# Definition:Differential Complex

## Definition

Let $R$ be a commutative ring with unity.

Let $\displaystyle M = \bigoplus_{n \in \Z} M^n$ be a $\Z$-graded $R$-module that is also a differential module with differential $d$.

Then $M$ is a differential complex if the differential $d$ satisfies:

$d \left({M^n}\right) \subseteq M^{n+1}$

for all $n \in \Z$.

The notation $d_n := d \restriction_{M_n}$ is often seen.