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Coboundary


In a cochain complex of modules

 ...->C^(i-1)->^(d^(i-1))C^i->^(d^i)C^(i+1)->...,

the module B^i of i-coboundaries is the image of d^(i-1). It is a submodule of C^i and is contained in the module of i-cocycles Z^i.

The cochain complex is called exact at C^i if B^i=Z^i.

In the right complex of Z-modules

 0->Z_2->^(·2)Z_4->^(·4)Z_8->...

for all i>0, the ith module is Z_(2^i), and the ith coboundary operator maps every element a of Z_(2^i) to the residue class of 2^ia in Z_(2^(i+1)). The module of i-coboundaries is the set of the residue classes of 0 and 2^(i-1) in Z_(2^i), and the module of i-cocycles the set of the residues classes of all even numbers 0,2,4,...,2^(i-1).


See also

Cohomology, Homology Boundary

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Coboundary." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Coboundary.html

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