Hochschild Homology and Formal Geometry

Question

My question concerns the degeneration of the spectral sequence computing Hochschild homology of differential operators on a smooth affine variety $X$.

The spectral sequence arises from the filtration by order of the differential operator and so has $E^{1}$ page $HH(\mathcal{O}(T^{*}X))$. The differentials on this page are well known to be given by $L_{\pi}$, where $\pi$ is the (symplectic) Poisson form on $T^{*}X$, and $L$ denotes Lie derivative wrt a vector field. The spectral sequence degenerates at the second page.

Here is one (very much sketchy) proof - compute the Hochschild homology of $D_{X}$ using general D-module 6-functors formalism. Compute Poisson cohomology of a cotangent bundle explicitly using the symplectic version of the Hodge star operator. Observe they're the same.

Here is how a proof could go in a complex analytic setting; do the computation for a disc and then use a cover by such and a Cech type argument.

It's this second that I'd like an algebraic version of. If we let $\Delta$ denote the formal $n$-disc, and $D_{\Delta}$ denote the algebra of differential operators on this. It's not hard to see that all homology of the $E^{1}$ page of the above spectral sequence for $D_{\Delta}$ is concentrated in degree $2n$, hence degeneration. I'd like to use some Gelfand-Kazhdan formal geometry type argument to conclude degeneration for an arbitrary smooth $X$ but I can't see how to do it. A reference or explanation would be great!

Edit: cf Ben Zvi's answer here for another less sketchy sketch of the d module argument. Microlocalizing Hochschild homology

Answer 1

I do not really know the six-functor proof. I assume you are talking about the paper by Wodzicki. The paper is rather dense and the proof is purely technical.

If you prefer, you may start with by localization to the affine case and prove some analog of the classical HRK theorem. This should not be difficult, but it is already non-trivial. Then you generalize this to the whole space using the generalized Mayer-Vietoris sequence. This avoids the use of spectral sequence in Wodzicki's paper. This type of local-global proof is quite classical in Hochschild homology. I think you may find Ginzuberg's book on it to be useful.

There are some details in your post omitted. Can you give a reference on what kind of reference you use (general D-module 6-functors formalism, the symplectic version of the Hodge star operator, etc)? My impression is that the notation has never been standardized....