# Tensor product of homology equivalences

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?

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.