In a recent question, 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 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.