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 ${\displaystyle \varepsilon }$, called the augmentation map, from the group ring ${\displaystyle R[G]}$ to ${\displaystyle R}$, defined by taking a (finite[Note 1]) sum ${\displaystyle \sum r_{i}g_{i}}$ to ${\displaystyle \sum r_{i}.}$ (Here ${\displaystyle r_{i}\in R}$ and ${\displaystyle g_{i}\in G}$.) In less formal terms, ${\displaystyle \varepsilon (g)=1_{R}}$ for any element ${\displaystyle g\in G}$, ${\displaystyle \varepsilon (r)=r}$ for any element ${\displaystyle r\in R}$, and ${\displaystyle \varepsilon }$ is then extended to a homomorphism of R-modules in the obvious way.

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

A is generated by the differences ${\displaystyle g-g'}$ of group elements. Equivalently, it is also generated by ${\displaystyle \{g-1:g\in G\}}$, 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 ${\displaystyle \mathbb {Z} [G]}$ the group ring over the integers. Let I denote the augmentation ideal of ${\displaystyle \mathbb {Z} [G]}$. 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 ${\displaystyle \mathbb {C} [G]}$ - module. The coinvariants of V can then be described as the quotient of V by IV, where I is the augmentation ideal in ${\displaystyle \mathbb {C} [G]}$.
• Another class of examples of augmentation ideal can be the kernel of the counit ${\displaystyle \varepsilon }$ of any Hopf algebra.

Notes

1. When constructing R[G], we restrict R[G] to only finite (formal) sums

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