The question is a special case of a [previous question][1].

Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\alpha}||\partial^{\beta}(f)||$.

Now let $E\to M$ be a $C^{\infty}$-fiber bundle with compact base and compact fiber. Then it is clear that $C^{\infty}(E)$, the space of  smooth functions on the total space of the fiber bundle,  is a Frechet $C^{\infty}(M)$-module.

>>My question is: is $C^{\infty}(E)$ always a *projective* Frechet $C^{\infty}(M)$-module?

I think this question is trivial for experts. Please let me know if there is any references or it the question is not suitable for mathoverflow.


  [1]: https://mathoverflow.net/questions/303964/is-c-inftym-a-projective-frechet-c-inftyn-module-for-a-smooth-map