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)?


1 Answer 1


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


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:


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


  • 3
    $\begingroup$ Heh. Despite appearances, I am not online all the time :) $\endgroup$ Commented Mar 2, 2012 at 1:22
  • $\begingroup$ 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? $\endgroup$
    – Alex
    Commented Mar 5, 2012 at 1:25

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