Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective?

I'm looking for conditions which apply in particular when $A = C^\infty(M)$ for a manifold $M$. In this case, the derivations are the vector fields and the module of derivations is finitely generated projective by Swan's theorem. Note that the module of Kähler differentials is *not* finitely generated unless $M$ consists only of isolated points.

injective? $\endgroup$Noncommutative Noetherian Rings, revised edition, 3.5.2 $\endgroup$10more comments