chain complex

Definition

A chain complex is a graded abelian group $A_{\bullet }$ with a differential

$d_{\bullet }=(d:A_n⟶A_{n-1}{)}_{n\in \mathbb{Z}}$

such that

1. $d:A_n\to A_{n-1}$ is a group homomorphism.

2. $d^2=0$ (that is $A_{n+1}\stackrel{d}{\to }{A}_n\stackrel{d}{\to }{A}_{n-1}$ is 0 map)

Examples

todo Add singular chain comples and de Rahm cohomology