# Characterizing non-singularity of varieties through properties of their derivations

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

1. Its spectrum is non-singular.
2. Its derivation module is projective and finitely generated.
3. 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?

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

2. 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.

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I assume that $A$ is finitely generated. Regarding your first question, $M$ nonsingular immediately implies projectivity of $der(A)$ (= the tangent module). The converse statement is the Zariski-Lipman conjecture. It is still open as far as I know, although partial results are known, see Flenner, "Extendibility of differential forms..." Invent 1988. –  Donu Arapura Nov 18 '11 at 16:58
There are more partial results on the Lipman-Zariski conjecture. It has been proven for klt spaces by Greb-Kebekus-K-Peternell. See here: springerlink.com/content/072664108771003x There are some older results as well see the link for references. –  Sándor Kovács Nov 18 '11 at 18:00
Thanks for your comments, knowing that my first question is not completely settled is already quite helpful. I will check the references you gave me. Any idea regarding the third question will also be very appreciated. –  Rodrigo Vargas Nov 18 '11 at 19:43