Let's consider the next theorem.
Theorem [The cohomology Leray-Serre Spectral sequence] Let $R$ be a commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{% \rightarrow }B$, where $B$ is path-connected, there is a first quadrant spectral sequence of algebras $\left\{ E_{r}^{\ast ,\ast },d_{r}\right\} $, with \begin{equation*} E_{2}^{p,q}\cong H^{p}\left( B;\mathcal{H}^{q}\left( F;R\right) \right) \end{equation*} and converging to $H^{\ast }\left( E;R\right) $ as an algebra, where $% \mathcal{H}^{q}\left( F;R\right) $ denotes the cohomology of $B$ with local coefficients in the cohomology of $F$, the fiber of $p$. Furthermore, this spectral sequence is natural with respect to fiber-preserving maps of fibrations. I want to calculate \begin{equation*} E_{2}^{p,0}\cong H^{p}\left( B;\mathcal{H}^{0}\left( F;R\right) \right) \end{equation*} Some sources claim that if $F$ is path-connected, then the local coefficients system $\mathcal{H}% ^{0}\left( F;R \right) $ over $B$ (here $R$ is a principal ideal domain, I don't know if the question has anything to do with this) is trivial and \begin{equation*} E_{2}^{p,0}=H^{p}\left( B;\mathcal{H}^{0}\left( F;R \right) \right) =H^{p}\left( B;H^{0}\left( F;R \right) \right) =H^{p}\left( B;R \right) \end{equation*} First of all, I'm curious about the proof of this, that is why the local coefficients system $\mathcal{H} ^{0}\left( F;R \right) $ over $B$ is trivial.
Secondly, is the path connectedness of $F$ related to the cohomology used?
I know that in the case of singular cohomology, $H^{0}\left( X;R\right) =R$ if and only if $X$ is path-connected, and in the case of Alexander-Spanier cohomology, $H^{0}\left( X;R\right) =R$ if and only if $X$ is connected.
So, If I use the Alexander-Spanier cohomology instead of the singular cohomology, is it enough for $F$ to be connected?
