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

for all $n \in \Z$.

We often write $d_n := d\big|_{M_n}$