# Hochschild Homology and Formal Geometry

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