Hochschild cohomology of a formal quantization of an associative algebra

Question

Let $A$ be a commutative associative $k$-algebra and let $A[[\hbar]]$ be the formal deformation of $A$. I would like to know if there is a relation between the Hochschild co-homologies $\mathrm{HH}^{\bullet}(A, A)$ and $\mathrm{HH}^{\bullet}(A[[\hbar]], A[[\hbar]])$, respectively. In particular, is it true that $\mathrm{HH}^{\bullet}(A[[\hbar]], A[[\hbar]]) = \mathrm{HH}^{\bullet}(A, A)[[\hbar]]$?

Answer 1

If $A[[\hbar]]$ actually means $A\otimes_kk[[h]]$ as an algebra, then yes, this is clear from the Hochschild complex.

It seems more likely that you mean that $A[[\hbar]]$ carries a star product $a \star b=ab+\frac{1}{2}\hbar \{a,b\}+\cdots$. In this case, this is totally false, as essentially any example will show. Let's say that $A=\mathbb{C}[x,y]$, and the star product is the usual Moyal star product (I think is also Weyl quantization), so $x\star y= xy +\frac{1}{2}\hbar\qquad y\star x=xy-\frac{1}{2}\hbar$.

Just look at $HH_0$, which is the center of the algebra. We have $HH_0(\mathbb{C}[x,y])=\mathbb{C}[x,y]$, but $HH_0(\mathbb{C}[x,y])_\hbar=\mathbb{C}[[\hbar]]$.

Answer 2

1) Ben Webster answered the question as it was asked, but it is not maybe the best version of the question. We can consider m the Maurer Cartan element giving rise to the deformation and formulate

H*(HC(A)[[t]] , d_ord + [m,-)

Where HC denotes the standard Hochschild complex. This is a much better version of the question.

2) Do you really mean A[[h]] = A\otimes k[[h]]. This is highly nonstandard and typically one requires a completion.