Let $R$ be a ring. A sequence of $R$ -modules and homomorphisms$$ \cdots \rightarrow A_{n+1} \buildrel {d_{n+1}} \over \longrightarrow A_n \buildrel {d_n} \over \longrightarrow A_{n-1} \rightarrow \cdots$$ is said to be a chain complex (or $R$ -complex, or just complex) if each pair of adjacent homomorphisms $(d_{n+1}, d_n)$ satisfies the relation $d_n\circ d_{n+1} = 0$ . This is equivalent to saying that $\im d_{n+1} \subseteq \ker d_n$ . We often denote such a complex by $({\bold A}, d)$ , or simply ${\bold A}$ .

Compare this to the notion of an exact sequence, which requires $\im d_{n+1} = \ker d_n$ .

The homomorphisms $d_n$ in the chain complex are called boundary operators, or boundary maps.