User:Bci21/Exact sequence

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.