Second hypercohomology spectral sequence

Fix resolution degree $p$. The horizontal sequences $0\to B^{q,p}\to Z^{q,p}\to H^{q,p}\to 0$ and $0\to Z^{q,p}\to I^{q,p}\to B^{q+1,p}\to 0$ split. An additive functor preserves a split sequence, since it preserves the identities of an inclusion and retraction. It follows that the horizontal kernel, image and quotient after $F$ are $F(Z^{q,p})$, $F(B^{q,p})$ and $F(H^{q,p})$. Thus horizontal cohomology of $F(I^{\bullet ,p})$ is canonically $F(H^{q,p})$; the canonical quotient map gives this identification independently of any chosen splitting.

The next differential on $F(H^{q,p})$ is $(-1{)}^{q}F(v)$. The column $H^{q,\bullet }$ resolves $H^{q}K$, so its degree-$p$ cohomology after $F$ is $R^{p}F(H^{q}K)$; the constant sign $(-1{)}^q$ leaves its kernels and images unchanged. This gives the asserted $E_2$, rather than an $E_1$ identification.

The total target is $H^n(\mathrm{Tot}(FI))$ and the filtration is induced by the subcomplex of resolution degrees at least $p$. In total degree $n\ge b$, its endpoints are $F^0{H}^n=H^n$ and $F^{n-b+1}{H}^n=0$. F2 therefore gives finite strong convergence. F3 supplies comparison maps preserving this filtration; their vertical homotopies give identical $E_2$ and target maps. Identity and composite comparisons prove naturality. For $K=0$ with zero data all terms vanish, and the case $n=b$ has only one possible graded quotient.