Standard complex

The name "bar complex" comes from the fact that Eilenberg & Mac Lane (1953) used a vertical bar | as a shortened form of the tensor product ${\displaystyle \otimes }$ in their notation for the complex.

Definition

If A is an associative algebra over a field K, the standard complex is

${\displaystyle \cdots \to A\otimes A\otimes A\to A\otimes A\to A\to 0,}$

with the differential given by

${\displaystyle d(a_0\otimes \cdots \otimes a_{n+1})=\sum _{i=0}^n(-1{)}^i{a}_0\otimes \cdots \otimes a_i{a}_{i+1}\otimes \cdots \otimes a_{n+1}.}$

If A is a unital K-algebra, the standard complex is exact. Moreover, ${\displaystyle [\cdots \to A\otimes A\otimes A\to A\otimes A]}$ is a free A-bimodule resolution of the A-bimodule A.