# Definition:Differential Complex

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## Definition

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.