Let $M$ be a smooth connected oriented without boundary non-compact manifold
of dimension n.


Let $k$ be a principal ideal, e. g. the integers $Z$


Let $H\_n(M)$ and $H^n(M)$ be the homology and cohomology in degree n of $M$
with coefficients in $k$.


It is well-known that $H\_n(M)=0$.


Is $H^n(M)$ also trivial ?


By the universal coefficient theorem for cohomology,
$H^n(M)=Ext(H\_{n-1}(M),k)$.
Therefore if $k$ is a field, $H^n(M)=0$. I would like to know this answer over
a principal ideal: $Z$.


It is also known (Bredon's book) that $H\_{n-1}(M)$ is without torsion.
But I believe that $H\_{n-1}(M)$ in the non-compact case, is a not a finitely 
generated $k$-module.
Therefore I don't know if $H\_{n-1}(M)$ is free, i. e. projective.
Since there exists abelians groups without torsion, non-free, e. g. the rationals $Q$


I suppose that this must be well-known.
But I could not find a reference.



n-th integral cohomology of a non-compact manifold of dimension n
By a version of Poincaré duality, $\mathrm H^n(M, \mathbb Z)$ is isomorphic to the $0$-th Borel-Moore homology group $\mathrm H\_0^{\mathrm BM}(M, \mathbb Z)$, (see, for example, IX-4 in Iversen's book on cohomology of sheaves). On the other hand $\mathrm H\_0^{\mathrm BM}(M, \mathbb Z) = 0$; this can easily be seen considering the one-point compactification $\overline M$, and the exact sequence
$$
0 \longrightarrow \mathrm H\_0^{\mathrm BM}(\overline M \smallsetminus M, \mathbb Z) \longrightarrow \mathrm H\_0^{\mathrm BM}(\overline M, \mathbb Z) \longrightarrow \mathrm H\_0^{\mathrm BM}(M, \mathbb Z)
\longrightarrow \mathrm H\_1^{\mathrm BM}(\overline M \smallsetminus M, \mathbb Z) = 0
$$
since $\mathrm H\_0^{\mathrm BM}(\overline M, \mathbb Z) = \mathbb Z$.


