Lemma 42.2.5. Let $R$ be a ring. Let $f : (M, \varphi , \psi ) \to (M', \varphi ', \psi ')$ be a map of $(2, 1)$-periodic complexes whose cohomology modules have finite length. If $\mathop{\mathrm{Ker}}(f)$ and $\mathop{\mathrm{Coker}}(f)$ have finite length, then $e_ R(M, \varphi , \psi ) = e_ R(M', \varphi ', \psi ')$.

**Proof.**
Apply the additivity of Lemma 42.2.3 and observe that $(\mathop{\mathrm{Ker}}(f), \varphi , \psi )$ and $(\mathop{\mathrm{Coker}}(f), \varphi ', \psi ')$ have vanishing multiplicity by Lemma 42.2.4.
$\square$

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