Augmentation ideal

In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If G is a group and R a commutative ring, there is a ring homomorphism [math]\displaystyle{ \varepsilon }[/math], called the augmentation map, from the group ring [math]\displaystyle{ R[G] }[/math] to [math]\displaystyle{ R }[/math], defined by taking a (finite) sum [math]\displaystyle{ \sum r_i g_i }[/math] to [math]\displaystyle{ \sum r_i. }[/math] (Here [math]\displaystyle{ r_i\in R }[/math] and [math]\displaystyle{ g_i\in G }[/math].) In less formal terms, [math]\displaystyle{ \varepsilon(g)=1_R }[/math] for any element [math]\displaystyle{ g\in G }[/math], [math]\displaystyle{ \varepsilon(rg)=r }[/math] for any elements [math]\displaystyle{ r\in R }[/math] and [math]\displaystyle{ g\in G }[/math], and [math]\displaystyle{ \varepsilon }[/math] is then extended to a homomorphism of R-modules in the obvious way.

The augmentation ideal A is the kernel of [math]\displaystyle{ \varepsilon }[/math] and is therefore a two-sided ideal in R[G].

A is generated by the differences [math]\displaystyle{ g - g' }[/math] of group elements. Equivalently, it is also generated by [math]\displaystyle{ \{g - 1 : g\in G\} }[/math], which is a basis as a free R-module.

For R and G as above, the group ring R[G] is an example of an augmented R-algebra. Such an algebra comes equipped with a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra.

The augmentation ideal plays a basic role in group cohomology, amongst other applications.

Examples of quotients by the augmentation ideal

• Let G a group and [math]\displaystyle{ \mathbb{Z}[G] }[/math] the group ring over the integers. Let I denote the augmentation ideal of [math]\displaystyle{ \mathbb{Z}[G] }[/math]. Then the quotient I/I² is isomorphic to the abelianization of G, defined as the quotient of G by its commutator subgroup.
• A complex representation V of a group G is a [math]\displaystyle{ \mathbb{C}[G] }[/math] - module. The coinvariants of V can then be described as the quotient of V by IV, where I is the augmentation ideal in [math]\displaystyle{ \mathbb{C}[G] }[/math].
• Another class of examples of augmentation ideal can be the kernel of the counit [math]\displaystyle{ \varepsilon }[/math] of any Hopf algebra.

References

• D. L. Johnson (1990). Presentations of groups. London Mathematical Society Student Texts. 15. Cambridge University Press. pp. 149–150. ISBN 0-521-37203-8. 
• Dummit and Foote, Abstract Algebra