Augmentation ideal


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'

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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