# 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:

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

## Relation with other properties

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal complex of groups chain complex of groups where the image of each of the homomorphisms is a normal subgroup of the next group |FULL LIST, MORE INFO
chain complex of groups chain complex of groups, where the groups need not be abelian Normal complex of groups|FULL LIST, MORE INFO