Chain complex of abelian groups

Definition
A chain complex of abelian groups is a chain complex of groups where all the groups involved are abelian groups. Explicitly, it is the following data:


 * A collection $$G_n$$ of abelian groups, where $$n$$ varies over the integers
 * A collection $$d_n:G_n \to G_{n-1}$$ of group homomorphisms

such that for any $$n$$:

$$d_{n-1} \circ d_n = 0$$

The chain complex is typically written as:

$$ \dots \to G_n \stackrel{d_n}{\to} G_{n-1} \stackrel{d_{n-1}}{\to} G_{n-2} \to \dots$$