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Long Exact Sequence


An infinite sequence of homomorphisms of modules or additive Abelian groups

 ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->...
(1)

such that, for all indices i in Z,

 Im(d_(i+1))=Ker(d_i).
(2)

In other words, a long exact sequence is a chain complex such that for all i, the ith homology module (or group)

 H^i(C)=Ker(d_i)/Im(d_(i+1))
(3)

is the zero module (or the zero group).


See also

Chain Complex, Homology, Long Exact Sequence of a Pair Axiom, Short Exact Sequence

This entry contributed by Margherita Barile

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Cite this as:

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

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