Hochschild cochain complex
= Hochschild cochain complex
{c}
The Hochschild cochains are $C^n(R,M)=\operatorname{Hom}_k(R^{\otimes_kn},M)$, with coboundary
$$
\begin{aligned}
(\delta f)(r_1,\ldots,r_{n+1})
={}&r_1f(r_2,\ldots,r_{n+1})\\
&+\sum_{i=1}^n(-1)^if(r_1,\ldots,r_ir_{i+1},\ldots,r_{n+1})\\
&+(-1)^{n+1}f(r_1,\ldots,r_n)r_{n+1}.
\end{aligned}
$$