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			|FULL LIST, MORE INFO