If we have two homomorphisms $f:A\to B$ and $g:B\to C$ in some category of modules, then we say that $f$ and $g$ are exact at $B$ if the image of $f$ is equal to the kernel of $g$.

A sequence of homomorphisms

is said to be exact if each pair of adjacent homomorphisms $(f_{{n+1}},f_{n})$ is exact – in other words if ${\rm im}f_{{n+1}}={\rm ker}f_{n}$ for all $n$.

Compare this to the notion of a chain complex.

Remark. The notion of exact sequences can be generalized to any abelian category $\mathcal{A}$, where $A_{i}$ and $f_{i}$ above are objects and morphisms in $\mathcal{A}$.