Corollary

Tag 0DSW, permanent. Cite it as [Bourbaki, Tag 0DSW].

Algebra, chapter X, § 2. Complexes de A-modules

Assume $\mathscr{C}$ stable, and let $u : C' \to C$ be a morphism of complexes such that $H(C)$ and $H(C')$ are bounded of type $\mathscr{C}$ . Then $H(\operatorname{Con}(u))$ is bounded of type $\mathscr{C}$ , and one has

\chi(H(\operatorname{Con}(u))) = \chi(H(C)) - \chi(H(C')) .

This follows from Prop. 12 applied to the exact sequence of complexes (X, p. 37, Prop. 7)

0 \to C \to \operatorname{Con}(u) \to C'(-1) \to 0 .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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