Lemma 42.2.3 (0EA7)—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 42.2.3 (cite)

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Lemma 42.2.3. Let $R$ be a ring. Suppose that we have a short exact sequence of $2$ -periodic complexes

$0 \to (M_1, N_1, \varphi _1, \psi _1) \to (M_2, N_2, \varphi _2, \psi _2) \to (M_3, N_3, \varphi _3, \psi _3) \to 0$

If two out of three have cohomology modules of finite length so does the third and we have

$e_ R(M_2, N_2, \varphi _2, \psi _2) = e_ R(M_1, N_1, \varphi _1, \psi _1) + e_ R(M_3, N_3, \varphi _3, \psi _3).$

Proof. We abbreviate $A = (M_1, N_1, \varphi _1, \psi _1)$ , $B = (M_2, N_2, \varphi _2, \psi _2)$ and $C = (M_3, N_3, \varphi _3, \psi _3)$ . We have a long exact cohomology sequence

$\ldots \to H^1(C) \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to H^1(C) \to \ldots$

This gives a finite exact sequence

$0 \to I \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to K \to 0$

with $0 \to K \to H^1(C) \to I \to 0$ a filtration. By additivity of the length function (Algebra, Lemma 10.52.3) we see the result. $\square$

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