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 (finiteWhen constructing {{math|R[G]}}, we restrict {{math|R[G]}} to only finite (formal) sums) 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 {{mvar|A}} is the kernel of $\varepsilon$ and is therefore a two-sided ideal in R[G].

{{mvar|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 {{math|I/I{{sup|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 $\mathbb{C}[G]$ - 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.

Notes

References

{{cite book | author=D. L. Johnson | title=Presentations of groups | series=London Mathematical Society Student Texts | volume=15 | publisher=Cambridge University Press | year=1990 | isbn=0-521-37203-8 | pages=149–150 }}
Dummit and Foote, Abstract Algebra

Category:Ideals (ring theory)

Category:Hopf algebras