# 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 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 *A*⊗*A*⊗...⊗*A*⊗*A* with
*A*⊗(*A*/*K*)⊗...⊗(*A*/*K*)⊗*A*.

## See also

## References

- Cartan, Henri; Eilenberg, Samuel (1956),
*Homological algebra*, Princeton Mathematical Series,**19**, Princeton University Press, ISBN 978-0-691-04991-5, MR 0077480 - Eilenberg, Samuel; Mac Lane, Saunders (1953), "On the groups of H(Π,n). I",
*Annals of Mathematics. Second Series*,**58**: 55–106, ISSN 0003-486X, JSTOR 1969820, MR 0056295 - Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry". arXiv:math.AG/0506603.