Let $F \rightarrow E \rightarrow B$ be a Serre fibration. Assume that the cohomological spectral sequence (with integer coefficients) $E_2^{p,q} = H^p(B; H^q(F)) \Rightarrow H^*(E)$ degenerates so that $E_2 \cong E_\infty$. Can we also conclude that the cohomological spectral sequence with cohomology over any field $\mathbb{F}$ degenerates? Or do we need further assumptions on the cohomology of the spaces,
I am confused about what is the relation between the differentials (from the two spectral sequences) under the UCT isomorphism $H^*(X;\mathbb{F}) \cong H^*(X) \otimes \mathbb{F} \oplus Tor(H^{*+1}(X), \mathbb{F})$.
Edit to add: I think it is true over rational coefficients as the Tor term vanishes in the UCT and looking at the surjectivity of the map $H^*(E)\otimes \mathbb{Q} \rightarrow H^*(F) \otimes \mathbb{Q}$ (since $E_2 \cong E_\infty$). A similiar reasoning must hold over $\mathbb{F}_p$ if the cohomology of the spaces cotains no $p$-torsion.
I am puzzled about the general case which I am inclined that it might not be true.