It's an often observed fact that the basic notions of analysis on manifolds and (pseudo-)Riemannian geometry, such as tensors, connections and curvature, can be defined in purely algebraic terms. The protagonist of this is the algebra of smooth functions $C^\infty(M)$ on a manifold $M$: the observation that tensors on $M$ can be defined as $C^\infty(M)$-multilinear maps enables the algebraic definitions of connection, Riemann curvature tensor, etc. This observation in turn derives from the fact that the module of derivations on this algebra is dualizable, or equivalently finitely generated projective.
Therefore, if $k$ is a field and $A$ a commutative $k$-algebra, then we can develop the basic notions of (pseudo-)Riemannian geometry over $A$ as soon as the module of derivations $\mathrm{Der}_k(A)$ is f.g. projective [1].
Besides $C^\infty(M)$, what are examples of algebras for which the module of derivations $\mathrm{Der}_k(A)$ is f.g. projective?
Some formulas in differential geometry also require division by $2$ or $3$, so let me restrict to the case $\mathrm{char}(k) \neq 2,3$.
Here's what I know so far:
For $\mathrm{char}(k) = 0$ and $A$ a finitely generated domain, this is essentially what the Zariski-Lipman conjecture is about.
One could hope to find natural candidates for further examples by looking at suitable subalgebras of $C^\infty(M)$, e.g. by imposing certain decay conditions on the functions. However, I have not yet found any decay condition that works.
The $\mathbb{C}$-algebra of holomorphic functions on a domain $U \subseteq \mathbb{C}^n$ is an example, since its module of derivations is free of finite rank.
Because of the first item, and because $C^\infty(M)$ is not finitely generated, I'm only interested in non-finitely generated examples, of which I hope to see more in the answers.
[1] I'm not sure who to attribute this too. It appears in slightly different form in Riemannian manifolds as Lie-Rinehart algebras by Pessers and van der Veken (2016), and in a stronger version in Geroch's Einstein algebras (1972) and a few follow-up papers in the philosophy of general relativity. Is there more work that I'm not aware of?
