Augmentation ideal

In mathematics, an augmentation ideal is an ideal 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 sum

: $sum\; r\_i\; g\_i$

to

: $sum\; r\_i$

Here "r"_"i" is an element of "R" and "g"_"i" an element of "G". The sums are finite, by definition of the group ring. In less formal terms,

: $varepsilon(g)$

is defined as 1_"R" whatever the element "g" in "G", and $varepsilon$ is then extended to a homomorphism of "R"-modules in the obvious way. The augmentation ideal is the kernel of $varepsilon$, and is therefore a two-sided ideal in "R" ["G"] . It is generated by the differences

: $g\; -\; g\text{'}$

of group elements.

Furthermore it is also generated by

: $g\; -\; 1\; ,\; 1\; eq\; gin\; G$

which is a basis for the augmentation ideal 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.

Another class of examples of augmentation ideal can be the kernel of the counit $varepsilon$ of any Hopf algebra.

The augmentation ideal plays a basic role in group cohomology, amongst other applications.

References

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