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This is related to Weibel Exercise 2.7.1 $C$ is periodic upper half plan complex $C_{pq}=Z_4$ for all $p\in Z,q\geq 0$ all differentials are multiplication by 2.

It is easy to apply acyclic assembly lemma to see $H_i(Tot^{\oplus}(C))=0$ for all $i$. It is not too hard to compute $H_0(Tot^{\prod}(C))=Z_2$.

$\textbf{Q:}$ Since vertical direction is not exact and I cannot cut off by truncation at $q=0$ to replace by kernel, I cannot use acyclic assembly lemma to conclude vanishing of $H_i(Tot^{\prod}(C))$ for $i\neq 0$. If I draw the line $p+q=n$, it will always hit the bottom $q=0$ part of the complex, so truncation does not make too much sense here. Is $H_i(Tot^{\prod}(C))=0$ for $i\neq 0$ or was there trick to compute them?

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    $\begingroup$ Usually it’s best to make your question as self-contained as possible, especially since this question has a number of (more specialized) moving parts. You may want to define some of the objects you’re asking about/give a direct statement of this lemma. $\endgroup$ – Santana Afton Jun 14 at 1:16

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