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 sum

$\sum r_{i}g_{i}$

to

$\sum r_{i}.$

Here r_i is an element of R and g_i an element of G. The sums are finite, by definition of the group ring. In less formal terms,

$\varepsilon (g)$

is defined as 1_R for any element g in G,

$\varepsilon (r)$

is defined as r for any element r in R, and $\varepsilon$ is then extended to a homomorphism of R-modules in the obvious way. The augmentation ideal is the kernel of $\varepsilon$ and is therefore a two-sided ideal in R[G]. It is generated by the differences

$g-g'$

of group elements.

Furthermore it is also generated by

$g-1,g\in G$

which is a basis for the augmentation ideal as a free R module.

Examples of Quotients by the Augmentation Ideal

• Let G a group and Z[G] the group ring over the integers. Let I denote the augmentation ideal of 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 C[G] - module. The coinvariants of V can then be described as the quotient of V by the augentation ideal in C[G].

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.

Another class of examples of augmentation ideal can be the kernel of the counit $\varepsilon$ of any Hopf algebra.

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

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