# Chain complex of groups

## Contents

## Definition

### Over the integers

A **chain complex of groups** is defined as the following data:

- A collection of groups, where varies over the integers
- A collection of group homomorphisms, called the
*differentials*.

such that for any :

The chain complex is typically written as:

### Over a contiguous segment of the integers

Suppose are integers. A chain complex with index from to is a chain complex of groups in the above sense (over all integers) but with the trivial group for or . This kind of chain complex is usually written in shorthand as:

where 1 denotes the trivial group.

### Abstract definition

A chain complex of groups is a group object in the category of chain complexes of pointed sets.

## Homomorphism of chain complexes

There are many notions of homomorphism of chain complexes, each discussed below. Each notion of homomorphism defines a corresponding notion of isomorphism.

### Homomorphism that preserves the positions

Given two chain complexes and of groups, a homomorphism , also called a **chain map** or **degree zero chain map**, is a collection of group homomorphisms:

such that, for all :

where both sides describe group homomorphisms from to .

### Homomorphism that allows for a shifting of positions

Let be an integer. Given two chain complexes and of groups, , a homomorphism of degree , also called a **degree chain map**, if it is a collection of group homomorphisms:

such that, for all :

.

## Terminology

- If the image of equals the kernel of , then the chain complex is said to be
*exact*at . If this is true for every , the chain complex is termed an exact sequence of groups. - If the image of is a normal subgroup of the kernel of , the quotient group is termed the homology group of the chain complex.
- If the image of is a normal subgroup of the whole next group , then the preceding condition holds and we can define the homology. A chain complex of groups where this holds for all is termed a normal complex of groups.