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

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.

• The question is confusing because it implies that you would like conditions for the case you already know about (smooth functions on a manifold)... are you saying that you would like to consider when $A/k$ is not necessarily of finite type? Jul 21, 2020 at 19:13
• Tobias: regarding your final statement, the dual of the Kahler module is ${\rm Der}(A,A^*)$, which is not going to be naturally isomorphic to ${\rm Der}(A)$ although perhaps they coincide for some reason when $A$ is regular Noetherian. Indeed, while there are natural instances where the Kahler module is projective, why would that make its dual projective? Surely you'd expect the dual of something projective to be injective ? Jul 22, 2020 at 0:01
• Also, as soon as you want to take $C^\infty(M)$ you probably want to be imposing topological constraints: there is an old MO question mathoverflow.net/questions/6074/… Jul 22, 2020 at 0:03
• A remark here: for commutative rings $A$, if $M$ is a finitely generated projective module, then so is $M^*= Hom_A(M,A)$ by the dual basis lemma: see McConnell, Robinson, Noncommutative Noetherian Rings, revised edition, 3.5.2 Jul 22, 2020 at 0:25
• @YemonChoi the existence of cut-off functions shows that all derivations are local and then Hadamard lemma (Taylor series up to order two) shows that a derivation is determined by its values on coordinate functions Jul 22, 2020 at 6:58

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.