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 , called the augmentation map, from the group ring to R, defined by taking a (finite[Note 1]) sum to (Here and .) In less formal terms, for any element , for any element , and is then extended to a homomorphism of R-modules in the obvious way.

The augmentation ideal A is the kernel of and is therefore a two-sided ideal in R[G].

A is generated by the differences of group elements. Equivalently, it is also generated by , 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/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 IV, where I is the augmentation ideal in C[G].
  • Another class of examples of augmentation ideal can be the kernel of the counit of any Hopf algebra.

Notes

  1. 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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