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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 R[G], we restrict 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 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 \mathbb{Z}[G] the group ring over the integers. Let I denote the augmentation ideal of \mathbb{Z}[G]. Then the quotient I/I 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
- Dummit and Foote, Abstract Algebra
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