In the situation of the Serre spectral sequence for a fibration $F \rightarrow E \rightarrow B$, when can I say that the cohomology of $E$ with coefficients in a field is the direct sum of the diagonal terms on the last page?
In Hatcher's Spectral Sequences book: https://pi.math.cornell.edu/~hatcher/SSAT/SSch1.pdf, on page 2, he says "For example if the coefficient group $G$ is a field, then $H_n(X;G)$ is the direct sum over p of $E_{p,n-p}^{\infty}$ of terms along the nth diagonal of the $E^{\infty}$ page"
But there is no clear statement of how this works for cohomology, rather than homology, and I also can't find anywhere specifically discussing the case of field coefficients.
Best Answer
One sufficient condition that the additive group of the cohomology with coefficients in an $R$-module $M$ is the direct sum of the diagonals is that all $Ext^1$ over $R$ vanish. This is equivalent to having $R$ be a product of fields. In the case $R$ is a field, it is easily seen that the $Ext^1$ terms all vanish because every vector space is free.
The reason that $Ext^1$ is relevant is that it counts extensions of modules, and if it vanishes this implies every extension is a direct sum.
A more relevant criterion if you are working over $\mathbb{Z}$ is that the relevant $Ext^1$ terms of your associated graded vanish. This happens, for example, when all the groups are free abelian.