This entry is about a section of the text
The Artin-Hasse exponential series? can be written as
with
The in the multiplication formula where of the Artin-Hasse exponential series can also be written as
We have
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
called translation.
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
In particular
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 ).
Last revised on May 27, 2012 at 13:42:18. See the history of this page for a list of all contributions to it.