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

$\sum r_i g_i$

$\sum r_i.$

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 whatever the element g in G, 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

$g - 1 , 1\neq g\in G$

which is a basis for the augmentation ideal 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.

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. 15. Cambridge University Press. pp. 149–150. ISBN 0-521-37203-8.