The Stacks project

12.32 Exactness of products

Lemma 12.32.1. Let $I$ be a set. For $i \in I$ let $L_ i \to M_ i \to N_ i$ be a complex of abelian groups. Let $H_ i = \mathop{\mathrm{Ker}}(M_ i \to N_ i)/\mathop{\mathrm{Im}}(L_ i \to M_ i)$ be the cohomology. Then

\[ \prod L_ i \to \prod M_ i \to \prod N_ i \]

is a complex of abelian groups with homology $\prod H_ i$.

Proof. Omitted. $\square$

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