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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.complicated.)

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

1

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.

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.