# Hochschild Cohomology of Differential Operators in characteristic 0

In Mariusz Wodzicki's paper "Cyclic homology of differential operators," the following result is mentioned: for $D_M$ the algebra of differential operators on a smooth manifold $M$ we have that $HH_n(D_M) \cong H_{DR}^{2m-n}(M)$ where $m=\dim M$. I'm having trouble finding a reference for the Hochschild Cohomology of $D_M$. Does anyone know of a paper with the result?

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Hi,

This is an instance of a more general fact concerning deformation quantization of symplectic varieties. The general theorem is:

`Let $X$ be an symplectic manifold, and let $A_\hbar$ be any quantization of the Poisson algebra $(C^\infty(X), \{, \})$. Then we have $HH^*(A_\hbar)\cong H_{DR}^*(X)$.

See the paper:

Hochschild cohomology and Characteristic Classes by Weinstein and Xu

for an early proof. As Jeremy noted, one proof of this goes through spectral sequences. Morally the point, as I understand it, is that there is a canonical quantization of a symplectic vector space, which is the Weyl algebra, and which has trivial Hochschild cohomology. Since every symplectic variety is locally symplectomorphic to a symplectic vector space, you can compute the Hochschild cohomology of $HH^*(A_\hbar)$ on some covering of $X$ via a Cech complex consisting of constant sheaves at each basic open set (constant sheaves because you choose the covering fine enough so that the symplectic form can be put in canonical form, and then you use the result about the Hochschild cohomology of the Weyl algebra). This is also how you compute DeRham comology, so the answers coincide.

The reason this relates to differential operators is that for a smooth manifold $M$, its cotangent space $T^*M$ is a symplectic manifold, and we can regard $D_M$ as a quantization of $C^\infty(T^*M)$. as above. So we find that $HH^*(D_M)\cong H_{DR}^*(T^*M)\cong H_{DR}^*(M)$, since $T^*M$ contracts onto $M$.

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There are many references for this fact, which is essentially a spectral sequence argument along with the Poincare Lemma. The full proof is in section 5 of "THE HOMOLOGY OF ALGEBRAS OF PSEUDO-DIFFERENTIAL SYMBOLS AND THE NON-COMMUTATIVE RESIDUE" by Brylinski and Getzler.

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Dear Jeremy, I am not finding any discussion of cohomology in this paper, just the homology. –  KReiser May 17 '12 at 4:50
I think about the same argument will work for cohomology but instead you should get something like $HH^*(D_M)\simeq H_{DR}^*(M)$ but this time with no shift in the grading. This is like a Poincare duality between Hochschild cohomology and Hochschild homology. –  Jeremy Pecharich May 17 '12 at 5:14