nLab
augmentation ideal

Context

Algebra

Group Theory

Contents

Idea

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

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

Examples

For group algebras

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

Definition

Write

ϵ:[G] \epsilon \colon \mathbb{Z}[G] \to \mathbb{Z}

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

ϵ:r gGr g. \epsilon \colon r \mapsto \sum_{g \in G} r_g \,.

This is called the augmentation map. Its kernel

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

is the augmentation ideal of [G]\mathbb{Z}[G].

Properties

General

Proposition

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

For group algebras

Proposition

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

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

in R[G]R[G].

Revised on October 14, 2012 18:48:41 by Tim Porter (95.147.237.39)