Demazure, lectures on p-divisible groups, II.3, affine k-groups
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Let , let .
Then the -group is called affine k-group.
A -biring is defined to be a -module which is a -ring and a -coring such that is a -ring morphism and is a -coring morphism.
The additive -group assigns to a -ring its underlying additive group. We have since by the Yoneda lemma we have
The multiplicative -group assigns to a -ring the multiplicative group of its invertible elements. We have .
There is a group homomorphism
its kernel we denote by . We have and . If is a field and is not in the -group is étale since is a separable polynomial. is the Galois module of the -th root of unity.
Let be a field of prime characteristic , let be an integer. Then there is a -group morphism.
We have and . For any field we have .