Harman Patil (Editor)

Standard complex

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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.

Contents

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

⋯ → A ⊗ A ⊗ A → A ⊗ A → A → 0 ,

with the differential given by

d ( a 0 ⊗ ⋯ ⊗ a n + 1 ) = ∑ i = 0 n ( − 1 ) i a 0 ⊗ ⋯ ⊗ a i a i + 1 ⊗ ⋯ ⊗ a n + 1 .

If A is a unital K-algebra, the standard complex is exact. [ ⋯ → A ⊗ A ⊗ A → A ⊗ 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.

References

Standard complex Wikipedia
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