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

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

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)