Standard complex

Template:Short description Template:Redirect Template:Format footnotes 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 Template:Harvs and Template:Harvs and has since been generalized in many ways.

The name "bar complex" comes from the fact that Template:Harvtxt 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

$\cdots \to A\otimes A\otimes A\to A\otimes A\to A\to 0,$

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, $[\cdots \to A\otimes A\otimes A\to A\otimes A]$ 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 $A\otimes (A/K)\otimes \cdots \otimes (A/K)\otimes A$.

See also

• Koszul complex

References

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