I am trying to understand the construction of the Kunneth spectral sequence from Weibel's "An Introduction to Homological Algebra" and one thing is driving me crazy. Throughout this post I will use Weibel's notation, these are sections 5.4 and 5.6 mostly. What I am mostly interested about is the explicit (as much as possible) description of $d_2$.
So the construction is going as follows. We have a bounded below complex of flat $R$-modules $P$ and a projective resolution of an $R$-module $M$, denoted by $Q$. We form a double complex $P\otimes Q$ and filter its total complex by row filtration, and its $E^2$ page will be identified with the $E^2$ page of the KSS. So we see that $E^2_{pq}=H^v_pH^h_q(P\otimes Q)\cong Tor^r_p(H_q(P),M)$.
And here my problem begins. According to the previous sections, $d_2$ should be induced by differential in the double complex on $A^2_p=\{c\in F_p(P\otimes Q)\: | \: d(c)\in F_{p-2}(P\otimes Q)\}$. However - the differential in a double complex is sending an element to the sum of elements from the column on the left and a row below. And since the filtration is given by rows here - how come the element from the $p$-th row might end in $F_{p-2}(Tot(P\otimes Q))$? If it is zero, then it would follow that all differentials on $E_2$ are zero, and the KSS collapses there - but it is not true, I suppose.
If it helps, I want $P$ to be a singular complex of a topological space. But I want to do the calculations through the method described above - whence the question.