# Exact sequence

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An **exact sequence** is defined as follows. The sequence of morphisms $f_i$ in $\mathcal{A}$:
$$
\cdots \rightarrow
A_{n+1} \buildrel {f_{n+1}} \over \longrightarrow
A_n \buildrel {f_n} \over \longrightarrow
A_{n-1} \rightarrow
\cdots
$$
is said to be {\em exact} if each pair of adjacent morphisms $(f_{n+1},
f_n)$ is exact; in other words, if $${\rm im} f_{n+1} = {\rm ker} f_n$$ for all $n$.
$A_i$ above are objects in $\mathcal{A}$.

One should also compare this concept of {\em exactness} with the notion of a homology chain complex.