# Hochschild homology of Weyl algebra

Could someone explain to me how one can compute the Hochschild homology of the Weyl algebra $$A_n$$ (i.e., algebra of differential operators with polynomial coefficients in $$n$$ variables)?

I'm surprised that Mariano hasn't replied. The cohomological version of this question has been asked on MO and answered by Mariano:

https://mathoverflow.net/questions/69059/a-simple-proof-of-the-weyl-algebras-rigidity

I think the answer you want is that $$HH_*(A_n(k)) = \begin{cases} 0 &\text{ if } \ast \ne 2n \\ k &\text{ if } \ast=2n\end{cases}$$

The reference is a paper of Sridharan:

http://www.ams.org/journals/tran/1961-100-03/S0002-9947-1961-0130900-1/S0002-9947-1961-0130900-1.pdf

And since I haven't yet included enough links here is another paper:

http://www.sciencedirect.com/science/article/pii/S0022404903001464#sec3.1

• Heh. Despite appearances, I am not online all the time :) Commented Mar 2, 2012 at 1:22
• Thanks, Mariano's answer on mathoverflow is nice, but is it really easier to proof that $A_n$ is CY then directly calculate Hochschild homology?
– Alex
Commented Mar 5, 2012 at 1:25