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

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

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

References

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