Harrison homology

Recall that when $A$ is an algebra over characteristic 0 field $k$, then

$$\HH_*(A)\xrightarrow{\isom}\Om^*_A.$$

In general there is a decomposition into direct sum
$$\begin{align*} \HH_n(A) &= \underbrace{\square\oplus\hdots\oplus \square}_{n\tex... ...ts & \\ \HH_2(A) &= \square \oplus \Om^2_A \\ \HH_1(A) &= \square \end{align*}$$
When one considers the first summands in each gradation then what one obtains is called Harrison homology of an algebra $A$. When $M$ is an $A$-bimodule, then $C_n(A, M)=M\ox_A A^{\ox n}$ gives a complex computing Hochschild homology of an algebra $A$ with coefficients in $M$. The complex for Harrison homology we obtain by taking a quotient by the shuffles in $C_n(A, M)$.