Could someone point me to a standard reference for the fact that the top cohomology $H^n(M,A)$ of an $n$-dimensional manifold $M$ is non-trivial for local coefficients $A$ if and only if the manifold is compact?
EDIT: It seems that there are some issues when $M$ is non-orientable. I would like to include the non-orientable case. I figure the result uses (twisted) Poincaré duality and some kind of pairing between the $n$th cohomology and compactly supported cohomology in degree $0$.
I am not sure of its validity, but I am looking for (a reference for) an isomorphism $H_c^0\cong H^n$ which holds for local systems.