nLab augmentation ideal

Idea

For $R \hookrightarrow A$ an associative algebra over a ring $R$ equipped with the structure of an augmented algebra $\epsilon \colon A \to R$, the augmentation ideal is the kernel of $\epsilon$.

Specifically for $G$ a group, and $R[G]$ its group algebra over a ring $R$, the augmentation ideal is the ideal in $R[G]$ which consists of those formal linear combinations over $R$ of elements in $G$ whose sum of coefficients vanishes in $R$.

Examples

For group algebras

Let $G$ be a discrete group and $R$ a ring. Write $R[G]$ for the group algebra of $G$ over $R$.

Definition

Write

$$\epsilon \colon \mathbb{Z}[G] \to \mathbb{Z}$$

for the homomorphism of abelian groups which forms the sum of $R$-coefficients of the formal linear combinations that constitute the group ring

$$\epsilon \colon r \mapsto \sum_{g \in G} r_g \,.$$

This is called the augmentation map. Its kernel

$$ker(\epsilon) \hookrightarrow \mathbb{Z}[G]$$

is the augmentation ideal of $\mathbb{Z}[G]$. (It is often denoted by $I(G)$.

Properties

General

Proposition

The augmentation ideal is indeed a left and right ideal in $R[G]$.

For group algebras

Proposition

The $R$-module underlying the augmentation ideal of a group algebra is a free module, free on the set of elements

$$\{ g - e | g \in G,\; g \neq e \}$$

in $R[G]$.

Proposition

(For the case $R= \mathbb{Z}$)

As a $\mathbb{Z}[G]$-module, considered with the same generators, the relations are generated by those of the form

$$g_1(g_2-e)= (g_1g_2-e)-(g_1-e).$$

Related entries

• derived module