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 $$ \cdots \rightarrow A_{n+1} \buildrel {f_{n+1}} \over \longrightarrow A_n \buildrel {f_n} \over \longrightarrow A_{n-1} \rightarrow \cdots $$ is said to be 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 $\mathcal{A}$ .