Let $C=\oplus_{p\geq0, q\geq0}C_{pq}$ be a double graded group with two differentials: $d^I_{pq}:C_{pq}\rightarrow C_{p-1,q}$ and $d^{II}_{pq}:C_{pq}\rightarrow C_{p,q-1}$, with the usual assumption on the anticommutator of $d^I, d^{II}$ being 0, which also gives a total complex. Define the filtration $F^I_pC=\oplus_{s\leq p}\left(\oplus_q C_{sq}\right)$, and $F^{II}$ symmetrically, so we get two spectral sequences $(^IE^r_{pq}, ^Id^r_{pq})$ and $(^{II}E^r_{pq},^{II}d^r_{pq})$.
Problem: Compute $^IE^r_{pq}$, $^{II}E^r_{pq}$ for $r=0,1,2,\infty$.
Attempt: $^IE^0_{pq}=\left(F^I_pC\cap\oplus_\ell C_{p+q,\ell}\right)\Big/ \left(F^I_{p-1}C\cap\oplus_\ell C_{p+q,\ell}\right)=\left(\oplus_{s\leq p}\oplus_q C_{sq}\cap\oplus_\ell C_{p+q,\ell}\right)\Big/\left(\oplus_{s\leq p-1}\oplus_q C_{sq}\cap \oplus_\ell C_{p+q,\ell}\right)=\oplus_q C_{pq}\cap\oplus_\ell C_{p+q,\ell}=\oplus_s\left(C_{ps}\cap C_{p+q,s}\right)$.
Except the answer ought to be $C_{pq}$. Now, if I assume that the answer is $C_{pq}$, the other three cases are easy, it's only the $r=0$ case that gives me trouble. As I see it, there are two places where I could be making a mistake:
The intersection is not with $\oplus_\ell C_{p+q,\ell}$ but with something else, for example, $C_{p+q,q}$, which still gives a wrong answer assuming everything else is right, or
The third equality, where everything but the $p^{th}$ term in the direct sum in the numerator gets quotiented out, is wrong, but it seems correct to me.
Thanks for any help.
