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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 $\mathrm{HH}^{\bullet}(A[[\hbar]], A[[\hbar]]) = \mathrm{HH}^{\bullet}(A, A)[[\hbar]]$?

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  • $\begingroup$ Yes,is it forbidden to pose the same question in different forums? $\endgroup$ Jun 23, 2014 at 15:09
  • $\begingroup$ It is suggested that you post the question at M.SE first, and then, if you do not get a (suitable) answer, post it at MO. $\endgroup$
    – user122283
    Jun 23, 2014 at 15:10
  • $\begingroup$ ok, I did not know this rule. And I badly need a fast answer to this question that is why i posed it twice on both platforms. I did not now that they are connected. How long do I need to wait for a response on M.SE before I post on MO? $\endgroup$ Jun 23, 2014 at 15:14
  • $\begingroup$ A few hours at least. $\endgroup$
    – user122283
    Jun 23, 2014 at 15:18
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    $\begingroup$ I personally feel that it would have been fine to post this at MO in the first place -- the level seems sufficiently high. Also, a lot of people do post at both sites; the main problem people have with such "cross-posting" is that it could lead to duplication of effort. Thus, as a minimum courtesy, one should at least link from each post to the other post if one decides to cross-post. $\endgroup$
    – user43208
    Jun 23, 2014 at 18:48

2 Answers 2

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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 $HH_0(\mathbb{C}[x,y])_\hbar=\mathbb{C}[[\hbar]]$.

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  • $\begingroup$ Sure, $A[[\hbar]]$ carries a star product. I suspected that the latter isomorphsims is too much to hope for. So basically I cannot compute the Hochschild cohomology of a formal deformation of an algebra knowing the Hochschild cohomology of the original algebra? Probably it depends on the concrete case. $\endgroup$ Jun 23, 2014 at 16:46
  • $\begingroup$ @FlaviusAetius, there is a spectral sequence going from the Hochschild cohomology of the algebra $A[[h]]$ with trivial star product to the cohomology of the one with the non-trival one, coming from the obvious filtration on the latter. This converges topologically and sometimes better. $\endgroup$ Jun 24, 2014 at 7:48
  • $\begingroup$ Hm, that is quite nice. You construct it probably using a filtration in $\hbar$ or not? $\endgroup$ Jun 24, 2014 at 21:42
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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.

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  • $\begingroup$ hm, I am not sure if I understand your question in 2.) $(A[[\hbar]],*)=(A\otimes k[[\hbar]],*)$ is what I meant. Here $*$ denotes the star product. $\endgroup$ Jun 24, 2014 at 21:38

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