Following on from my last two questions [link text](https://mathoverflow.net/questions/5865/classical-calculi-as-universal-quotients) and [link text](https://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials): Is it correct (and useful) to say that the relationship between Connes' cyclic cohomology approach to de Rham cohomology and Woronowicz's differential calculi approach, is a noncommutative generalisation of the difference between ordinary differentials and Kahler differentials respectively?



Connes v Woronowicz - Cyclic Cohomology v Diff Calculi
I'll answer here instead of in a comment, because of the character limit...


If $A$ is the coordinate algebra of an affine variety which is *smooth* and the base field $k$ contains $\mathbb{Q}$, then $$HC\\_n(A) \cong \Omega^n\\_{A/k} / d\Omega^{n-1}\\_{A/k} \oplus H\\_{\mathrm{dR}}^{n-2}(A) \oplus H\\_{\mathrm{dR}}^{n-4}(A) \oplus \cdots$$ for all $n\geq0$. Here $\Omega^n\\_{A/k}$ is the $n$-th exterior power of the $A$-module $\Omega^1\\_{A/k}$ of Kähler differentials of $A$ over $k$, and $H\\_{\mathrm{dR}}^\bullet(A)$ denotes the cohomology of the complex $$A\to \Omega^1\\_{A/k} \to \Omega^2\\_{A/k} \to \Omega^3\\_{A/k} \to\cdots $$ whose differential is the exterior differential. 


The summand $\Omega^n\\_{A/k} / d\Omega^{n-1}\\_{A/k}$ appearing in $HC\\_\bullet(A)$ is slightly ugly. If we consider instead periodic cyclic homology, we get instead $$HC^{\mathrm{per}}\\_n(A) \cong \bigoplus\\_{i\in\mathbb{Z}}H\\_{\mathrm{dR}}^{n+2i}(A),$$ which is manifestly nicer.


(If $k$ is not of characteristic zero you only have a spectral sequences going from de Rham cohomology to the cyclic homology). If, on the other hand, $A$ is not smooth then André-Quillen cohomology intervenes, and everythng is rather more complicated.)


You should really take a look at Loday's book.


