I am trying to understand the construction of a spectral sequence of a filtered complex. After reading through the entry in the nLab I came up with an example, that I don't understand: Consider the double complex $C_{p,q}$
$\require{AMScd}$ \begin{CD} \mathbb{Z} @<5<< \mathbb{Z} @<<<0@<<<0\\ @V V V @VV 3 V@V V V @VV V\\ 0 @<<< \mathbb{Z} @<3<<\mathbb{Z}@<<<0\\ @V V V @VV V@V 4 V V @VV V\\ 0 @<<< 0 @<<<\mathbb{Z}@<4<<\mathbb{Z}\\ \end{CD} (with $C_{0,0}$ being in the lower left corner). Then the total complex $Tot(C)$ is as usually given by summing the antidiagonals and it is filtered by the value of $p$. Following the construction from the nLab (using r-almost zycles and boundaries) I get, that the $E^1$ page has the form
$\require{AMScd}$ \begin{CD} \mathbb{Z} @<<< 0 @<<<0@<<<0\\ @. @. @. @. \\ 0 @<<< \mathbb{Z} @<<<0@<<<0\\ @. @. @. @. \\ 0 @<<< 0 @<<<\mathbb{Z}/4@<<<\mathbb{Z}\\ \end{CD}
And the $E^2$-page has the form
$\require{AMScd}$ \begin{CD} \mathbb{Z}/5 @. 0 @[email protected]\\ @. @.@. @.\\ 0 @. \mathbb{Z}/3 @[email protected]\\ @. @. @. @. \\ 0 @. 0 @. \mathbb{Z}/[email protected]\\ \end{CD}
And althought the $E^2$ page clearly is the homology of the double complex it is not the homology of the $E^1$ page.
What am I missing?