I am trying to understand more precisely some aspects of the statement of Leray's theorem on spectral sequences, as stated in Loring Tu's "Introduction to Equivariant Cohomology". The statement is this:
EDITED: From the comments, I understand that the spectral sequence doesn't have anything to do with $E$ directly and also that $GH^*(E) = \bigoplus\limits_{i = 1}^\infty D_i/ D_{i+1}$.
We have for each $p \in \mathbb{N}$ an $F_p := \bigoplus\limits_{i \geq p} \bigoplus\limits_{q\geq 0} E_2^{i,q}$. Part of the statement of the theorem is that $F_p = \bigoplus\limits_{i \geq p} \bigoplus\limits_{q\geq 0} \left( H^i(B) \otimes H^q(F) \right)$.
What I realised still confuses me is this: How is the filtration $D_p$ is derived from $F_p$, and does it even have anything to do with Leray's theorem at all?
I am guessing it's not part of the theorem, but it's something to do with singular cohomology and fiber bundles in general, i.e. if we have a filtration on $\bigoplus\limits_{p, q \in \mathbb{N}} (H^p(B) \otimes H^q(F))$, then this somehow translates to one in $H^*(E)$. Intuitively it seems like this should be the case, at least, but I am not familiar with singular cohomology.