Homology sequence

An exact sequence, infinite on both sides, of homology groups of three complexes, connected by a short exact sequence. Let $ 0 \rightarrow K _ {\mathbf . } \rightarrow L _ {\mathbf . } \rightarrow M _ {\mathbf . } \rightarrow 0 $ be an exact sequence of chain complexes in an Abelian category. Then there are morphisms

$$ \partial _ {n} : H _ {n} ( M _ {\mathbf . } ) \rightarrow H _ {n - 1 } ( K _ {\mathbf . } ) $$

defined for all $ n $. They are called connecting (or boundary) morphisms. Their definition in the category of modules is especially simple: For $ h \in H _ {n} ( M _ {\mathbf . } ) $ a pre-image $ x \in L _ {n} $ is chosen; $ d x $ will then be the image of an element $ z \in Z _ {n-} 1 ( K _ {\mathbf . } ) $ whose homology class is $ \partial _ {n} ( h) $. The sequence of homology groups

$$ \dots \rightarrow ^ { {\partial _ { n} + 1 } } \ H _ {n} ( K _ {\mathbf . } ) \rightarrow H _ {n} ( L _ {\mathbf . } ) \rightarrow \ H _ {n} ( M _ {\mathbf . } ) \rightarrow ^ { {\partial _ n } } \ H _ {n - 1 } ( K _ {\mathbf . } ) \rightarrow \dots , $$

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)