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The Artin-Hasse exponential series? can be written as
The in the multiplication formula where of the Artin-Hasse exponential series can also be written as
There exists a unique commutative group law on the -group scheme called -group of Witt-vectors of finite length relative to satisfying the following properties:
is a morphism of -groups.
Each is a morphism of -groups.
For , is called the -th component of and is called the -th phantom component of . The phantom components of define a group isomorphism
There is an endomorphism of the group of Witt vectors
We define the group of Witt vectors of length by or equivalently by the exact sequence
and we have
The underlying scheme of is , the projection morphism is .
The group law on is
The snake lemma gives from diagram (1) translation morphisms such that , projection morphisms such that and exact sequence
and the projections give rise to an isomorphism
Let be an inclusion.
Then we have and).
There is a unique ring-structure on the -group such that either of the two following conditions is satisfied:
Each is a ring homomorphism.
The -ring is called Witt-Ring, each is a quotient ring of . The canonical morphisms and are ring homomorphisms (but not ).