Hochschild cocycle ($Z^n(A,M)$)

A Hochschild cocycle is a cochain $f$ with $\delta f=0$ in the Hochschild cochain complex. For degree two, this is $a\mu (b,c)-\mu (ab,c)+\mu (a,bc)-\mu (a,b)c=0$. It is the condition for associativity of a square-zero extension of an algebra.