TITLE: Hochschild (co)homology of differential operators
QUESTION [16 upvotes]: I googled the title on the internet, and arrived at the following result
$$HH_k(D)\cong H_{DR}^{2n-k}(M).$$
Here $M$ is a smooth manifold of dimension $n$, and $D$ is the ring of differential operators on $M$. My first question is
(A): is there a known chain map realizing the above isomorphism?
I think the above isomorphism remains true if we replace $D$ by the ring of differential operators on $M$ with coefficients in a vector bundle $E$. But I am not so sure. If this is indeed the case, my second question is
(B): question (A) with coefficients in $E$?

REPLY [2 votes]: It is not literally what you want, but very close: check Proposition 2.3 and related results in "A Riemann-Roch-Hirzebruch formula for traces of differential operators" by Markus Engeli and Giovanni Felder (http://arxiv.org/abs/math/0702461).