Hochschild and cyclic cohomology of commutative algebra? I'm wondering how to compute the Hochschild cohomology ${\rm HH}^n(A,A)$ and the cyclic cohomology ${\rm HC}^n(A)$ where $A = \mathbb{C}[t_i]$ is the algebra of polynomials in a finite set of commutative variables $t_i$? A related question is, given an abelian Lie algebra ${\bf g}$, and a general ${\bf g}$-module $N$, how do I compute the Lie algebra homology ${\rm H}_n({\bf g}, N)$? 
Sorry for the rather elementary questions but I'd very much appreciate help on this!
 A: For the Hochschild cohomology, the celebrated Hochschild-Kostant-Rosenberg theorem gives a canonical algebra isomorphism, for a smooth $k$-algebra $A$, $\ HH^*(A,A)\cong \wedge^*T_A$, where $T_A$ is the $A$-module of $k$-derivations of $A$. If $A=\mathbb{C}[t_1,\ldots ,t_p]$ $\ T_A$ is a free $A$-module of rank $p$, so $HH^*(A,A)\cong \wedge^*A^p$. 
A: Concerning the related question, which has not been addressed so far: the space of $p$-chains for a Lie algebra $\mathfrak{g}$ over a field $K$ is given by $C_p(\mathfrak{g},N)=\Lambda^p\mathfrak{g}\otimes_K N$ for any $\mathfrak{g}$-module $N$ and any integer $p\ge 0$. The homology is given by the homology of the complex of $p$-chains together with the usual boundary maps $\partial_p\colon C_p(\mathfrak{g},N)\rightarrow C_{p-1}(\mathfrak{g},N)$, i.e.,
$H_p(\mathfrak{g},N)=\ker \partial_p/{\rm im} \,\partial_{p+1}$.
Now if $\mathfrak{g}$ is abelian of dimension $n$ then the boundary map  simplifies quite a lot, and one can use this to compute the homology. For example, for the
trivial module $K$ we obtain
$H_p(\mathfrak{g},K)=\Lambda^p\mathfrak{g}$
for $p\le n$. Alternatively, $H_n(\mathfrak{g},N)=Tor_n^{U(\mathfrak{g})}(N,K)=Tor_n^{S(\mathfrak{g})}(N,K)$, because $U(\mathfrak{g})=S(\mathfrak{g})$ for abelian $\mathfrak{g}$.
