I am interested in knowing about the possible implications between the following properties of a commutative, complex algebra:

- Its spectrum is non-singular.
- Its derivation module is projective and finitely generated.
- All
*pointwise derivations*(as explained below) admit an extension to an open set of the spectrum.

Let me note, in case it is not already clear, that I have a weak to non-existent background in commutative algebra and algebraic geometry.

## Notational preliminaries

Let $A$ be a commutative algebra over $\mathbb C$. I want to think of it as an algebra of functions on its character space
$$
M = \left\{ \chi\in A^*\ \vert\ (\forall a,b\in A)\ \chi(ab) = \chi(a)\chi(b) \right\},
$$
so I will suppose that the Gelfand transform
$$
a\in A\mapsto \hat a\in \{\text{functions } M\rightarrow \mathbb C\},\quad \hat a(\chi)=\chi(a)
$$
is injective (or equivalently, that the Jacobson radical of $A$ is trivial). A *derivation* is a linear application $\delta:A\rightarrow A$ such that
$$
\delta(ab) = \delta(a)b + a\delta(b),\quad a,b\in A,
$$
and a *derivation over a character $\chi$* is a linear application $\delta_\chi:A\rightarrow \mathbb C$ such that
$$
\delta_\chi(ab) = \delta_\chi(a)\chi(b) + \chi(a)\delta_\chi(b),\quad a,b\in A.
$$
The $A$-modules of derivations and derivations over $\chi$ will be denoted by $\text{der}(A)$ and $\text{der}_\chi(A)$, respectively.

I will suppose that $M$ is endowed with the trace of the Zariski topology (I hope that there are no problems in considering only maximal ideals), and I will write $\mathcal O_A$ for the structure sheaf of $M$.

## Questions

The Serre-Swan theorem establishes an equivalence between the categories of vector bundles and projective, finitely generated modules. So, here is my first question:

Is the non-singularity of $M$ equivalent to $\text{der}(A)$ being projective and finitely generated?

Given $\chi\in U\subseteq M$, observe that there is a natural map $\text{der}(\mathcal O_A(U)) \rightarrow \text{der}_\chi(A)$ given by $$ \delta \mapsto \delta_\chi = \chi\circ \delta. $$ It seems to me geometrically plausible that there is an affirmative answer to my second question:

Is the non-singularity of $M$ equivalent to the surjectivity of the map above, for every $\chi\in M$ and a suitable neighborhood $U\ni\chi$?

Now, if both questions above have affirmative answers, my third one is superfluous, but it must be stated anyways. It is actually the problem that interests me the most, independently of singularity considerations:

Under what conditions are the following two properties equivalent? Does at least one imply the other?

$\text{der}(A)$ is projective and finitely generated.

For every $\chi\in M$, there exists a neighborhood $U\ni\chi$ such that $\text{der}(\mathcal O_A(U))\rightarrow \text{der}_\chi(A)$ is surjective.