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 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, is a free A-bimodule resolution of the A-bimodule A.

Normalized standard complex

The normalized standard complex replaces with.

Monads