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 and  and has since been generalized in many ways.

The name "bar complex" comes from the fact that 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
 * $$d(a_0\otimes \cdots\otimes a_{n+1})=\sum_{i=0}^n (-1)^i a_0\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_{n+1}\,.$$

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