Lemma 42.2.5 (0EA9)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.2: Periodic complexes and Herbrand quotients
Lemma 42.2.5 (cite)

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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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