# 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 $$\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \begin{array}{ccc} B \times F & \xrightarrow{\pi_1} & B \\ \da{\pi_2} & & \da{a_B} \\ F & \xrightarrow{a_F} & \{*\} \end{array}$$ 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.

## 1 Answer

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.