In a [recent question](https://mathoverflow.net/questions/81446/for-which-algebras-does-differential-operators-satisfy-a-pbw-like-theorem), I recalled the notion of *differential operator*, *polyderivation*, and *principal symbol* for a commutative algebra $A$ over some fixed commutative ring $k$. (I will not repeat those definitions here, because you can read them there.) In that question, I asked for sufficient conditions to assure that $A$ satisfied the following version of the "PBW theorem": whether the principal symbol map (which I called $s\_n$) from ($n$th order) differential operators to ($n$th order) polyderivations is a surjection.


I thought I had an example to show that it is not always a surjection, but in the comments [Vladimir Dotsenko](https://mathoverflow.net/users/1306/vladimir-dotsenko) pointed out that I was wrong. In fact, I have since been unable to come up with any counterexample to this "PBW theorem".


Because for some other part of my project I must restrict to the situation when $k\supseteq \mathbb Q$ (essentially because I only know how to define the commutator of polyderivations in that case), I will ask my question there:



> 
> **Question:** What's an explicit example of a commutative algebra $A$ over a commutative ring $k\supseteq \mathbb Q$ such that there exists a polyderivation $A^{\otimes n} \to A$ which is not the principal symbol of any differential operator.
> 
> 
> 


If you can show that no such algebra exists, then I will gladly accept your answer both here and as an answer to my previous question.



What's an example of a commutative algebra over $\mathbb Q$ that fails to satisfy this version of the "PBW theorem"
Let me try to give a naive answer. Consider the following symmetric triderivation on $A=k[x]/x^2$: 
$$
x\partial\_x\otimes\partial\_x\otimes\partial\_x:(x,x,x)\mapsto x
$$
How could it be in the image of $s\_3$, considering that any differential operator on $A$ is of order $\leq 2$?


