# Is $H_i(Tot(C))\neq 0$($i\geq 0$) with $C$ double complex on upper half plane every arrows are $Z_4\to Z_4$ by 2 multiplication

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?

• 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. – Santana Afton Jun 14 at 1:16