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
:$d(a_0\otimes \cdots\otimes a_)=\sum_^n (-1)^i a_0\otimes\cdots\otimes a_ia_\otimes\cdots\otimes a_\,.$
If ''A'' is a unital ''K''-algebra, the standard complex is exact. Moreover, cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A/math> 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 $A\otimes(A/K) \otimes \cdots \otimes (A/K)\otimes A$.

Monads

See also

* Koszul complex

References

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Homological algebra

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