A global fiber basis trivializes Serre monodromy

Let $\gamma :b\to c$ be a path. Fiber transport is represented, up to the equivalence built into [F1], by a map $T_{\gamma }:F_b\to F_c$ lying over the path. The inclusion maps of the endpoint fibers into $E$ are homotopic after composing the first with $T_{\gamma }$. Therefore [F2] gives $T_{\gamma }^{\ast }(e_i{\mathrm{\mid }}_{F_c})=e_i{\mathrm{\mid }}_{F_b}$ for every $i$, in the variance convention of [F1].

Since the restricted $e_i$ are a basis in both endpoint stalks, step 1.1 says that the transport matrix sends every named basis vector to the corresponding named basis vector. It is therefore the identity matrix. This holds for every path class, so the basis identifies the local system with the constant graded $R$-module $H^{\ast }(F_b;R)$.

If $m=0$, the basis hypothesis says every stalk is the zero module and the conclusion is the constant zero system. The zero ring and zero-degree classes obey the same calculation. A path-connected base is not needed: the argument applies independently on each component on which the same finite list is a basis. No basis or representative is chosen; the list is part of the hypotheses. AC is used exactly through [A1] in [F1].