Sufficient conditions for $mathrm{Der}_k(A)$ to be f.g. projective

Question

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.

jg1896 · Answer

For finitely generated domains over a base field $k$ of characteristic 0, we have that if $A$ is regular, then both $Der_k , A$ and the module of Khäler differentials are finitely generated projective (McConnell, Robinson, Noncommutative Noetherian Rings, revised edition, 15.2.11).
Zariski-Lipman's Conjecture says that if $Der_k , A$ is finitely generated projective (or, in a more modest version, free), then $A$ is regular.
So for this class of algebras (roughly, regular functions on smooth affine varieties), it is expected that $A$ is regular if and only if $Der_k , A$ is a finitey generated projective module.
Your example does not fit here as smooth functions on a real manifold are not even domains, but I think that the literature on this subject (i.e. Zariski-Lipman's Conjecture) might be a good direction to look.

Sufficient conditions for $mathrm{Der}_k(A)$ to be f.g. projective

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