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 , called the augmentation map, from the group ring to , defined by taking a (finite) sum to (Here and .) In less formal terms, for any element , for any elements and , and is then extended to a homomorphism of R-modules in the obvious way.

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

is generated by the differences of group elements. Equivalently, it is also generated by , which is a basis for 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 the group ring over the integers. Let I denote the augmentation ideal of . Then the quotient 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 - module. The coinvariants of V can then be described as the quotient of V by IV, where I is the augmentation ideal in .
• Another class of examples of augmentation ideal can be the kernel of the counit of any Hopf algebra.

Notes

References

• Dummit and Foote, Abstract Algebra