Short exact sequence of chain complexes gives long exact sequence of homology

Statement

Suppose $A,B,C$ are chain complexes and we have a short exact sequence of chain complexes, i.e., a short exact sequence in the category of chain complexes with chain maps given by:

$0\to A{\stackrel {i}{\to }}B{\stackrel {p}{\to }}C\to 0$

Here, exactness means that for each integer $n$, the induced sequence of maps:

$0\to A_{n}{\stackrel {i_{n}}{\to }}B_{n}{\stackrel {p_{n}}{\to }}C_{n}\to 0$

is a short exact sequence of abelian groups (or modules, if this is a chain complex of modules).

Then, we can obtain (canonically) a long exact sequence of homology groups (respectively, modules):

$\dots {\stackrel {\operatorname {connecting} }{\to }}H_{n}(A){\stackrel {H_{n}(i)}{\to }}H_{n}(B){\stackrel {H_{n}(p)}{\to }}H_{n}(C){\stackrel {\operatorname {connecting} }{\to }}H_{n-1}(A){\stackrel {H_{n-1}(i)}{\to }}H_{n-1}(B){\stackrel {H_{n-1}(p)}{\to }}H_{n-1}(C){\stackrel {\operatorname {connecting} }{\to }}\dots$

The maps from a given homology of $C$ to the lower homology of $A$ are connecting homomorphisms and arise via an application of the snake lemma.