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

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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.