What is the analog of HKR theorem in the noncommutative world?

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, 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?

  • $\begingroup$ Typically, in non-commutative geometry, one thinks of the left-hand side as a definition for the right-hand side. $\endgroup$ – Daniel Pomerleano Jul 22 '14 at 1:13
  • $\begingroup$ @DanielPomerleano I don't know.. I have seen only the definition where $\Omega^1_{nc}(A)=ker(m\colon A\otimes A\to A)$ is the kernel of the multiplication map, and then $\Omega^\bullet(A)=T_A^\bullet(\Omega^1_{nc}(A))$ is the tensor algebra of $\Omega^1_{nc}(A)$. $\endgroup$ – Sasha Patotski Jul 22 '14 at 1:37
  • $\begingroup$ A long time ago, I read something about it in Loday's book "cyclic homology", see Section 3.4. $\endgroup$ – user23860 Jul 22 '14 at 2:36
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    $\begingroup$ Daniel is right in that a sensible definition of forms in the nc world is through Hochschild homology (or the actual Hochschild complex); this is the point of the noncommutative calculi of Tsygan, Tamarkin and others. It is also true that for other (closely related!) purpuses one uses the definition mentioned by Sasha in his comment above. You can read about it in Max Karoubi's Astérique on cyclic homology, among other places. $\endgroup$ – Mariano Suárez-Álvarez Jul 22 '14 at 7:13
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    $\begingroup$ They are not the same. $\endgroup$ – Mariano Suárez-Álvarez Jul 23 '14 at 21:35

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

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