Let $C$ be a smooth projective curve over an algebraically closed field $k$. 


The tangent lie algeborid $\mathcal{T}\_C$ of $C$ is just sheaf of vector fields on $C$ equipped with the usual lie bracket (commutator of derivations).



> 
> **Definition:** We'll say a **lie algebroid** $\mathcal{E}$ over a regular ring $R$ **split** iff it is free as a module over $R$ and there
>  exists a basis for $\mathcal{E}$ consisting of mutually commuting
>  sections (commuting for the bracket on $\mathcal{E}$). Is there already a name
>  for this property in the literature?
> 
> 
> 


For instance the tangent sheaf of a curve is locally split when it is locally spanned by mutually commuting vector fields.


Clearly the tangent lie algebroid of $\mathbb{A}^1$ is split and hence the tangent lie algebroid of $\mathbb{P}^1$ is locally split in the zariski topology.


Since every smooth variety is etale locally isomorphic to affine space we get that (at least for smooth varieties) **the tangent lie algebroid is always locally split in the etale topology**.


My hunch is that for the Zariski topology this fails miserably however i'm unsure about how to come up with an example. Hence the question:



> 
> Is $\mathbb{P}^1$ the only **smooth projective curve with a locally split** (in the zariski topology) **tangent lie algebroid?** If it is, **how can this be shown?** If not is there some kind of classification of those with this property?
> 
> 
> 



Is $\mathbb{P}^1$ the only smooth projective curve with a locally split tangent lie algebroid?
If $f: X \to \mathbb A^n$ is an etale map, then we can pull back vector fields on $\mathbb A^n$ to vector fields on $X$. This pullback operation is a lie algebroid homomorphism. Hence if we pull back a commuting basis of vector fields from $\mathbb A^n$, we obtain a commuting basis on $X$.


Every smooth $n$-dimensional variety Zariski-locally admits an etale map to $\mathbb A^n$: If you choose a basis for the maximal ideal of the local ring of a point, these functions will extend to some Zariski neighborhood of the point, defining a map to $\mathbb A^n$ on that neighborhood, which is etale at that point.


So every smooth variety has a locally split tangent Lie algebroid.


