Standard complex

In mathematics, the standard complex, also called standard resolution, bar resolution, bar complex, bar construction, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by and and has since been generalized in many ways.
The name "bar complex" comes from the fact that used a vertical bar | as a shortened form of the tensor product formula_1 in their notation for the complex.
Definition.
If "A" is an associative algebra over a field "K", the standard complex is
with the differential given by
If "A" is a unital "K"-algebra, the standard complex is exact. Moreover, formula_4 is a free "A"-bimodule resolution of the "A"-bimodule "A".
Normalized standard complex.
The normalized (or reduced) standard complex replaces formula_5 with formula_6.