# Higher direct image with compact support of a constant sheaf

Let $$f: X \to Y$$ be a locally trivial fibration between locally compact spaces with fiber $$F$$. It is well known that for a constant sheaf $$A_X$$ on $$X$$, the higher direct images $$R^n f_* A_X$$ are locally constant, with stalk $$H^n(F, A)$$. This can be seen from the description of said higher direct images as the sheaf associated to the presheaf $$U \mapsto H^n(f^{-1}(U), A_X|_{f^{-1}(U)}).$$

According to Iversen, Cohomology of Sheaves, Theorem VII.1.4, the stalks of the higher direct images with compact support have a similar description (for an arbitrary continuous map $$f: X \to Y$$ and an arbitrary sheaf $$\mathcal{F}$$) $$(R^n f_! \mathcal{F})_y = H^n_c(f^{-1}(y), \mathcal{F}).$$

My question is then: if $$f$$ is a locally trivial fibration, does the same result hold? That is, are the sheaves $$R^n f_! A_X$$ local systems on Y? The same method of proof will not do, since the analogous candidate for a presheaf does not have well-defined restriction maps. And if you try to modify the proof that Iversen gives for the result on the stalks, things seem to break down from the begining, so I don't know how I could prove this.

UPDATE: Since we want to prove a local result, we may assume that we have a trivial fibration $$B \times F \to B$$. Using proper base change on the diagram $$\newcommand{\da}{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \begin{array}{ccc} B \times F & \xrightarrow{\pi_1} & B \\ \da{\pi_2} & & \da{a_B} \\ F & \xrightarrow{a_F} & \{*\} \end{array}$$ gives an isomorphism $$a_B^* \circ R^n a_{F,!} (A_F) \cong R^n \pi_{1,!} \circ \pi_2^* (A_F),$$ which by adjointness gives a map $$H^n_c(F, A) = R^n a_{F,!} (A_F) \to a_{B,*} \circ R^n \pi_{1,!} \circ \pi_2^* (A_F) = \Gamma(B, R^n \pi_{1,!} (A_{B\times F})).$$ This map should be an isomorphism if the result were true (it is equivalent to the result) but I don't see how to prove that it actually is an isomorphism.

With the information added in the last update, the result is already proved once you realize that the unit of the adjunction $$id \to a_{B,*} a_B^*$$ is actually an isomorphism on the category of abelian groups if the space $$B$$ is connected.