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

$\cdots \rightarrow A\otimes A\otimes A\rightarrow A\otimes A\rightarrow A\rightarrow 0\,,$

with the differential given by

$d(a_{0}\otimes \cdots \otimes a_{{n+1}})=\sum _{{i=0}}^{n}(-1)^{i}a_{0}\otimes \cdots \otimes a_{i}a_{{i+1}}\otimes \cdots \otimes a_{{n+1}}\,.$

If A is a unital K-algebra, the standard complex is exact. $[\cdots \rightarrow A\otimes A\otimes A\rightarrow A\otimes A]$ 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

• Koszul complex

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.