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''binary complex, double complex''
 
''binary complex, double complex''
  
A graded module, i.e. one representable as the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160801.png" /> of its submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160802.png" />, together with a pair of differentials
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A graded module, i.e. one representable as the direct sum $  \sum A  ^ {m,n} $
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of its submodules $  A  ^ {m,n} $,  
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together with a pair of differentials
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160803.png" /></td> </tr></table>
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$$
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d _ {1} : A  ^ {m,n}  \rightarrow  A  ^ {m+1,n} ,
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160804.png" /></td> </tr></table>
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$$
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d _ {2} : A  ^ {m,n}  \rightarrow  A  ^ {m,n+1} ,
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$$
  
 
which satisfy the conditions
 
which satisfy the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160805.png" /></td> </tr></table>
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$$
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d _ {1} \cdot d _ {1}  = 0,\ \
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d _ {2} \cdot d _ {2}  = 0 ,\ \
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d _ {2} d _ {1} +d _ {1} d _ {2}  = 0 .
 +
$$
  
Instead of the direct sum, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160806.png" /> and the differentials
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Instead of the direct sum, the set $  \{ A  ^ {m,n} \} $
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and the differentials
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160807.png" /></td> </tr></table>
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$$
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d _ {1} : A ^ {m, n }  \rightarrow  A ^ {m - 1, n } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016080/b0160808.png" /></td> </tr></table>
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$$
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d _ {2} : A ^ {m, n }  \rightarrow  A ^ {m, n - 1 } ,
 +
$$
  
 
satisfying the above conditions, may be considered.
 
satisfying the above conditions, may be considered.
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>

Revision as of 10:59, 29 May 2020


binary complex, double complex

A graded module, i.e. one representable as the direct sum $ \sum A ^ {m,n} $ of its submodules $ A ^ {m,n} $, together with a pair of differentials

$$ d _ {1} : A ^ {m,n} \rightarrow A ^ {m+1,n} , $$

$$ d _ {2} : A ^ {m,n} \rightarrow A ^ {m,n+1} , $$

which satisfy the conditions

$$ d _ {1} \cdot d _ {1} = 0,\ \ d _ {2} \cdot d _ {2} = 0 ,\ \ d _ {2} d _ {1} +d _ {1} d _ {2} = 0 . $$

Instead of the direct sum, the set $ \{ A ^ {m,n} \} $ and the differentials

$$ d _ {1} : A ^ {m, n } \rightarrow A ^ {m - 1, n } , $$

$$ d _ {2} : A ^ {m, n } \rightarrow A ^ {m, n - 1 } , $$

satisfying the above conditions, may be considered.

Comments

References

[a1] S. MacLane, "Homology" , Springer (1963)
How to Cite This Entry:
Bicomplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicomplex&oldid=18032
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article