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!

• You can find details for the polynomial ring in Weibel's book (or, as suggested below, Loday for the cyclic case). – Yemon Choi Mar 1 '14 at 3:28

2 Answers

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 $HH^*(A,A)\cong \wedge^*A^p$.

• Thanks! What about the cyclic cohomology of A? I know there is some general theorem on the cyclic cohomology of the ring of functions on an affine variety, but I didn't have enough background to understand the notions involved in the statements. Please help! – user36075 Feb 11 '14 at 11:30
• @user36075: Did you consult with Loday's book? – user23860 Feb 11 '14 at 11:42
• are you sure you want cyclic cohomology and not homology? If it is cyclic homology, what sort HC^{-}, HC^{per}. This is not pedantic at all because the different theories are very different. – user36931 Feb 11 '14 at 11:44
• For instance I don't think cyclic cohomology of A necessarily has a nice description because it is not naturally an A module... maybe something like algebraic distributions ? – user36931 Feb 11 '14 at 11:45
• The cyclic cohomology I'm interested in is defined using the cyclic complex that is the cyclically invariant part of ${\rm Hom}(\otimes^{k+1}A,\mathbb{C})$. Indeed I don't want ${\rm HC}^*(A)$ to be an $A$-module. Rather, I want it to be a module of $h^*$ where $h=\langle t_i\rangle$ (i.e. $h^*$ acts on $A=\mathbb{C}[t_i]$ by taking derivatives). – user36075 Feb 11 '14 at 11:59

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 $\mathfrak{g}$.