The Stacks project

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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