Homology sequence
An exact sequence, infinite on both sides, of homology groups of three complexes, connected by a short exact sequence. Let be an exact sequence of chain complexes in an Abelian category. Then there are morphisms
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defined for all . They are called connecting (or boundary) morphisms. Their definition in the category of modules is especially simple: For
a pre-image
is chosen;
will then be the image of an element
whose homology class is
. The sequence of homology groups
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constructed with the aid of the connecting morphisms, is exact; it is called the homology sequence. Thus, the homology groups form a homology functor on the category of complexes.
Cohomology sequences are defined in a dual manner.
References
[1] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) |
Homology sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_sequence&oldid=47262