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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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