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Let $f : C \to C'$ and $g : D \to D'$ be chain maps of non-negative chain complexes of $R$-modules, where $R$ is any commutative ring. Assume that $f$ and $g$ are homology equivalences. Is the same true for $f \otimes g : C \otimes D \to C' \otimes D'$?

The answer is yes if $C$ and $C'$ consist of flat modules, because then we may use the Künneth spectral sequence and its functoriality. I wonder what happens in the general case?

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Take $R=\mathbb{Z}$, $C=0, C'$ the standard extension of $\mathbb{Z}/2$ by $\mathbb{Z}$, and $g$ the identity of $\mathbb{Z}/2$ in grade 0. Then $C\otimes D=0$ but $C'\otimes D'$ has homology $\mathbb{Z}/2$ and $f\otimes g=0$. So $f\otimes g$ need not be surjective on homology, and running the same example in the other direction shows it need not be injective.

It would be interesting to say something more detailed, but I think the canonical response at this point would be "this is why we have derived functors."

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