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 ε {\displaystyle \varepsilon }, called the augmentation map, from the group ring R [ G ] {\displaystyle R[G]} to R {\displaystyle R}, defined by taking a (finite) sum ∑ r i g i {\displaystyle \sum r_{i}g_{i}} to ∑ r i . {\displaystyle \sum r_{i}.} (Here r i ∈ R {\displaystyle r_{i}\in R} and g i ∈ G {\displaystyle g_{i}\in G}.) In less formal terms, ε ( g ) = 1 R {\displaystyle \varepsilon (g)=1_{R}} for any element g ∈ G {\displaystyle g\in G}, ε ( r g ) = r {\displaystyle \varepsilon (rg)=r} for any elements r ∈ R {\displaystyle r\in R} and g ∈ G {\displaystyle g\in G}, and ε {\displaystyle \varepsilon } is then extended to a homomorphism of R-modules in the obvious way.

The augmentation ideal A is the kernel of ε {\displaystyle \varepsilon } and is therefore a two-sided ideal in R[G].

A is generated by the differences g − g ′ {\displaystyle g-g'} of group elements. Equivalently, it is also generated by { g − 1 : g ∈ G } {\displaystyle \{g-1:g\in G\}}, which is a basis for A 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 ] {\displaystyle \mathbb {Z} [G]} the group ring over the integers. Let I denote the augmentation ideal of Z [ G ] {\displaystyle \mathbb {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 ] {\displaystyle \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 C [ G ] {\displaystyle \mathbb {C} [G]}.
  • Another class of examples of augmentation ideal can be the kernel of the counit ε {\displaystyle \varepsilon } of any Hopf algebra.

Notes

  • D. L. Johnson (1990). Presentations of groups. London Mathematical Society Student Texts. Vol. 15. Cambridge University Press. pp. 149–150. ISBN 0-521-37203-8.
  • Dummit and Foote, Abstract Algebra