# Standard complex

"Standard resolution" redirects here. For the television monitor size, see Standard definition.

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 Eilenberg & Mac Lane (1953) and Cartan & Eilenberg (1956, IX.6) and has since been generalized in many ways.

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

## Normalized standard complex

The normalized (or reduced) standard complex replaces AA⊗...⊗AA with A⊗(A/K)⊗...⊗(A/K)⊗A.