Definition 29.7:

A short exact sequence of chain complexes consists of three chain complexes of abelian groups $L, G$ and $K$ and chain maps $f:L\longrightarrow G$ and $g:G\longrightarrow K$ such that

(i)

For each $n$, the map $f_n$ is injective.

(ii)

For each $n$, the map $g_n$ is surjective.

(iii)

For each $n$, ker$\;g_n =$Im$\;f_n$.

Thus for each $n$ we have the diagram

$\begin{CD} \{0\} @> >> L_n @> f_n >> G_n @> g_n >> K_n @> >> \{0\} \\ \end{CD}$(29.16)

We now write out two more parallel rows with $n$ replaced by $n-1$ and $n+1$ and the boundary maps going across the rows:

$\begin{CD} @. @VV{} V @VV{}V @VV{}V\\ {0}@> >> L_{n+1} @> {f_{n+1}} >> G_{n... ...n-1} @> {g_{n-1}} >> K_{n-1} @> >> {0}\\ @. @VV{}V @VV{} V @VV{}V \\ \end{CD}$

We now state and prove the fundamental result.