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| :''Standard resolution redirects here, for the television monitor size, see [[standard definition]]''
| | Jeweller Hosea Corredor from Brooks, enjoys to spend time bowls, ganhando dinheiro na internet and drawing. Gets inspiration by visiting L'viv – the Ensemble of the Historic Centre.<br><br>Here is my website :: [http://comoganhardinheiro.comoganhardinheiro101.com ganhe dinheiro] |
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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 {{harvtxt|Eilenberg|Mac Lane|1953}} and {{harvtxt|Cartan|Eilenberg|1956|loc=IX.6}} and has since been generalized in many ways.
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| 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 ⊗ in their notation for the complex.
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| ==Definition==
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| If ''A'' is an algebra over a field ''K'', the standard complex is
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| :<math>\cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A\rightarrow A \rightarrow 0</math>
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| with the differential given by
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| :<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> | |
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| ==Normalized standard complex==
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| The normalized (or reduced) standard complex replaces ''A''⊗''A''⊗...⊗''A''⊗''A'' with
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| ''A''⊗(''A''/''K'')⊗...⊗(''A''/''K'')⊗''A''.
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| ==Monads==
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| {{Empty section|date=June 2011}}
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| ==See also==
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| *[[Koszul complex]]
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| ==References==
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| *{{Citation | last1=Cartan | first1=Henri | last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg | title=Homological algebra | url=http://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}}
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| *{{Citation | last1=Eilenberg | first1=Samuel | author1-link=Samuel Eilenberg | last2=Mac Lane | first2=Saunders | author2-link=Saunders Mac Lane | title=On the groups of H(Π,n). I | jstor=1969820 | mr=0056295 | year=1953 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=58 | pages=55–106}}
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| *{{cite arxiv | last1=Ginzburg | first1=Victor | title=Lectures on Noncommutative Geometry | eprint=math.AG/0506603 | year=2005}}
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| {{DEFAULTSORT:Standard Complex}}
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| [[Category:Homological algebra]]
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Jeweller Hosea Corredor from Brooks, enjoys to spend time bowls, ganhando dinheiro na internet and drawing. Gets inspiration by visiting L'viv – the Ensemble of the Historic Centre.
Here is my website :: ganhe dinheiro