\begin{definition}[Definition:Differential Complex]
Let $R$ be a commutative ring with unity.
Let $\ds M = \bigoplus_{n \mathop \in \Z} M^n$ be a $\Z$-graded $R$-module that is also a differential module with differential $\d$.
Then $M$ is a '''differential complex''' if $\d$ satisfies:
:$\map \d {M^n} \subseteq M^{n + 1}$
for all $n \in \Z$.
The notation $\d_n := \d \restriction_{M_n}$ is often seen.
\end{definition}