}}(\overline{\alpha }) \mapsto abx' \in K$. We omit the verification. $\square$