Bar Resolution

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 $A\otimes (A/K)\otimes \cdots \otimes (A/K)\otimes A$ .