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^{[Note 1]}) sum $\sum r_{i}g_{i}$ to $\sum r_{i}.$ (Here $r i \in R$ and $g i \in G$ .) In less formal terms, $ε (g)=1 R$ for any element $g \in G$ , $\varepsilon (r)=r$ for any element $r \in R$ , and $\varepsilon$ is then extended to a homomorphism of R-modules in the obvious way.

The augmentation ideal $A$ 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 Z[G] the group ring over the integers. Let I denote the augmentation ideal of Z[G]. Then the quotient $I / I 2$ 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 IV, where I is the augmentation ideal in C[G].
Another class of examples of augmentation ideal can be the kernel of the counit $\varepsilon$ of any Hopf algebra.

Notes

↑ 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

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