# Bar resolution

## Contents

## Definition

Suppose is a group. The **bar resolution** of is a long exact sequence of -modules:

defined by the following information.

We denote the identity element of by .

### The groups

The group is defined as the free abelian group on the set , with acting on it diagonally:

This group can thus be regarded as a -module.

As a -module, has a free generating set identified by by:

The notation with bars is termed the *bar notation*.

### The derivation in the original notation

The derivation in the original notation is given by:

### The derivation with the bar notation

The map is defined as follows:

In the more precise summation notation:

## Particular cases

In the comma notation, we have:

The source of is , which is a free module on . The target of is , which is simply .

In the comma notation, we have:

The source of is , which is a free -module on . The target of is , which is the free -module on .

In the comma notation, we have:

In the bar notation, we have: