= Bar resolution of an associative algebra
{title2=$B_n(A)$}
The bar resolution of an <associative algebra> $A$ over a <field> has degree-$n$ term $A^{\otimes(n+2)}$ as an $A^e$-<module>, with the outer factors carrying the left and right actions. Its boundary multiplies each consecutive pair with alternating signs, and its augmentation is multiplication $A\otimes A\to A$. It is exact: the $k$-linear contracting homotopy inserts $1$ at the left. Applying $\operatorname{Hom}_{A^e}(-,M)$ gives the <Hochschild cochain complex> of the <bimodule> $M$.
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