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More specifically, let $(K,\partial^K,\varepsilon^K)$ and $(L,\partial^L,\varepsilon^L)$ be augmented aclycic complexes of free abelian groups with augmentation module $\mathbb{Z}$; that is, $\varepsilon^K:K_0 \rightarrow \mathbb{Z}$. Does the tensor product of these complexes $(K \otimes L,\partial^{\otimes})$ have a acyclic augmentation?

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Let us be a bit more simple minded. Suppose $K,L$ are two free $R$ resolutions of $\mathbb{Z}$ where we can take $R$ to be polynomial over $\mathbb{Z}$ with a countable number of generators. Then $K \otimes L$ will not be acyclic, in fact its homology computes $Tor^R(\mathbb{Z}, \mathbb{Z})$ which in this situation is and exterior algebra over $\mathbb{Z}$ on countably many generators. It seems like your question, if we don't worry about the augmentation for a moment, is asking if $Tor$ is always zero, which it isn't. Granted you are asking about acyclic things, well free resolutions are certainly acyclic.

Does this help?

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