Hochschild homology - motivation and examples

Question

I'm currently trying to learn about Hochschild homology of differential graded algebras. After reading the definition, the notion of Hochschild homology is somewhat unmotivated and myterious to me. What is the motivation to define Hochschild homology and what are some nice examples?

I'm particularly interested in the Hochschild homology of truncated polynomial algebras $$k[x]/(x^{n+1})$$ where $k$ is a field of characteristic zero and $x$ is of some degree $d$.

Are there any nice references for Hochschild homology?

Answer 1

Set $R = k[x]/(x^{n+1}),\,u=x\otimes 1-1\otimes x,\,v=\sum_{i=0}^n x^i\otimes x^{n-i} \in R^e := R \otimes_k R$.

First, let's recall from Weibel (Ex. 9.1.4) that in the ungraded case a projective resolution of $R$ over $R^e$ is given by the periodic complex $$\cdots \xrightarrow[]{v} R^e \xrightarrow[]{u} R^e \xrightarrow[]{v} R^e \xrightarrow[]{u} R^e \xrightarrow[]{\mu} R \to 0$$

Now suppose $R$ is a DGA with $\deg(x)=d$ and zero differentials. The latter implies that the notions of the Hochschild homology of $R$ as DGA and as graded algebra agree. Hence we can compute the Hochschild homology of $R$ by a projective resolution of $R$ over $R^e$ in the category of graded $R^e$-modules.

For a graded $R^e$-module $M$ let $\Sigma^kM$ be the shifted graded $R^e$-module given by $(\Sigma^kM)_i := M_{i-k}$. Set $e_k := (0,\ldots,1\otimes 1,\ldots 0) \in (\Sigma^kR^e)_k$. Then $\Sigma^kR^e=R^e\cdot e_k$ is a free graded $R^e$-module (in particular it's a projective object in the category of graded $R^e$-modules).

Taking into account $\deg u = d, \,\deg v=nd$, we can adjust the projective resolution from Weibel above and find the following projective resolution of $R$ over $R^e$ (taken in the category of graded $R^e$-modules): $$\cdots \to \Sigma^{(n+1)d}R^e \xrightarrow[]{d_2} \Sigma^dR^e \xrightarrow[]{d_1} R^e \to R \to 0$$ $$\cdots \to \Sigma^{(n+1)di}R^e\xrightarrow[]{d_{2i}}\Sigma^{(n+1)di-nd}R^e \xrightarrow[]{d_{2i-1}}\Sigma^{(n+1)d(i-1)}R^e\to\cdots $$ where $d_{2i}: e_{(n+1)di} \mapsto v\cdot e_{(n+1)di-nd},\,d_{2i-1}: e_{(n+1)di-nd} \mapsto u \cdot e_{(n+1)d(i-1)}$.

Now $HH_\ast(R,M)$ can be computed by tensoring this complex with $M$ (over $R^e$) and taking the homology. Using the relation $M \otimes_{R^e}\Sigma^kR^e=\Sigma^k M$ we obtain, for example, for $M=R$ the complex $$\displaystyle\cdots \to \Sigma^{(n+1)di}R\xrightarrow[]{d_{2i}}\Sigma^{(n+1)di-nd}R \xrightarrow[]{0}\Sigma^{(n+1)d(i-1)}R\to\cdots $$ where $d_{2i}: e_{(n+1)di} \mapsto (n+1)x^n\cdot e_{(n+1)di-nd}$. Hence

If $n+1$ is invertible in $k$ then (as graded $R$-module) $$HH_{2i}(R,R)=\Sigma^{(n+1)di}Rx,\quad HH_{2i-1}(R,R)=\Sigma^{(n+1)di-nd}R/(x^n), \quad H_0(R,R)=R.$$