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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 $M\to N$ be a smooth map between compact smooth manifolds. Then it is clear that $C^{\infty}(M)$ is a Frechet $C^{\infty}(N)$-module.

My question is: is $C^{\infty}(M)$ always a projective Frechet $C^{\infty}(N)$-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.

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In his Ph.D. thesis "Проблема существования инъективных модулей над классическими топологическими алгебрами и инъективные гомологические размерности", A.Yu Pirkovskii showed that $\mathrm{dh}_{C^\infty(N)}C^\infty(M)=\mathrm{codim}_N(M)$, if $M$ is a closed submanifold of $N$.

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    $\begingroup$ More precisely, this result is Corollary 3.3.28 in Pirkovskii's thesis. Just to clarify the meaning of this answer: $\mathrm{dh}_A X$ refers to the projective dimension of the $A$-module $X$. The projectivity of $X$ is equivalent to $dh_A X = 0$. So $C^\infty(M)$ is projective over $C^\infty(N)$ only if $M\subset N$ is codimension-0. But in that case a much older result applies: $C^\infty(M)$ is projective over $C^\infty(N)$ for any open $M \subset N$, due to Ogneva (1986). $\endgroup$ – Igor Khavkine Jun 30 '18 at 12:33
  • $\begingroup$ @IgorKhavkine It seems that the answer is trivially negative. Is it also a well-known result that if $M\to N$ is a submersion of compact smooth manifolds, then $C^{\infty}(M)$ is a projective $C^{\infty}(N)$ module? $\endgroup$ – Zhaoting Wei Jun 30 '18 at 13:41
  • $\begingroup$ @ZhaotingWei, unfortunately I don't know. This doesn't seem to be covered in Pirkovskii's thesis. An important keyword for this kind of work seems to be "topological homology" and the name Helemskii appears central to the subject. $\endgroup$ – Igor Khavkine Jul 1 '18 at 7:05
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The relevant notion is that of a $C^\infty$-algebra; one can evaluate functions in $C^\infty(\mathbb R^n,\mathbb R)$ in the algebra. The definite source on these is:

  • MR1083355 (91m:18017) Moerdijk, Ieke(NL-UTRE); Reyes, Gonzalo E.(3-MTRL-R) Models for smooth infinitesimal analysis. Springer-Verlag, New York, 1991. x+399 pp. ISBN: 0-387-97489-X

There is also (if I remember correctly) the notion of a $C^\infty$-module developed there.

A functional analytic characterization of $C^\infty$-algebras is in:

  • Gerd Kainz, Andreas Kriegl, Peter W. Michor: C∞-algebras from the functional analytic viewpoint. J. Pure Appl. Algebra 46 (1987), 89-107. (pdf)
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    $\begingroup$ How does this address the question? I couldn't find (by a keyword search) a discussion of projectivity of modules over $C^\infty(M)$ in either reference. $\endgroup$ – Igor Khavkine Jul 1 '18 at 7:09

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