I'm trying to understand the differentials in the Leray spectral sequence but I must be making a mistake because I'm getting that certain differentials are always zero. I am wondering if I have misunderstood the setup:
If $f:X\to Y$ is continuous map between topological spaces and $S$ is a sheaf of abelian groups on $X$ then the Leray spectral sequence basically says $$H^p(Y; R^qf_*S)\Rightarrow H^{p+q}(X;S).$$ I read that it is derived in the following way: First take an injective resolution $S\to A^\bullet$ and then push it forward via $f_*$ and cut off the first term to obtain the complex $$f_*A^0\to f_*A^1\to f_*A^²\to...$$ Now we need a Cartan-Eilenberg resolution of this complex: For each $p\geq 0$ let $B^p, Z^p, H^p$ denote the coboundaries, the cocycles and the cohomology of $f_*A^\bullet$ at the $p$-th spot. Take injective resolutions $I^{p\bullet}_B$ and $I^{p\bullet}_H$ of the coboundaries $B^p$ and the cohomology $H^p$. Then by the horseshoe lemma we obtain an injective resolution $I^{p\bullet}_B\oplus I^{p\bullet}_H$ of the cocycles $Z^p$ that fits into the diagram $$\require{AMScd} \begin{CD} 0 @>>> I^{p\bullet}_B @>>> I^{p\bullet}_B\oplus I^{p\bullet}_H @>>> I^{p\bullet}_H @>>> 0\\ @. @AAA @AAA @AAA\\ 0 @>>> B^p @>>> Z^p @>>> H^p @>>> 0 \end{CD}$$ where the top row is exact (as a sequence of complexes). Now apply the horseshoe lemma a second time to obtain an injective resolution of $f_*A^p$: $$\require{AMScd} \begin{CD} 0 @>>> I^{p\bullet}_B\oplus I^{p\bullet}_H @>>> I^{p\bullet}_B\oplus I^{p\bullet}_H\oplus I^{p+1\bullet}_B @>>> I^{p+1\bullet}_B @>>> 0\\ @. @AAA @AAA @AAA\\ 0 @>>> Z^p @>>> f_*A^p @>>> B^{p+1} @>>> 0 \end{CD}$$ The Cartan-Eilenberg resolution is the first quadrant double complex $I^{pq}=I^{pq}_B\oplus I^{pq}_H\oplus I^{p+1,q}_B$ together with the augmentation maps $f_*A^p\to I^{p0}$. The horizontal differentials of this double complex are the projection of the third summand on the left to the first summand on the right: $$I^{pq}_B\oplus I^{pq}_H\oplus I^{p+1,q}_B \to I^{p+1,q}_B\oplus I^{p+1,q}_H\oplus I^{p+2,q}_B$$ $$(a,b,c)\mapsto (c,0,0)$$ The vertical differentials are the sums of the differentials in the resolutions $I^{p\bullet}_B$, $I^{p\bullet}_H$ and $I^{p+1\bullet}_B$ (multiplied with $(-1)^p$ to make each square in the double complex anticommute). Lastly I apply the global sections functor $\Gamma$ and get the double complex $$\Gamma(I^{pq}) = \Gamma(I^{pq}_B)\oplus \Gamma(I^{pq}_H)\oplus \Gamma(I^{p+1,q}_B)$$ where still the horizontal differentials are maps $(a,b,c)\mapsto (c,0,0)$.
My question: Have I set up everything correctly up to here? Because I am getting the feeling my error in the end (I explain it in another post if the setup is okay up to here) is due to wrong horizontal differentials.