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If you take the two quotients of a chain complex by two quasi-isomorphic chain subcomplexes, are the results quasi-isomorphic as well?

I think it can be proved by making use of long exact sequences and the five-lemma, but I cannot find a proper chain map in the middle.

WJL
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1 Answers1

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No. This doesn’t work at the level of complexes concentrated in degree $0$ (ie modules): consider the two isomorphic submodules $2\mathbb{Z},3\mathbb{Z}$ of $\mathbb{Z}$…

Aphelli
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