I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman:
<https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf>


I am currently confused about the convergence of the zig-zag double complex $D$ mentioned here as one of the answers:
<https://math.stackexchange.com/questions/1368777/double-complex-with-exact-rows>


Firstly Let $C$ be the total complex of $D$. By Theorem 10.1 in Boardman's paper the spectral sequences $E^{\alpha}\_2=H(H(D,\delta),d)$ and $E^{\beta}\_2=H(H(D,d),\delta)$ converges conditionally to the cohomology $H(\widehat{C})$ where $\widehat{C}$ is the completion of $C$. Furthermore it is clear that $E^{\alpha}$ collapses at page 1 and $E^{\beta}$ collapses at page 2 - so by the remark after Theorem 7.1 we see that $RE\_{\infty}=0$ for both spectral sequences. The fact that both spectral sequences collapse imply that the condition for Theorem 8.1 is also satisfied - which means that both of them have $W=0$. Theorem 8.2 then implies that both spectral sequences converge strongly to $H(\widehat{C})$. 


Is this a contradiction? Because $E^{\alpha}$ implies that all the filter quotients of $H^n(\widehat{C})$ is zero for all $n$, whereas $E^{\beta}$ tells that that the filter quotients of $H^n(\widehat{C})$ are non-zero. 


Thanks!



Strong convergence of whole-plane spectral sequences
The spectral sequences $E^\alpha$ and $E^\beta$ arise from two different filtrations of $C$, which have different completions, say $\widehat C^\alpha$ and $\widehat C^\beta$. The first converges strongly to $H^\*(\widehat C^\alpha) = 0$, and the second converges strongly to $H^\*(\widehat C^\beta) \ne 0$. The apparent contradiction arises because you assume that the two completions are the same.


