Let $E$ be a smooth oriented vector bundle over a smooth manifold $M$ and let $E^0$ be the complement of the zero section in $E$. I would like a reasonably explicit isomorphism between the relative De Rham cohomology group $H^p(E,E^0)$ and $H^p(M)$. Perhaps this is some version of the Thom isomorphism?
In case anyone needs a refresher, $H^*(E,E^0)$ is the cohomology of the complex whose chain groups are $\Omega^p(E,E^0) := \Omega^p(E) \oplus \Omega^{p-1}(E^0)$ and whose differential is given by $d(\omega_1, \omega_2) = (d \omega_1, i^* \omega_1 - d \omega_2)$.
EDIT: I actually want to prove that $H_{cv}^P(E,E^0) \cong H^p(M)$, where $H_{cv}^p(E,E_0)$ is the cohomology of the complex described above where all forms have compact support in the vertical direction.