# 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)
How to Cite This Entry:
Homology sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_sequence&oldid=47262
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article