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 Samuel Eilenberg and Saunders Mac Lane (1953) and Henri Cartan and 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 $\otimes$ 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. Moreover, $[\cdots \rightarrow A\otimes A\otimes A\rightarrow A\otimes A]$ is a free A-bimodule resolution of the A-bimodule A.

Normalized standard complex

The normalized (or reduced) standard complex replaces $A\otimes A\otimes \cdots \otimes A\otimes A$ with $A\otimes (A/K)\otimes \cdots \otimes (A/K)\otimes A$ .