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> What is the analog of HKR theorem in the noncommutative world?
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Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a *smooth* *commutative* algebra $A$ of finite type over a field $k$ (let's assume of characteristic zero) there is an isomorphism $$HH\_\bullet(A)\simeq \Omega^\bullet(A)$$ where $HH\_\bullet(A)$ denotes the Hochschild homology of $A$ with coefficients in itself, and $\Omega^\bullet(A):=\Lambda^\bullet(\Omega^1\_k(A))$ is the space of algebraic differential forms on $A$, and $\Omega^1\_k(A)$ denotes the module of Kahler differentials. 


There is a notion of *smoothness* for any associative algebra $A$, not necessarily commutative (formal smoothness). Moreover, one can define noncommutative differential forms $\Omega^\bullet\_{nc}(A)$. So we have all the ingredients for noncommutative HKR, but I can't find the precise statement. I found the page on [nLab](http://ncatlab.org/nlab/show/Hochschild-Kostant-Rosenberg+theorem#for_noncommutative_algebras), but the section is missing.


Thank you very much for your help!


**Edit:** As Mariano pointed out in the comments, $HH\_\bullet(A)$ is not the same as $\Omega^\bullet\_{nc}(A)$ for formally smooth $A$. I was indeed stupid here. Formal smoothness is equivalent to saying that $\Omega^1\_{nc}(A)$ is a projective bimodule. But then the complex $0\to \Omega^1\_{nc}(A)\to A\otimes A\to 0$ is a projective resolution of $A$, and so Hochschild homology $HH\_n(A)$, defined as $Tor\_n^{A^e}(A,A)$, vanish for $n>1$, while noncommutative differential forms do not vanish. 


But the question is still valid. What is the NC analog that the nLab page is talking about?



Noncommutative HKR theorem
Maybe the following paper helps: 


* Andreas Cap, Andreas Kriegl, Peter W. Michor, Jiri Vanzura: The Frölicher-Nijenhuis bracket in non commutative differential geometry, Acta Math. Univ. Comenianae 62(1993), 17--49 [pdf](http://www.mat.univie.ac.at/~michor/ncdg.pdf)


