I am currently reading up on cohomology with compact supports, and am looking for a reference as to whether the De Rham isomorphism $H_\textrm{DR}^*(M)\simeq H^*(M;\mathbb R)$ exists for the compactly supported cohomologies $H_\textrm{DR,c}^*(M)$ and $H_c^*(M;\mathbb R)$ as well. I think it probably does, because I suspect two more things to be true as well:
- that $H_\textrm{DR,c}^*(M)$ is the direct limit of relative cohomology groups $H_\textrm{DR}^*(M,M\setminus K)$ in the same way that $H_c^*(M;\mathbb R)$ is the direct limit of $H^*(M,M\setminus K;\mathbb R)$,
- and that the De Rham isomorphism $H_\textrm{DR}^*(M)\simeq H^*(M;\mathbb R)$ exists for the relative cohomology groups $H_\textrm{DR}^*(M,M\setminus K)$ and $H^*(M,M\setminus K;\mathbb R)$ too.
I have however not been able to find a source for that at all, neither for the De Rham isomorphism on compactly supported cohomology nor for the two weaker statements that together would imply it. The only thing I was able to find is this answer on MathOverflow, which refers to a similar-looking theorem in a book on sheaf theory that I don't really understand. Can anyone maybe point me to a more elementary proof, or outline one here?