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Let be a fixed prime number.
Let be a ring. Then the assignation
There is an exact sequence
We have .
Each is is an -fold extension of the additive group .
If is a field then is a unipotent group.
For there is an isomorphism of -schemes
If and we have and is an isomorphism .
If is a field with characteristic it is not possible an such that . However there is always a formula where . This is verified by Möbius inversion.
The Artin-Hasse exponential is defined by the morphism
We have where
, can be uniquely written as
The -group is isomorphic to the n/(n,p)=1-power of the subgroup image of .
By base-change a similar statement applies to . This shows that the Artin-Hasse exponential plays over a similar role as the usual exponential over .