HOME

TheInfoList



OR:

In mathematics, the standard complex, also called standard resolution, bar resolution, bar complex, bar construction, is a way of constructing resolutions in
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
. It was first introduced for the special case of algebras over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
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 In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ho ...


References

* * * Homological algebra {{algebra-stub