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
⊗
If A is an associative algebra over a field K, the standard complex is
… → A ⊗ A ⊗ A → A ⊗ A → A → 0,
d(a0 ⊗ … ⊗ an+1
| n | |
| )=\sum | |
| i=0 |
(-1)ia0 ⊗ … ⊗ aiai+1 ⊗ … ⊗ an+1.
[ … → A ⊗ A ⊗ A → A ⊗ A]
The normalized (or reduced) standard complex replaces
A ⊗ A ⊗ … ⊗ A ⊗ A
A ⊗ (A/K) ⊗ … ⊗ (A/K) ⊗ A