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{{Short description|Technique for constructing resolutions in homological algebra}}
{{redirect|Standard resolution|the television monitor size|Standard-definition television}}
{{format footnotes |date=May 2024}}
In mathematics, the '''standard complex''', also called '''standard resolution''', '''bar resolution''', '''bar complex''', '''bar construction''', is a way of constructing [[resolution (algebra)|resolutions]] in [[homological algebra]].  It was first introduced for the special case of algebras over a [[commutative ring]] by {{harvs|txt|last1=Eilenberg | first1=Samuel | author1-link=Samuel Eilenberg | last2=Mac Lane | first2=Saunders | author2-link=Saunders Mac Lane |year=1953}} and {{harvs|txt|last1=Cartan | first1=Henri | author1-link=Henri Cartan|last2=Eilenberg |year=1956|loc=IX.6}} and has since been generalized in many ways.

The name "bar complex" comes from the fact that {{harvtxt|Eilenberg|Mac Lane|1953}} used a vertical bar | as a shortened form of the tensor product <math>\otimes</math> in their notation for the complex.

==Definition==

If ''A'' is an associative algebra over a field ''K'', the standard complex is
:<math>\cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A\rightarrow A \rightarrow 0\,,</math>
with the differential given by
:<math>d(a_0\otimes \cdots\otimes a_{n+1})=\sum_{i=0}^n (-1)^i a_0\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_{n+1}\,.</math>
If ''A'' is a unital ''K''-algebra, the standard complex is exact.  Moreover, <math>[\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 <math>A\otimes A\otimes \cdots \otimes A\otimes A</math> with <math>A\otimes(A/K) \otimes \cdots \otimes (A/K)\otimes A</math>.

==Monads==
{{Empty section|date=June 2011}}

==See also==

*[[Koszul complex]]

==References==

*{{Citation | last1=Cartan | first1=Henri | author1-link=Henri Cartan|last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg | title=Homological algebra | url=https://books.google.com/books?id=0268b52ghcsC | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series | isbn=978-0-691-04991-5  | mr=0077480 | year=1956 | volume=19}}
*{{Citation | last1=Eilenberg | first1=Samuel | author1-link=Samuel Eilenberg | last2=Mac Lane | first2=Saunders | author2-link=Saunders Mac Lane | title=On the groups of <math>H(\Pi,n)</math>. I | jstor=1969820 | mr=0056295 | year=1953 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=58 | pages=55–106 | doi=10.2307/1969820}}
*{{cite arXiv | last=Ginzburg | first=Victor | authorlink=Victor Ginzburg| title=Lectures on Noncommutative Geometry | eprint=math.AG/0506603 | year=2005}}

[[Category:Homological algebra]]


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